Work, Energy and Power AP style Energy Energy: the currency of the universe. Everything has to be...

37

Transcript of Work, Energy and Power AP style Energy Energy: the currency of the universe. Everything has to be...

Page 1: Work, Energy and Power AP style Energy Energy: the currency of the universe. Everything has to be “paid for” with energy. Energy can’t be created or.

Work Energy and PowerAP style

Energy

Energy the currency of the universe

Everything has to be ldquopaid forrdquo with energy

Energy canrsquot be created or destroyed but it can be transformed from one kind to another and it can be transferred from one object to another

How do you know an object has mechanical energy (kinetic potential)

If it can change itself or change its environment then it certainly has energy

bull Doing WORK is one way to transfer energy from one object to another

Work = Force x displacement

W = Fd

bull Unit for work is Newton x meter One Newton-meter is also called a Joule J

Work = Force x displacement

bull Work is not done unless there is a displacement

bull If you hold an object a long time you may get tired but NO work was done

bull If you push against a solid wall for hours there is still NO work done

bull For work to be done the displacement of the object must be in the same direction as the applied force They must be parallel

bull If the force and the displacement are perpendicular to each other NO work is done by the force

So using vector multiplication

W = F bull d

(In many university texts as well as the AP test the displacement is represented by ldquosrdquo

and not ldquodrdquo

W = F bull s

An object is subject to a force given byF = 6i ndash 8j as it moves from the position r = -4i + 3j

to the position r = i + 7j

What work was done by this forceFirst find the displacement s

s = r =

rf ndash ro =(i + 7j) - (-4i + 3j) =

5i + 4j

Then do the dot product W = F middot s(6i ndash 8j) middot (5i + 4j) =

30 ndash 32 = -2J of work

bull In lifting a book the force exerted by your hands is upward and the displacement is upward- work is done

bull Similarly in lowering a book the force exerted by your hands is still upward and the displacement is downward

bull The force and the displacement are STILL parallel so work is still done

bull But since they are in opposite directions now it is NEGATIVE work

bull On the other hand while carrying a book down the hallway the force from your hands is vertical and the displacement of the book is horizontal

bull Therefore NO work is done by your hands

bull Sohellipwhile climbing stairs or walking up an incline only the vertical component of the displacement is used to calculate the work done in moving the object from the bottom to the top Horizontal component of d

Ver

tical

com

pone

nt o

f d

Yo

ur

Fo

rce

Example

How much work is done to carry a 5 kg cat to the top of a ramp that is 7 meters long and 3 meters tall

W = Force x displacementForce = weight of the catd = height NOT lengthW = mg x hW = 5 x 10 x 3 W = 150 J

7 m

3 m

ExampleA boy pushes a

lawnmower 20 meters across the yard If he pushed with a force of 200 N and the angle between the handle and the ground was 50 degrees how much work did he do

F

Displacement = 20 m

F cos

W = (F cos )dW = (200 cos 50) 20W = 2571 J

NOTE If while pushing an object it is moving at a constant velocity

the NET force must be zero

Sohellip Your applied force must be exactly equal to any resistant forces like friction

How much work do you do to carry a 30 kg cat from one side of the room to the other if the room is 10 meters long

ZERO because your Force is vertical but the displacement is horizontal

bull Energy and Work have no direction associated with them and are therefore scalar quantities not vectors

Power

The rate at which work is done

1 Power = Work divide time

Unit for power = J divide s

= Watt W

What is a Watt in ldquofundamental unitsrdquo

Since work is also the energy transferred or transformed ldquopowerrdquo is the rate at which energy is transferred or transformed

2 P = Energy divide time

This energy could be in ANY form heat light potential chemical nuclear

Since NET work = K

3 P = K divide t

And yet another approach

P = W divide t = (Fd) divide t = F middot (d divide t)

P = F middot v

Kinetic Energy

the energy of motion

K = frac12 mv2

Kinetic Energy

the energy of motion

K = frac12 mv2

Where does K = frac12 mv2 come from

Did your amazing teacher just arbitrarily make that equation up

Hmmmhellip

The ldquoWork- Kinetic Energy TheoremrdquoWorknet = K

Fnet bulld = K = frac12 mvf2 - frac12 mvo

2

You are supposed to be able touse Fnet = ma kinematics equations and Wnet = Fnet bulld to

derive the work-kinetic energy theoremhellip

Sohellip do it Then yoursquoll see where the equation we call ldquokinetic energyrdquo comes from

(Hint start with Fnet = ma and use the kinematics equation that doesnrsquot involve time)

Potential EnergyStored energy

It is called potential energy because it has the potential to do work

bull Example 1 Spring potential energy in the stretched string of a bow or spring or rubber band SPE = frac12 kx2

bull Example 2 Chemical potential energy in fuels- gasoline propane batteries food

bull Example 3 Gravitational potential energy- stored in an object due to its position from a chosen reference point

Gravitational potential energy

GPE = weight x height

GPE = mgh

bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative

bull The GPE only depends on the weight and the height not on the path that it took to get to that height

Work and EnergyOften some force must do work

to give an object potential or kinetic energy

You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy

Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy

Work = Force x distance = change in energy

Example

How much more distance is required to stop if a car is going twice as fast

Fd = frac12 mv2

The work done by the brakes = the change in the kinetic energy

With TWICE the speed the car has

FOUR times the kinetic energy

Therefore it takes FOUR times the stopping distance

The Work-Kinetic Energy Theorem

NET Work done = Kinetic Energy

Wnet = frac12 mv2f ndash frac12 mv2

o

Varying Forces

The rule ishellip ldquoIf the Force varies you must integraterdquo

If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done

If the force is a function of velocity you must integrate to find the power output

Examples

If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5

If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms

Examples of Integration

An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force

W F x dx x dx x 33

43 4

Examples of Integration

At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)

v t a t dt tdt t C t( ) ( )

1515

20 75 32 2

x t v t dt t dt t t C t t( ) ( ) ( )

0 75 30 75

23 0 375 3 02 2

Examples of IntegrationA particle of mass m moves along the y-axis as

y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force

P = Fmiddotv

We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )

P F v m at b at bty y ( )( )12 2 4 2 22 3

Examples of Definite Integration

An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s

x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2

62

2

63 2

2

6 3 2 3 22 8 62

3

8

26

2

36

8

26 6 6

2

32

8

22 6 2 45 33

Examples of Definite Integration

To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x

What work is done to stretch the spring 2 meters beyond its equilibrium position

W F ds x x ds x x Jmm

m ( ) ( ) ( ) 4

4

3

1

2

4

32

1

22

4

30

1

20 8 6672

0

23

0

22

0

2 3 2 2

Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under

the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m

P = Fmiddotv but what is the velocity at x = 3m

Hmmmhellip

W F x dx and W K so ( )

F x dx K mv mvx

x

f o

o

f

( ) 12

12

2 2 ( )6 4 3 12

2

0

3

32 x x dx K mv x

( ) ( )6 4 3 64

2

3

36 3

4

23

3

33 0 9 1

22

0

3

2 3

0

3 2 33

2 x x dx x x x J mv x

v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5

  • Slide 1
  • Work Energy and Power AP style
  • Energy
  • Slide 4
  • Slide 5
  • Work = Force x displacement
  • Slide 7
  • Slide 8
  • Slide 9
  • Slide 10
  • Slide 11
  • Example
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Power
  • Slide 18
  • Slide 19
  • Slide 20
  • Slide 21
  • The ldquoWork- Kinetic Energy Theoremrdquo
  • Potential Energy
  • Slide 24
  • Gravitational potential energy
  • Slide 26
  • Work and Energy
  • Slide 28
  • The Work-Kinetic Energy Theorem
  • Varying Forces
  • Slide 31
  • Examples of Integration
  • Slide 33
  • Slide 34
  • Examples of Definite Integration
  • Slide 36
  • Slide 37
Page 2: Work, Energy and Power AP style Energy Energy: the currency of the universe. Everything has to be “paid for” with energy. Energy can’t be created or.

Energy

Energy the currency of the universe

Everything has to be ldquopaid forrdquo with energy

Energy canrsquot be created or destroyed but it can be transformed from one kind to another and it can be transferred from one object to another

How do you know an object has mechanical energy (kinetic potential)

If it can change itself or change its environment then it certainly has energy

bull Doing WORK is one way to transfer energy from one object to another

Work = Force x displacement

W = Fd

bull Unit for work is Newton x meter One Newton-meter is also called a Joule J

Work = Force x displacement

bull Work is not done unless there is a displacement

bull If you hold an object a long time you may get tired but NO work was done

bull If you push against a solid wall for hours there is still NO work done

bull For work to be done the displacement of the object must be in the same direction as the applied force They must be parallel

bull If the force and the displacement are perpendicular to each other NO work is done by the force

So using vector multiplication

W = F bull d

(In many university texts as well as the AP test the displacement is represented by ldquosrdquo

and not ldquodrdquo

W = F bull s

An object is subject to a force given byF = 6i ndash 8j as it moves from the position r = -4i + 3j

to the position r = i + 7j

What work was done by this forceFirst find the displacement s

s = r =

rf ndash ro =(i + 7j) - (-4i + 3j) =

5i + 4j

Then do the dot product W = F middot s(6i ndash 8j) middot (5i + 4j) =

30 ndash 32 = -2J of work

bull In lifting a book the force exerted by your hands is upward and the displacement is upward- work is done

bull Similarly in lowering a book the force exerted by your hands is still upward and the displacement is downward

bull The force and the displacement are STILL parallel so work is still done

bull But since they are in opposite directions now it is NEGATIVE work

bull On the other hand while carrying a book down the hallway the force from your hands is vertical and the displacement of the book is horizontal

bull Therefore NO work is done by your hands

bull Sohellipwhile climbing stairs or walking up an incline only the vertical component of the displacement is used to calculate the work done in moving the object from the bottom to the top Horizontal component of d

Ver

tical

com

pone

nt o

f d

Yo

ur

Fo

rce

Example

How much work is done to carry a 5 kg cat to the top of a ramp that is 7 meters long and 3 meters tall

W = Force x displacementForce = weight of the catd = height NOT lengthW = mg x hW = 5 x 10 x 3 W = 150 J

7 m

3 m

ExampleA boy pushes a

lawnmower 20 meters across the yard If he pushed with a force of 200 N and the angle between the handle and the ground was 50 degrees how much work did he do

F

Displacement = 20 m

F cos

W = (F cos )dW = (200 cos 50) 20W = 2571 J

NOTE If while pushing an object it is moving at a constant velocity

the NET force must be zero

Sohellip Your applied force must be exactly equal to any resistant forces like friction

How much work do you do to carry a 30 kg cat from one side of the room to the other if the room is 10 meters long

ZERO because your Force is vertical but the displacement is horizontal

bull Energy and Work have no direction associated with them and are therefore scalar quantities not vectors

Power

The rate at which work is done

1 Power = Work divide time

Unit for power = J divide s

= Watt W

What is a Watt in ldquofundamental unitsrdquo

Since work is also the energy transferred or transformed ldquopowerrdquo is the rate at which energy is transferred or transformed

2 P = Energy divide time

This energy could be in ANY form heat light potential chemical nuclear

Since NET work = K

3 P = K divide t

And yet another approach

P = W divide t = (Fd) divide t = F middot (d divide t)

P = F middot v

Kinetic Energy

the energy of motion

K = frac12 mv2

Kinetic Energy

the energy of motion

K = frac12 mv2

Where does K = frac12 mv2 come from

Did your amazing teacher just arbitrarily make that equation up

Hmmmhellip

The ldquoWork- Kinetic Energy TheoremrdquoWorknet = K

Fnet bulld = K = frac12 mvf2 - frac12 mvo

2

You are supposed to be able touse Fnet = ma kinematics equations and Wnet = Fnet bulld to

derive the work-kinetic energy theoremhellip

Sohellip do it Then yoursquoll see where the equation we call ldquokinetic energyrdquo comes from

(Hint start with Fnet = ma and use the kinematics equation that doesnrsquot involve time)

Potential EnergyStored energy

It is called potential energy because it has the potential to do work

bull Example 1 Spring potential energy in the stretched string of a bow or spring or rubber band SPE = frac12 kx2

bull Example 2 Chemical potential energy in fuels- gasoline propane batteries food

bull Example 3 Gravitational potential energy- stored in an object due to its position from a chosen reference point

Gravitational potential energy

GPE = weight x height

GPE = mgh

bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative

bull The GPE only depends on the weight and the height not on the path that it took to get to that height

Work and EnergyOften some force must do work

to give an object potential or kinetic energy

You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy

Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy

Work = Force x distance = change in energy

Example

How much more distance is required to stop if a car is going twice as fast

Fd = frac12 mv2

The work done by the brakes = the change in the kinetic energy

With TWICE the speed the car has

FOUR times the kinetic energy

Therefore it takes FOUR times the stopping distance

The Work-Kinetic Energy Theorem

NET Work done = Kinetic Energy

Wnet = frac12 mv2f ndash frac12 mv2

o

Varying Forces

The rule ishellip ldquoIf the Force varies you must integraterdquo

If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done

If the force is a function of velocity you must integrate to find the power output

Examples

If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5

If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms

Examples of Integration

An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force

W F x dx x dx x 33

43 4

Examples of Integration

At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)

v t a t dt tdt t C t( ) ( )

1515

20 75 32 2

x t v t dt t dt t t C t t( ) ( ) ( )

0 75 30 75

23 0 375 3 02 2

Examples of IntegrationA particle of mass m moves along the y-axis as

y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force

P = Fmiddotv

We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )

P F v m at b at bty y ( )( )12 2 4 2 22 3

Examples of Definite Integration

An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s

x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2

62

2

63 2

2

6 3 2 3 22 8 62

3

8

26

2

36

8

26 6 6

2

32

8

22 6 2 45 33

Examples of Definite Integration

To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x

What work is done to stretch the spring 2 meters beyond its equilibrium position

W F ds x x ds x x Jmm

m ( ) ( ) ( ) 4

4

3

1

2

4

32

1

22

4

30

1

20 8 6672

0

23

0

22

0

2 3 2 2

Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under

the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m

P = Fmiddotv but what is the velocity at x = 3m

Hmmmhellip

W F x dx and W K so ( )

F x dx K mv mvx

x

f o

o

f

( ) 12

12

2 2 ( )6 4 3 12

2

0

3

32 x x dx K mv x

( ) ( )6 4 3 64

2

3

36 3

4

23

3

33 0 9 1

22

0

3

2 3

0

3 2 33

2 x x dx x x x J mv x

v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5

  • Slide 1
  • Work Energy and Power AP style
  • Energy
  • Slide 4
  • Slide 5
  • Work = Force x displacement
  • Slide 7
  • Slide 8
  • Slide 9
  • Slide 10
  • Slide 11
  • Example
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Power
  • Slide 18
  • Slide 19
  • Slide 20
  • Slide 21
  • The ldquoWork- Kinetic Energy Theoremrdquo
  • Potential Energy
  • Slide 24
  • Gravitational potential energy
  • Slide 26
  • Work and Energy
  • Slide 28
  • The Work-Kinetic Energy Theorem
  • Varying Forces
  • Slide 31
  • Examples of Integration
  • Slide 33
  • Slide 34
  • Examples of Definite Integration
  • Slide 36
  • Slide 37
Page 3: Work, Energy and Power AP style Energy Energy: the currency of the universe. Everything has to be “paid for” with energy. Energy can’t be created or.

How do you know an object has mechanical energy (kinetic potential)

If it can change itself or change its environment then it certainly has energy

bull Doing WORK is one way to transfer energy from one object to another

Work = Force x displacement

W = Fd

bull Unit for work is Newton x meter One Newton-meter is also called a Joule J

Work = Force x displacement

bull Work is not done unless there is a displacement

bull If you hold an object a long time you may get tired but NO work was done

bull If you push against a solid wall for hours there is still NO work done

bull For work to be done the displacement of the object must be in the same direction as the applied force They must be parallel

bull If the force and the displacement are perpendicular to each other NO work is done by the force

So using vector multiplication

W = F bull d

(In many university texts as well as the AP test the displacement is represented by ldquosrdquo

and not ldquodrdquo

W = F bull s

An object is subject to a force given byF = 6i ndash 8j as it moves from the position r = -4i + 3j

to the position r = i + 7j

What work was done by this forceFirst find the displacement s

s = r =

rf ndash ro =(i + 7j) - (-4i + 3j) =

5i + 4j

Then do the dot product W = F middot s(6i ndash 8j) middot (5i + 4j) =

30 ndash 32 = -2J of work

bull In lifting a book the force exerted by your hands is upward and the displacement is upward- work is done

bull Similarly in lowering a book the force exerted by your hands is still upward and the displacement is downward

bull The force and the displacement are STILL parallel so work is still done

bull But since they are in opposite directions now it is NEGATIVE work

bull On the other hand while carrying a book down the hallway the force from your hands is vertical and the displacement of the book is horizontal

bull Therefore NO work is done by your hands

bull Sohellipwhile climbing stairs or walking up an incline only the vertical component of the displacement is used to calculate the work done in moving the object from the bottom to the top Horizontal component of d

Ver

tical

com

pone

nt o

f d

Yo

ur

Fo

rce

Example

How much work is done to carry a 5 kg cat to the top of a ramp that is 7 meters long and 3 meters tall

W = Force x displacementForce = weight of the catd = height NOT lengthW = mg x hW = 5 x 10 x 3 W = 150 J

7 m

3 m

ExampleA boy pushes a

lawnmower 20 meters across the yard If he pushed with a force of 200 N and the angle between the handle and the ground was 50 degrees how much work did he do

F

Displacement = 20 m

F cos

W = (F cos )dW = (200 cos 50) 20W = 2571 J

NOTE If while pushing an object it is moving at a constant velocity

the NET force must be zero

Sohellip Your applied force must be exactly equal to any resistant forces like friction

How much work do you do to carry a 30 kg cat from one side of the room to the other if the room is 10 meters long

ZERO because your Force is vertical but the displacement is horizontal

bull Energy and Work have no direction associated with them and are therefore scalar quantities not vectors

Power

The rate at which work is done

1 Power = Work divide time

Unit for power = J divide s

= Watt W

What is a Watt in ldquofundamental unitsrdquo

Since work is also the energy transferred or transformed ldquopowerrdquo is the rate at which energy is transferred or transformed

2 P = Energy divide time

This energy could be in ANY form heat light potential chemical nuclear

Since NET work = K

3 P = K divide t

And yet another approach

P = W divide t = (Fd) divide t = F middot (d divide t)

P = F middot v

Kinetic Energy

the energy of motion

K = frac12 mv2

Kinetic Energy

the energy of motion

K = frac12 mv2

Where does K = frac12 mv2 come from

Did your amazing teacher just arbitrarily make that equation up

Hmmmhellip

The ldquoWork- Kinetic Energy TheoremrdquoWorknet = K

Fnet bulld = K = frac12 mvf2 - frac12 mvo

2

You are supposed to be able touse Fnet = ma kinematics equations and Wnet = Fnet bulld to

derive the work-kinetic energy theoremhellip

Sohellip do it Then yoursquoll see where the equation we call ldquokinetic energyrdquo comes from

(Hint start with Fnet = ma and use the kinematics equation that doesnrsquot involve time)

Potential EnergyStored energy

It is called potential energy because it has the potential to do work

bull Example 1 Spring potential energy in the stretched string of a bow or spring or rubber band SPE = frac12 kx2

bull Example 2 Chemical potential energy in fuels- gasoline propane batteries food

bull Example 3 Gravitational potential energy- stored in an object due to its position from a chosen reference point

Gravitational potential energy

GPE = weight x height

GPE = mgh

bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative

bull The GPE only depends on the weight and the height not on the path that it took to get to that height

Work and EnergyOften some force must do work

to give an object potential or kinetic energy

You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy

Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy

Work = Force x distance = change in energy

Example

How much more distance is required to stop if a car is going twice as fast

Fd = frac12 mv2

The work done by the brakes = the change in the kinetic energy

With TWICE the speed the car has

FOUR times the kinetic energy

Therefore it takes FOUR times the stopping distance

The Work-Kinetic Energy Theorem

NET Work done = Kinetic Energy

Wnet = frac12 mv2f ndash frac12 mv2

o

Varying Forces

The rule ishellip ldquoIf the Force varies you must integraterdquo

If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done

If the force is a function of velocity you must integrate to find the power output

Examples

If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5

If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms

Examples of Integration

An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force

W F x dx x dx x 33

43 4

Examples of Integration

At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)

v t a t dt tdt t C t( ) ( )

1515

20 75 32 2

x t v t dt t dt t t C t t( ) ( ) ( )

0 75 30 75

23 0 375 3 02 2

Examples of IntegrationA particle of mass m moves along the y-axis as

y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force

P = Fmiddotv

We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )

P F v m at b at bty y ( )( )12 2 4 2 22 3

Examples of Definite Integration

An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s

x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2

62

2

63 2

2

6 3 2 3 22 8 62

3

8

26

2

36

8

26 6 6

2

32

8

22 6 2 45 33

Examples of Definite Integration

To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x

What work is done to stretch the spring 2 meters beyond its equilibrium position

W F ds x x ds x x Jmm

m ( ) ( ) ( ) 4

4

3

1

2

4

32

1

22

4

30

1

20 8 6672

0

23

0

22

0

2 3 2 2

Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under

the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m

P = Fmiddotv but what is the velocity at x = 3m

Hmmmhellip

W F x dx and W K so ( )

F x dx K mv mvx

x

f o

o

f

( ) 12

12

2 2 ( )6 4 3 12

2

0

3

32 x x dx K mv x

( ) ( )6 4 3 64

2

3

36 3

4

23

3

33 0 9 1

22

0

3

2 3

0

3 2 33

2 x x dx x x x J mv x

v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5

  • Slide 1
  • Work Energy and Power AP style
  • Energy
  • Slide 4
  • Slide 5
  • Work = Force x displacement
  • Slide 7
  • Slide 8
  • Slide 9
  • Slide 10
  • Slide 11
  • Example
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Power
  • Slide 18
  • Slide 19
  • Slide 20
  • Slide 21
  • The ldquoWork- Kinetic Energy Theoremrdquo
  • Potential Energy
  • Slide 24
  • Gravitational potential energy
  • Slide 26
  • Work and Energy
  • Slide 28
  • The Work-Kinetic Energy Theorem
  • Varying Forces
  • Slide 31
  • Examples of Integration
  • Slide 33
  • Slide 34
  • Examples of Definite Integration
  • Slide 36
  • Slide 37
Page 4: Work, Energy and Power AP style Energy Energy: the currency of the universe. Everything has to be “paid for” with energy. Energy can’t be created or.

bull Doing WORK is one way to transfer energy from one object to another

Work = Force x displacement

W = Fd

bull Unit for work is Newton x meter One Newton-meter is also called a Joule J

Work = Force x displacement

bull Work is not done unless there is a displacement

bull If you hold an object a long time you may get tired but NO work was done

bull If you push against a solid wall for hours there is still NO work done

bull For work to be done the displacement of the object must be in the same direction as the applied force They must be parallel

bull If the force and the displacement are perpendicular to each other NO work is done by the force

So using vector multiplication

W = F bull d

(In many university texts as well as the AP test the displacement is represented by ldquosrdquo

and not ldquodrdquo

W = F bull s

An object is subject to a force given byF = 6i ndash 8j as it moves from the position r = -4i + 3j

to the position r = i + 7j

What work was done by this forceFirst find the displacement s

s = r =

rf ndash ro =(i + 7j) - (-4i + 3j) =

5i + 4j

Then do the dot product W = F middot s(6i ndash 8j) middot (5i + 4j) =

30 ndash 32 = -2J of work

bull In lifting a book the force exerted by your hands is upward and the displacement is upward- work is done

bull Similarly in lowering a book the force exerted by your hands is still upward and the displacement is downward

bull The force and the displacement are STILL parallel so work is still done

bull But since they are in opposite directions now it is NEGATIVE work

bull On the other hand while carrying a book down the hallway the force from your hands is vertical and the displacement of the book is horizontal

bull Therefore NO work is done by your hands

bull Sohellipwhile climbing stairs or walking up an incline only the vertical component of the displacement is used to calculate the work done in moving the object from the bottom to the top Horizontal component of d

Ver

tical

com

pone

nt o

f d

Yo

ur

Fo

rce

Example

How much work is done to carry a 5 kg cat to the top of a ramp that is 7 meters long and 3 meters tall

W = Force x displacementForce = weight of the catd = height NOT lengthW = mg x hW = 5 x 10 x 3 W = 150 J

7 m

3 m

ExampleA boy pushes a

lawnmower 20 meters across the yard If he pushed with a force of 200 N and the angle between the handle and the ground was 50 degrees how much work did he do

F

Displacement = 20 m

F cos

W = (F cos )dW = (200 cos 50) 20W = 2571 J

NOTE If while pushing an object it is moving at a constant velocity

the NET force must be zero

Sohellip Your applied force must be exactly equal to any resistant forces like friction

How much work do you do to carry a 30 kg cat from one side of the room to the other if the room is 10 meters long

ZERO because your Force is vertical but the displacement is horizontal

bull Energy and Work have no direction associated with them and are therefore scalar quantities not vectors

Power

The rate at which work is done

1 Power = Work divide time

Unit for power = J divide s

= Watt W

What is a Watt in ldquofundamental unitsrdquo

Since work is also the energy transferred or transformed ldquopowerrdquo is the rate at which energy is transferred or transformed

2 P = Energy divide time

This energy could be in ANY form heat light potential chemical nuclear

Since NET work = K

3 P = K divide t

And yet another approach

P = W divide t = (Fd) divide t = F middot (d divide t)

P = F middot v

Kinetic Energy

the energy of motion

K = frac12 mv2

Kinetic Energy

the energy of motion

K = frac12 mv2

Where does K = frac12 mv2 come from

Did your amazing teacher just arbitrarily make that equation up

Hmmmhellip

The ldquoWork- Kinetic Energy TheoremrdquoWorknet = K

Fnet bulld = K = frac12 mvf2 - frac12 mvo

2

You are supposed to be able touse Fnet = ma kinematics equations and Wnet = Fnet bulld to

derive the work-kinetic energy theoremhellip

Sohellip do it Then yoursquoll see where the equation we call ldquokinetic energyrdquo comes from

(Hint start with Fnet = ma and use the kinematics equation that doesnrsquot involve time)

Potential EnergyStored energy

It is called potential energy because it has the potential to do work

bull Example 1 Spring potential energy in the stretched string of a bow or spring or rubber band SPE = frac12 kx2

bull Example 2 Chemical potential energy in fuels- gasoline propane batteries food

bull Example 3 Gravitational potential energy- stored in an object due to its position from a chosen reference point

Gravitational potential energy

GPE = weight x height

GPE = mgh

bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative

bull The GPE only depends on the weight and the height not on the path that it took to get to that height

Work and EnergyOften some force must do work

to give an object potential or kinetic energy

You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy

Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy

Work = Force x distance = change in energy

Example

How much more distance is required to stop if a car is going twice as fast

Fd = frac12 mv2

The work done by the brakes = the change in the kinetic energy

With TWICE the speed the car has

FOUR times the kinetic energy

Therefore it takes FOUR times the stopping distance

The Work-Kinetic Energy Theorem

NET Work done = Kinetic Energy

Wnet = frac12 mv2f ndash frac12 mv2

o

Varying Forces

The rule ishellip ldquoIf the Force varies you must integraterdquo

If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done

If the force is a function of velocity you must integrate to find the power output

Examples

If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5

If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms

Examples of Integration

An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force

W F x dx x dx x 33

43 4

Examples of Integration

At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)

v t a t dt tdt t C t( ) ( )

1515

20 75 32 2

x t v t dt t dt t t C t t( ) ( ) ( )

0 75 30 75

23 0 375 3 02 2

Examples of IntegrationA particle of mass m moves along the y-axis as

y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force

P = Fmiddotv

We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )

P F v m at b at bty y ( )( )12 2 4 2 22 3

Examples of Definite Integration

An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s

x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2

62

2

63 2

2

6 3 2 3 22 8 62

3

8

26

2

36

8

26 6 6

2

32

8

22 6 2 45 33

Examples of Definite Integration

To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x

What work is done to stretch the spring 2 meters beyond its equilibrium position

W F ds x x ds x x Jmm

m ( ) ( ) ( ) 4

4

3

1

2

4

32

1

22

4

30

1

20 8 6672

0

23

0

22

0

2 3 2 2

Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under

the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m

P = Fmiddotv but what is the velocity at x = 3m

Hmmmhellip

W F x dx and W K so ( )

F x dx K mv mvx

x

f o

o

f

( ) 12

12

2 2 ( )6 4 3 12

2

0

3

32 x x dx K mv x

( ) ( )6 4 3 64

2

3

36 3

4

23

3

33 0 9 1

22

0

3

2 3

0

3 2 33

2 x x dx x x x J mv x

v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5

  • Slide 1
  • Work Energy and Power AP style
  • Energy
  • Slide 4
  • Slide 5
  • Work = Force x displacement
  • Slide 7
  • Slide 8
  • Slide 9
  • Slide 10
  • Slide 11
  • Example
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Power
  • Slide 18
  • Slide 19
  • Slide 20
  • Slide 21
  • The ldquoWork- Kinetic Energy Theoremrdquo
  • Potential Energy
  • Slide 24
  • Gravitational potential energy
  • Slide 26
  • Work and Energy
  • Slide 28
  • The Work-Kinetic Energy Theorem
  • Varying Forces
  • Slide 31
  • Examples of Integration
  • Slide 33
  • Slide 34
  • Examples of Definite Integration
  • Slide 36
  • Slide 37
Page 5: Work, Energy and Power AP style Energy Energy: the currency of the universe. Everything has to be “paid for” with energy. Energy can’t be created or.

Work = Force x displacement

bull Work is not done unless there is a displacement

bull If you hold an object a long time you may get tired but NO work was done

bull If you push against a solid wall for hours there is still NO work done

bull For work to be done the displacement of the object must be in the same direction as the applied force They must be parallel

bull If the force and the displacement are perpendicular to each other NO work is done by the force

So using vector multiplication

W = F bull d

(In many university texts as well as the AP test the displacement is represented by ldquosrdquo

and not ldquodrdquo

W = F bull s

An object is subject to a force given byF = 6i ndash 8j as it moves from the position r = -4i + 3j

to the position r = i + 7j

What work was done by this forceFirst find the displacement s

s = r =

rf ndash ro =(i + 7j) - (-4i + 3j) =

5i + 4j

Then do the dot product W = F middot s(6i ndash 8j) middot (5i + 4j) =

30 ndash 32 = -2J of work

bull In lifting a book the force exerted by your hands is upward and the displacement is upward- work is done

bull Similarly in lowering a book the force exerted by your hands is still upward and the displacement is downward

bull The force and the displacement are STILL parallel so work is still done

bull But since they are in opposite directions now it is NEGATIVE work

bull On the other hand while carrying a book down the hallway the force from your hands is vertical and the displacement of the book is horizontal

bull Therefore NO work is done by your hands

bull Sohellipwhile climbing stairs or walking up an incline only the vertical component of the displacement is used to calculate the work done in moving the object from the bottom to the top Horizontal component of d

Ver

tical

com

pone

nt o

f d

Yo

ur

Fo

rce

Example

How much work is done to carry a 5 kg cat to the top of a ramp that is 7 meters long and 3 meters tall

W = Force x displacementForce = weight of the catd = height NOT lengthW = mg x hW = 5 x 10 x 3 W = 150 J

7 m

3 m

ExampleA boy pushes a

lawnmower 20 meters across the yard If he pushed with a force of 200 N and the angle between the handle and the ground was 50 degrees how much work did he do

F

Displacement = 20 m

F cos

W = (F cos )dW = (200 cos 50) 20W = 2571 J

NOTE If while pushing an object it is moving at a constant velocity

the NET force must be zero

Sohellip Your applied force must be exactly equal to any resistant forces like friction

How much work do you do to carry a 30 kg cat from one side of the room to the other if the room is 10 meters long

ZERO because your Force is vertical but the displacement is horizontal

bull Energy and Work have no direction associated with them and are therefore scalar quantities not vectors

Power

The rate at which work is done

1 Power = Work divide time

Unit for power = J divide s

= Watt W

What is a Watt in ldquofundamental unitsrdquo

Since work is also the energy transferred or transformed ldquopowerrdquo is the rate at which energy is transferred or transformed

2 P = Energy divide time

This energy could be in ANY form heat light potential chemical nuclear

Since NET work = K

3 P = K divide t

And yet another approach

P = W divide t = (Fd) divide t = F middot (d divide t)

P = F middot v

Kinetic Energy

the energy of motion

K = frac12 mv2

Kinetic Energy

the energy of motion

K = frac12 mv2

Where does K = frac12 mv2 come from

Did your amazing teacher just arbitrarily make that equation up

Hmmmhellip

The ldquoWork- Kinetic Energy TheoremrdquoWorknet = K

Fnet bulld = K = frac12 mvf2 - frac12 mvo

2

You are supposed to be able touse Fnet = ma kinematics equations and Wnet = Fnet bulld to

derive the work-kinetic energy theoremhellip

Sohellip do it Then yoursquoll see where the equation we call ldquokinetic energyrdquo comes from

(Hint start with Fnet = ma and use the kinematics equation that doesnrsquot involve time)

Potential EnergyStored energy

It is called potential energy because it has the potential to do work

bull Example 1 Spring potential energy in the stretched string of a bow or spring or rubber band SPE = frac12 kx2

bull Example 2 Chemical potential energy in fuels- gasoline propane batteries food

bull Example 3 Gravitational potential energy- stored in an object due to its position from a chosen reference point

Gravitational potential energy

GPE = weight x height

GPE = mgh

bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative

bull The GPE only depends on the weight and the height not on the path that it took to get to that height

Work and EnergyOften some force must do work

to give an object potential or kinetic energy

You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy

Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy

Work = Force x distance = change in energy

Example

How much more distance is required to stop if a car is going twice as fast

Fd = frac12 mv2

The work done by the brakes = the change in the kinetic energy

With TWICE the speed the car has

FOUR times the kinetic energy

Therefore it takes FOUR times the stopping distance

The Work-Kinetic Energy Theorem

NET Work done = Kinetic Energy

Wnet = frac12 mv2f ndash frac12 mv2

o

Varying Forces

The rule ishellip ldquoIf the Force varies you must integraterdquo

If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done

If the force is a function of velocity you must integrate to find the power output

Examples

If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5

If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms

Examples of Integration

An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force

W F x dx x dx x 33

43 4

Examples of Integration

At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)

v t a t dt tdt t C t( ) ( )

1515

20 75 32 2

x t v t dt t dt t t C t t( ) ( ) ( )

0 75 30 75

23 0 375 3 02 2

Examples of IntegrationA particle of mass m moves along the y-axis as

y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force

P = Fmiddotv

We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )

P F v m at b at bty y ( )( )12 2 4 2 22 3

Examples of Definite Integration

An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s

x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2

62

2

63 2

2

6 3 2 3 22 8 62

3

8

26

2

36

8

26 6 6

2

32

8

22 6 2 45 33

Examples of Definite Integration

To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x

What work is done to stretch the spring 2 meters beyond its equilibrium position

W F ds x x ds x x Jmm

m ( ) ( ) ( ) 4

4

3

1

2

4

32

1

22

4

30

1

20 8 6672

0

23

0

22

0

2 3 2 2

Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under

the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m

P = Fmiddotv but what is the velocity at x = 3m

Hmmmhellip

W F x dx and W K so ( )

F x dx K mv mvx

x

f o

o

f

( ) 12

12

2 2 ( )6 4 3 12

2

0

3

32 x x dx K mv x

( ) ( )6 4 3 64

2

3

36 3

4

23

3

33 0 9 1

22

0

3

2 3

0

3 2 33

2 x x dx x x x J mv x

v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5

  • Slide 1
  • Work Energy and Power AP style
  • Energy
  • Slide 4
  • Slide 5
  • Work = Force x displacement
  • Slide 7
  • Slide 8
  • Slide 9
  • Slide 10
  • Slide 11
  • Example
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Power
  • Slide 18
  • Slide 19
  • Slide 20
  • Slide 21
  • The ldquoWork- Kinetic Energy Theoremrdquo
  • Potential Energy
  • Slide 24
  • Gravitational potential energy
  • Slide 26
  • Work and Energy
  • Slide 28
  • The Work-Kinetic Energy Theorem
  • Varying Forces
  • Slide 31
  • Examples of Integration
  • Slide 33
  • Slide 34
  • Examples of Definite Integration
  • Slide 36
  • Slide 37
Page 6: Work, Energy and Power AP style Energy Energy: the currency of the universe. Everything has to be “paid for” with energy. Energy can’t be created or.

bull For work to be done the displacement of the object must be in the same direction as the applied force They must be parallel

bull If the force and the displacement are perpendicular to each other NO work is done by the force

So using vector multiplication

W = F bull d

(In many university texts as well as the AP test the displacement is represented by ldquosrdquo

and not ldquodrdquo

W = F bull s

An object is subject to a force given byF = 6i ndash 8j as it moves from the position r = -4i + 3j

to the position r = i + 7j

What work was done by this forceFirst find the displacement s

s = r =

rf ndash ro =(i + 7j) - (-4i + 3j) =

5i + 4j

Then do the dot product W = F middot s(6i ndash 8j) middot (5i + 4j) =

30 ndash 32 = -2J of work

bull In lifting a book the force exerted by your hands is upward and the displacement is upward- work is done

bull Similarly in lowering a book the force exerted by your hands is still upward and the displacement is downward

bull The force and the displacement are STILL parallel so work is still done

bull But since they are in opposite directions now it is NEGATIVE work

bull On the other hand while carrying a book down the hallway the force from your hands is vertical and the displacement of the book is horizontal

bull Therefore NO work is done by your hands

bull Sohellipwhile climbing stairs or walking up an incline only the vertical component of the displacement is used to calculate the work done in moving the object from the bottom to the top Horizontal component of d

Ver

tical

com

pone

nt o

f d

Yo

ur

Fo

rce

Example

How much work is done to carry a 5 kg cat to the top of a ramp that is 7 meters long and 3 meters tall

W = Force x displacementForce = weight of the catd = height NOT lengthW = mg x hW = 5 x 10 x 3 W = 150 J

7 m

3 m

ExampleA boy pushes a

lawnmower 20 meters across the yard If he pushed with a force of 200 N and the angle between the handle and the ground was 50 degrees how much work did he do

F

Displacement = 20 m

F cos

W = (F cos )dW = (200 cos 50) 20W = 2571 J

NOTE If while pushing an object it is moving at a constant velocity

the NET force must be zero

Sohellip Your applied force must be exactly equal to any resistant forces like friction

How much work do you do to carry a 30 kg cat from one side of the room to the other if the room is 10 meters long

ZERO because your Force is vertical but the displacement is horizontal

bull Energy and Work have no direction associated with them and are therefore scalar quantities not vectors

Power

The rate at which work is done

1 Power = Work divide time

Unit for power = J divide s

= Watt W

What is a Watt in ldquofundamental unitsrdquo

Since work is also the energy transferred or transformed ldquopowerrdquo is the rate at which energy is transferred or transformed

2 P = Energy divide time

This energy could be in ANY form heat light potential chemical nuclear

Since NET work = K

3 P = K divide t

And yet another approach

P = W divide t = (Fd) divide t = F middot (d divide t)

P = F middot v

Kinetic Energy

the energy of motion

K = frac12 mv2

Kinetic Energy

the energy of motion

K = frac12 mv2

Where does K = frac12 mv2 come from

Did your amazing teacher just arbitrarily make that equation up

Hmmmhellip

The ldquoWork- Kinetic Energy TheoremrdquoWorknet = K

Fnet bulld = K = frac12 mvf2 - frac12 mvo

2

You are supposed to be able touse Fnet = ma kinematics equations and Wnet = Fnet bulld to

derive the work-kinetic energy theoremhellip

Sohellip do it Then yoursquoll see where the equation we call ldquokinetic energyrdquo comes from

(Hint start with Fnet = ma and use the kinematics equation that doesnrsquot involve time)

Potential EnergyStored energy

It is called potential energy because it has the potential to do work

bull Example 1 Spring potential energy in the stretched string of a bow or spring or rubber band SPE = frac12 kx2

bull Example 2 Chemical potential energy in fuels- gasoline propane batteries food

bull Example 3 Gravitational potential energy- stored in an object due to its position from a chosen reference point

Gravitational potential energy

GPE = weight x height

GPE = mgh

bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative

bull The GPE only depends on the weight and the height not on the path that it took to get to that height

Work and EnergyOften some force must do work

to give an object potential or kinetic energy

You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy

Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy

Work = Force x distance = change in energy

Example

How much more distance is required to stop if a car is going twice as fast

Fd = frac12 mv2

The work done by the brakes = the change in the kinetic energy

With TWICE the speed the car has

FOUR times the kinetic energy

Therefore it takes FOUR times the stopping distance

The Work-Kinetic Energy Theorem

NET Work done = Kinetic Energy

Wnet = frac12 mv2f ndash frac12 mv2

o

Varying Forces

The rule ishellip ldquoIf the Force varies you must integraterdquo

If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done

If the force is a function of velocity you must integrate to find the power output

Examples

If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5

If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms

Examples of Integration

An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force

W F x dx x dx x 33

43 4

Examples of Integration

At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)

v t a t dt tdt t C t( ) ( )

1515

20 75 32 2

x t v t dt t dt t t C t t( ) ( ) ( )

0 75 30 75

23 0 375 3 02 2

Examples of IntegrationA particle of mass m moves along the y-axis as

y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force

P = Fmiddotv

We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )

P F v m at b at bty y ( )( )12 2 4 2 22 3

Examples of Definite Integration

An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s

x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2

62

2

63 2

2

6 3 2 3 22 8 62

3

8

26

2

36

8

26 6 6

2

32

8

22 6 2 45 33

Examples of Definite Integration

To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x

What work is done to stretch the spring 2 meters beyond its equilibrium position

W F ds x x ds x x Jmm

m ( ) ( ) ( ) 4

4

3

1

2

4

32

1

22

4

30

1

20 8 6672

0

23

0

22

0

2 3 2 2

Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under

the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m

P = Fmiddotv but what is the velocity at x = 3m

Hmmmhellip

W F x dx and W K so ( )

F x dx K mv mvx

x

f o

o

f

( ) 12

12

2 2 ( )6 4 3 12

2

0

3

32 x x dx K mv x

( ) ( )6 4 3 64

2

3

36 3

4

23

3

33 0 9 1

22

0

3

2 3

0

3 2 33

2 x x dx x x x J mv x

v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5

  • Slide 1
  • Work Energy and Power AP style
  • Energy
  • Slide 4
  • Slide 5
  • Work = Force x displacement
  • Slide 7
  • Slide 8
  • Slide 9
  • Slide 10
  • Slide 11
  • Example
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Power
  • Slide 18
  • Slide 19
  • Slide 20
  • Slide 21
  • The ldquoWork- Kinetic Energy Theoremrdquo
  • Potential Energy
  • Slide 24
  • Gravitational potential energy
  • Slide 26
  • Work and Energy
  • Slide 28
  • The Work-Kinetic Energy Theorem
  • Varying Forces
  • Slide 31
  • Examples of Integration
  • Slide 33
  • Slide 34
  • Examples of Definite Integration
  • Slide 36
  • Slide 37
Page 7: Work, Energy and Power AP style Energy Energy: the currency of the universe. Everything has to be “paid for” with energy. Energy can’t be created or.

An object is subject to a force given byF = 6i ndash 8j as it moves from the position r = -4i + 3j

to the position r = i + 7j

What work was done by this forceFirst find the displacement s

s = r =

rf ndash ro =(i + 7j) - (-4i + 3j) =

5i + 4j

Then do the dot product W = F middot s(6i ndash 8j) middot (5i + 4j) =

30 ndash 32 = -2J of work

bull In lifting a book the force exerted by your hands is upward and the displacement is upward- work is done

bull Similarly in lowering a book the force exerted by your hands is still upward and the displacement is downward

bull The force and the displacement are STILL parallel so work is still done

bull But since they are in opposite directions now it is NEGATIVE work

bull On the other hand while carrying a book down the hallway the force from your hands is vertical and the displacement of the book is horizontal

bull Therefore NO work is done by your hands

bull Sohellipwhile climbing stairs or walking up an incline only the vertical component of the displacement is used to calculate the work done in moving the object from the bottom to the top Horizontal component of d

Ver

tical

com

pone

nt o

f d

Yo

ur

Fo

rce

Example

How much work is done to carry a 5 kg cat to the top of a ramp that is 7 meters long and 3 meters tall

W = Force x displacementForce = weight of the catd = height NOT lengthW = mg x hW = 5 x 10 x 3 W = 150 J

7 m

3 m

ExampleA boy pushes a

lawnmower 20 meters across the yard If he pushed with a force of 200 N and the angle between the handle and the ground was 50 degrees how much work did he do

F

Displacement = 20 m

F cos

W = (F cos )dW = (200 cos 50) 20W = 2571 J

NOTE If while pushing an object it is moving at a constant velocity

the NET force must be zero

Sohellip Your applied force must be exactly equal to any resistant forces like friction

How much work do you do to carry a 30 kg cat from one side of the room to the other if the room is 10 meters long

ZERO because your Force is vertical but the displacement is horizontal

bull Energy and Work have no direction associated with them and are therefore scalar quantities not vectors

Power

The rate at which work is done

1 Power = Work divide time

Unit for power = J divide s

= Watt W

What is a Watt in ldquofundamental unitsrdquo

Since work is also the energy transferred or transformed ldquopowerrdquo is the rate at which energy is transferred or transformed

2 P = Energy divide time

This energy could be in ANY form heat light potential chemical nuclear

Since NET work = K

3 P = K divide t

And yet another approach

P = W divide t = (Fd) divide t = F middot (d divide t)

P = F middot v

Kinetic Energy

the energy of motion

K = frac12 mv2

Kinetic Energy

the energy of motion

K = frac12 mv2

Where does K = frac12 mv2 come from

Did your amazing teacher just arbitrarily make that equation up

Hmmmhellip

The ldquoWork- Kinetic Energy TheoremrdquoWorknet = K

Fnet bulld = K = frac12 mvf2 - frac12 mvo

2

You are supposed to be able touse Fnet = ma kinematics equations and Wnet = Fnet bulld to

derive the work-kinetic energy theoremhellip

Sohellip do it Then yoursquoll see where the equation we call ldquokinetic energyrdquo comes from

(Hint start with Fnet = ma and use the kinematics equation that doesnrsquot involve time)

Potential EnergyStored energy

It is called potential energy because it has the potential to do work

bull Example 1 Spring potential energy in the stretched string of a bow or spring or rubber band SPE = frac12 kx2

bull Example 2 Chemical potential energy in fuels- gasoline propane batteries food

bull Example 3 Gravitational potential energy- stored in an object due to its position from a chosen reference point

Gravitational potential energy

GPE = weight x height

GPE = mgh

bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative

bull The GPE only depends on the weight and the height not on the path that it took to get to that height

Work and EnergyOften some force must do work

to give an object potential or kinetic energy

You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy

Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy

Work = Force x distance = change in energy

Example

How much more distance is required to stop if a car is going twice as fast

Fd = frac12 mv2

The work done by the brakes = the change in the kinetic energy

With TWICE the speed the car has

FOUR times the kinetic energy

Therefore it takes FOUR times the stopping distance

The Work-Kinetic Energy Theorem

NET Work done = Kinetic Energy

Wnet = frac12 mv2f ndash frac12 mv2

o

Varying Forces

The rule ishellip ldquoIf the Force varies you must integraterdquo

If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done

If the force is a function of velocity you must integrate to find the power output

Examples

If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5

If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms

Examples of Integration

An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force

W F x dx x dx x 33

43 4

Examples of Integration

At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)

v t a t dt tdt t C t( ) ( )

1515

20 75 32 2

x t v t dt t dt t t C t t( ) ( ) ( )

0 75 30 75

23 0 375 3 02 2

Examples of IntegrationA particle of mass m moves along the y-axis as

y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force

P = Fmiddotv

We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )

P F v m at b at bty y ( )( )12 2 4 2 22 3

Examples of Definite Integration

An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s

x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2

62

2

63 2

2

6 3 2 3 22 8 62

3

8

26

2

36

8

26 6 6

2

32

8

22 6 2 45 33

Examples of Definite Integration

To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x

What work is done to stretch the spring 2 meters beyond its equilibrium position

W F ds x x ds x x Jmm

m ( ) ( ) ( ) 4

4

3

1

2

4

32

1

22

4

30

1

20 8 6672

0

23

0

22

0

2 3 2 2

Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under

the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m

P = Fmiddotv but what is the velocity at x = 3m

Hmmmhellip

W F x dx and W K so ( )

F x dx K mv mvx

x

f o

o

f

( ) 12

12

2 2 ( )6 4 3 12

2

0

3

32 x x dx K mv x

( ) ( )6 4 3 64

2

3

36 3

4

23

3

33 0 9 1

22

0

3

2 3

0

3 2 33

2 x x dx x x x J mv x

v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5

  • Slide 1
  • Work Energy and Power AP style
  • Energy
  • Slide 4
  • Slide 5
  • Work = Force x displacement
  • Slide 7
  • Slide 8
  • Slide 9
  • Slide 10
  • Slide 11
  • Example
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Power
  • Slide 18
  • Slide 19
  • Slide 20
  • Slide 21
  • The ldquoWork- Kinetic Energy Theoremrdquo
  • Potential Energy
  • Slide 24
  • Gravitational potential energy
  • Slide 26
  • Work and Energy
  • Slide 28
  • The Work-Kinetic Energy Theorem
  • Varying Forces
  • Slide 31
  • Examples of Integration
  • Slide 33
  • Slide 34
  • Examples of Definite Integration
  • Slide 36
  • Slide 37
Page 8: Work, Energy and Power AP style Energy Energy: the currency of the universe. Everything has to be “paid for” with energy. Energy can’t be created or.

bull In lifting a book the force exerted by your hands is upward and the displacement is upward- work is done

bull Similarly in lowering a book the force exerted by your hands is still upward and the displacement is downward

bull The force and the displacement are STILL parallel so work is still done

bull But since they are in opposite directions now it is NEGATIVE work

bull On the other hand while carrying a book down the hallway the force from your hands is vertical and the displacement of the book is horizontal

bull Therefore NO work is done by your hands

bull Sohellipwhile climbing stairs or walking up an incline only the vertical component of the displacement is used to calculate the work done in moving the object from the bottom to the top Horizontal component of d

Ver

tical

com

pone

nt o

f d

Yo

ur

Fo

rce

Example

How much work is done to carry a 5 kg cat to the top of a ramp that is 7 meters long and 3 meters tall

W = Force x displacementForce = weight of the catd = height NOT lengthW = mg x hW = 5 x 10 x 3 W = 150 J

7 m

3 m

ExampleA boy pushes a

lawnmower 20 meters across the yard If he pushed with a force of 200 N and the angle between the handle and the ground was 50 degrees how much work did he do

F

Displacement = 20 m

F cos

W = (F cos )dW = (200 cos 50) 20W = 2571 J

NOTE If while pushing an object it is moving at a constant velocity

the NET force must be zero

Sohellip Your applied force must be exactly equal to any resistant forces like friction

How much work do you do to carry a 30 kg cat from one side of the room to the other if the room is 10 meters long

ZERO because your Force is vertical but the displacement is horizontal

bull Energy and Work have no direction associated with them and are therefore scalar quantities not vectors

Power

The rate at which work is done

1 Power = Work divide time

Unit for power = J divide s

= Watt W

What is a Watt in ldquofundamental unitsrdquo

Since work is also the energy transferred or transformed ldquopowerrdquo is the rate at which energy is transferred or transformed

2 P = Energy divide time

This energy could be in ANY form heat light potential chemical nuclear

Since NET work = K

3 P = K divide t

And yet another approach

P = W divide t = (Fd) divide t = F middot (d divide t)

P = F middot v

Kinetic Energy

the energy of motion

K = frac12 mv2

Kinetic Energy

the energy of motion

K = frac12 mv2

Where does K = frac12 mv2 come from

Did your amazing teacher just arbitrarily make that equation up

Hmmmhellip

The ldquoWork- Kinetic Energy TheoremrdquoWorknet = K

Fnet bulld = K = frac12 mvf2 - frac12 mvo

2

You are supposed to be able touse Fnet = ma kinematics equations and Wnet = Fnet bulld to

derive the work-kinetic energy theoremhellip

Sohellip do it Then yoursquoll see where the equation we call ldquokinetic energyrdquo comes from

(Hint start with Fnet = ma and use the kinematics equation that doesnrsquot involve time)

Potential EnergyStored energy

It is called potential energy because it has the potential to do work

bull Example 1 Spring potential energy in the stretched string of a bow or spring or rubber band SPE = frac12 kx2

bull Example 2 Chemical potential energy in fuels- gasoline propane batteries food

bull Example 3 Gravitational potential energy- stored in an object due to its position from a chosen reference point

Gravitational potential energy

GPE = weight x height

GPE = mgh

bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative

bull The GPE only depends on the weight and the height not on the path that it took to get to that height

Work and EnergyOften some force must do work

to give an object potential or kinetic energy

You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy

Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy

Work = Force x distance = change in energy

Example

How much more distance is required to stop if a car is going twice as fast

Fd = frac12 mv2

The work done by the brakes = the change in the kinetic energy

With TWICE the speed the car has

FOUR times the kinetic energy

Therefore it takes FOUR times the stopping distance

The Work-Kinetic Energy Theorem

NET Work done = Kinetic Energy

Wnet = frac12 mv2f ndash frac12 mv2

o

Varying Forces

The rule ishellip ldquoIf the Force varies you must integraterdquo

If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done

If the force is a function of velocity you must integrate to find the power output

Examples

If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5

If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms

Examples of Integration

An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force

W F x dx x dx x 33

43 4

Examples of Integration

At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)

v t a t dt tdt t C t( ) ( )

1515

20 75 32 2

x t v t dt t dt t t C t t( ) ( ) ( )

0 75 30 75

23 0 375 3 02 2

Examples of IntegrationA particle of mass m moves along the y-axis as

y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force

P = Fmiddotv

We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )

P F v m at b at bty y ( )( )12 2 4 2 22 3

Examples of Definite Integration

An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s

x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2

62

2

63 2

2

6 3 2 3 22 8 62

3

8

26

2

36

8

26 6 6

2

32

8

22 6 2 45 33

Examples of Definite Integration

To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x

What work is done to stretch the spring 2 meters beyond its equilibrium position

W F ds x x ds x x Jmm

m ( ) ( ) ( ) 4

4

3

1

2

4

32

1

22

4

30

1

20 8 6672

0

23

0

22

0

2 3 2 2

Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under

the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m

P = Fmiddotv but what is the velocity at x = 3m

Hmmmhellip

W F x dx and W K so ( )

F x dx K mv mvx

x

f o

o

f

( ) 12

12

2 2 ( )6 4 3 12

2

0

3

32 x x dx K mv x

( ) ( )6 4 3 64

2

3

36 3

4

23

3

33 0 9 1

22

0

3

2 3

0

3 2 33

2 x x dx x x x J mv x

v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5

  • Slide 1
  • Work Energy and Power AP style
  • Energy
  • Slide 4
  • Slide 5
  • Work = Force x displacement
  • Slide 7
  • Slide 8
  • Slide 9
  • Slide 10
  • Slide 11
  • Example
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Power
  • Slide 18
  • Slide 19
  • Slide 20
  • Slide 21
  • The ldquoWork- Kinetic Energy Theoremrdquo
  • Potential Energy
  • Slide 24
  • Gravitational potential energy
  • Slide 26
  • Work and Energy
  • Slide 28
  • The Work-Kinetic Energy Theorem
  • Varying Forces
  • Slide 31
  • Examples of Integration
  • Slide 33
  • Slide 34
  • Examples of Definite Integration
  • Slide 36
  • Slide 37
Page 9: Work, Energy and Power AP style Energy Energy: the currency of the universe. Everything has to be “paid for” with energy. Energy can’t be created or.

bull On the other hand while carrying a book down the hallway the force from your hands is vertical and the displacement of the book is horizontal

bull Therefore NO work is done by your hands

bull Sohellipwhile climbing stairs or walking up an incline only the vertical component of the displacement is used to calculate the work done in moving the object from the bottom to the top Horizontal component of d

Ver

tical

com

pone

nt o

f d

Yo

ur

Fo

rce

Example

How much work is done to carry a 5 kg cat to the top of a ramp that is 7 meters long and 3 meters tall

W = Force x displacementForce = weight of the catd = height NOT lengthW = mg x hW = 5 x 10 x 3 W = 150 J

7 m

3 m

ExampleA boy pushes a

lawnmower 20 meters across the yard If he pushed with a force of 200 N and the angle between the handle and the ground was 50 degrees how much work did he do

F

Displacement = 20 m

F cos

W = (F cos )dW = (200 cos 50) 20W = 2571 J

NOTE If while pushing an object it is moving at a constant velocity

the NET force must be zero

Sohellip Your applied force must be exactly equal to any resistant forces like friction

How much work do you do to carry a 30 kg cat from one side of the room to the other if the room is 10 meters long

ZERO because your Force is vertical but the displacement is horizontal

bull Energy and Work have no direction associated with them and are therefore scalar quantities not vectors

Power

The rate at which work is done

1 Power = Work divide time

Unit for power = J divide s

= Watt W

What is a Watt in ldquofundamental unitsrdquo

Since work is also the energy transferred or transformed ldquopowerrdquo is the rate at which energy is transferred or transformed

2 P = Energy divide time

This energy could be in ANY form heat light potential chemical nuclear

Since NET work = K

3 P = K divide t

And yet another approach

P = W divide t = (Fd) divide t = F middot (d divide t)

P = F middot v

Kinetic Energy

the energy of motion

K = frac12 mv2

Kinetic Energy

the energy of motion

K = frac12 mv2

Where does K = frac12 mv2 come from

Did your amazing teacher just arbitrarily make that equation up

Hmmmhellip

The ldquoWork- Kinetic Energy TheoremrdquoWorknet = K

Fnet bulld = K = frac12 mvf2 - frac12 mvo

2

You are supposed to be able touse Fnet = ma kinematics equations and Wnet = Fnet bulld to

derive the work-kinetic energy theoremhellip

Sohellip do it Then yoursquoll see where the equation we call ldquokinetic energyrdquo comes from

(Hint start with Fnet = ma and use the kinematics equation that doesnrsquot involve time)

Potential EnergyStored energy

It is called potential energy because it has the potential to do work

bull Example 1 Spring potential energy in the stretched string of a bow or spring or rubber band SPE = frac12 kx2

bull Example 2 Chemical potential energy in fuels- gasoline propane batteries food

bull Example 3 Gravitational potential energy- stored in an object due to its position from a chosen reference point

Gravitational potential energy

GPE = weight x height

GPE = mgh

bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative

bull The GPE only depends on the weight and the height not on the path that it took to get to that height

Work and EnergyOften some force must do work

to give an object potential or kinetic energy

You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy

Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy

Work = Force x distance = change in energy

Example

How much more distance is required to stop if a car is going twice as fast

Fd = frac12 mv2

The work done by the brakes = the change in the kinetic energy

With TWICE the speed the car has

FOUR times the kinetic energy

Therefore it takes FOUR times the stopping distance

The Work-Kinetic Energy Theorem

NET Work done = Kinetic Energy

Wnet = frac12 mv2f ndash frac12 mv2

o

Varying Forces

The rule ishellip ldquoIf the Force varies you must integraterdquo

If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done

If the force is a function of velocity you must integrate to find the power output

Examples

If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5

If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms

Examples of Integration

An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force

W F x dx x dx x 33

43 4

Examples of Integration

At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)

v t a t dt tdt t C t( ) ( )

1515

20 75 32 2

x t v t dt t dt t t C t t( ) ( ) ( )

0 75 30 75

23 0 375 3 02 2

Examples of IntegrationA particle of mass m moves along the y-axis as

y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force

P = Fmiddotv

We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )

P F v m at b at bty y ( )( )12 2 4 2 22 3

Examples of Definite Integration

An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s

x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2

62

2

63 2

2

6 3 2 3 22 8 62

3

8

26

2

36

8

26 6 6

2

32

8

22 6 2 45 33

Examples of Definite Integration

To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x

What work is done to stretch the spring 2 meters beyond its equilibrium position

W F ds x x ds x x Jmm

m ( ) ( ) ( ) 4

4

3

1

2

4

32

1

22

4

30

1

20 8 6672

0

23

0

22

0

2 3 2 2

Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under

the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m

P = Fmiddotv but what is the velocity at x = 3m

Hmmmhellip

W F x dx and W K so ( )

F x dx K mv mvx

x

f o

o

f

( ) 12

12

2 2 ( )6 4 3 12

2

0

3

32 x x dx K mv x

( ) ( )6 4 3 64

2

3

36 3

4

23

3

33 0 9 1

22

0

3

2 3

0

3 2 33

2 x x dx x x x J mv x

v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5

  • Slide 1
  • Work Energy and Power AP style
  • Energy
  • Slide 4
  • Slide 5
  • Work = Force x displacement
  • Slide 7
  • Slide 8
  • Slide 9
  • Slide 10
  • Slide 11
  • Example
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Power
  • Slide 18
  • Slide 19
  • Slide 20
  • Slide 21
  • The ldquoWork- Kinetic Energy Theoremrdquo
  • Potential Energy
  • Slide 24
  • Gravitational potential energy
  • Slide 26
  • Work and Energy
  • Slide 28
  • The Work-Kinetic Energy Theorem
  • Varying Forces
  • Slide 31
  • Examples of Integration
  • Slide 33
  • Slide 34
  • Examples of Definite Integration
  • Slide 36
  • Slide 37
Page 10: Work, Energy and Power AP style Energy Energy: the currency of the universe. Everything has to be “paid for” with energy. Energy can’t be created or.

bull Sohellipwhile climbing stairs or walking up an incline only the vertical component of the displacement is used to calculate the work done in moving the object from the bottom to the top Horizontal component of d

Ver

tical

com

pone

nt o

f d

Yo

ur

Fo

rce

Example

How much work is done to carry a 5 kg cat to the top of a ramp that is 7 meters long and 3 meters tall

W = Force x displacementForce = weight of the catd = height NOT lengthW = mg x hW = 5 x 10 x 3 W = 150 J

7 m

3 m

ExampleA boy pushes a

lawnmower 20 meters across the yard If he pushed with a force of 200 N and the angle between the handle and the ground was 50 degrees how much work did he do

F

Displacement = 20 m

F cos

W = (F cos )dW = (200 cos 50) 20W = 2571 J

NOTE If while pushing an object it is moving at a constant velocity

the NET force must be zero

Sohellip Your applied force must be exactly equal to any resistant forces like friction

How much work do you do to carry a 30 kg cat from one side of the room to the other if the room is 10 meters long

ZERO because your Force is vertical but the displacement is horizontal

bull Energy and Work have no direction associated with them and are therefore scalar quantities not vectors

Power

The rate at which work is done

1 Power = Work divide time

Unit for power = J divide s

= Watt W

What is a Watt in ldquofundamental unitsrdquo

Since work is also the energy transferred or transformed ldquopowerrdquo is the rate at which energy is transferred or transformed

2 P = Energy divide time

This energy could be in ANY form heat light potential chemical nuclear

Since NET work = K

3 P = K divide t

And yet another approach

P = W divide t = (Fd) divide t = F middot (d divide t)

P = F middot v

Kinetic Energy

the energy of motion

K = frac12 mv2

Kinetic Energy

the energy of motion

K = frac12 mv2

Where does K = frac12 mv2 come from

Did your amazing teacher just arbitrarily make that equation up

Hmmmhellip

The ldquoWork- Kinetic Energy TheoremrdquoWorknet = K

Fnet bulld = K = frac12 mvf2 - frac12 mvo

2

You are supposed to be able touse Fnet = ma kinematics equations and Wnet = Fnet bulld to

derive the work-kinetic energy theoremhellip

Sohellip do it Then yoursquoll see where the equation we call ldquokinetic energyrdquo comes from

(Hint start with Fnet = ma and use the kinematics equation that doesnrsquot involve time)

Potential EnergyStored energy

It is called potential energy because it has the potential to do work

bull Example 1 Spring potential energy in the stretched string of a bow or spring or rubber band SPE = frac12 kx2

bull Example 2 Chemical potential energy in fuels- gasoline propane batteries food

bull Example 3 Gravitational potential energy- stored in an object due to its position from a chosen reference point

Gravitational potential energy

GPE = weight x height

GPE = mgh

bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative

bull The GPE only depends on the weight and the height not on the path that it took to get to that height

Work and EnergyOften some force must do work

to give an object potential or kinetic energy

You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy

Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy

Work = Force x distance = change in energy

Example

How much more distance is required to stop if a car is going twice as fast

Fd = frac12 mv2

The work done by the brakes = the change in the kinetic energy

With TWICE the speed the car has

FOUR times the kinetic energy

Therefore it takes FOUR times the stopping distance

The Work-Kinetic Energy Theorem

NET Work done = Kinetic Energy

Wnet = frac12 mv2f ndash frac12 mv2

o

Varying Forces

The rule ishellip ldquoIf the Force varies you must integraterdquo

If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done

If the force is a function of velocity you must integrate to find the power output

Examples

If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5

If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms

Examples of Integration

An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force

W F x dx x dx x 33

43 4

Examples of Integration

At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)

v t a t dt tdt t C t( ) ( )

1515

20 75 32 2

x t v t dt t dt t t C t t( ) ( ) ( )

0 75 30 75

23 0 375 3 02 2

Examples of IntegrationA particle of mass m moves along the y-axis as

y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force

P = Fmiddotv

We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )

P F v m at b at bty y ( )( )12 2 4 2 22 3

Examples of Definite Integration

An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s

x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2

62

2

63 2

2

6 3 2 3 22 8 62

3

8

26

2

36

8

26 6 6

2

32

8

22 6 2 45 33

Examples of Definite Integration

To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x

What work is done to stretch the spring 2 meters beyond its equilibrium position

W F ds x x ds x x Jmm

m ( ) ( ) ( ) 4

4

3

1

2

4

32

1

22

4

30

1

20 8 6672

0

23

0

22

0

2 3 2 2

Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under

the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m

P = Fmiddotv but what is the velocity at x = 3m

Hmmmhellip

W F x dx and W K so ( )

F x dx K mv mvx

x

f o

o

f

( ) 12

12

2 2 ( )6 4 3 12

2

0

3

32 x x dx K mv x

( ) ( )6 4 3 64

2

3

36 3

4

23

3

33 0 9 1

22

0

3

2 3

0

3 2 33

2 x x dx x x x J mv x

v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5

  • Slide 1
  • Work Energy and Power AP style
  • Energy
  • Slide 4
  • Slide 5
  • Work = Force x displacement
  • Slide 7
  • Slide 8
  • Slide 9
  • Slide 10
  • Slide 11
  • Example
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Power
  • Slide 18
  • Slide 19
  • Slide 20
  • Slide 21
  • The ldquoWork- Kinetic Energy Theoremrdquo
  • Potential Energy
  • Slide 24
  • Gravitational potential energy
  • Slide 26
  • Work and Energy
  • Slide 28
  • The Work-Kinetic Energy Theorem
  • Varying Forces
  • Slide 31
  • Examples of Integration
  • Slide 33
  • Slide 34
  • Examples of Definite Integration
  • Slide 36
  • Slide 37
Page 11: Work, Energy and Power AP style Energy Energy: the currency of the universe. Everything has to be “paid for” with energy. Energy can’t be created or.

Example

How much work is done to carry a 5 kg cat to the top of a ramp that is 7 meters long and 3 meters tall

W = Force x displacementForce = weight of the catd = height NOT lengthW = mg x hW = 5 x 10 x 3 W = 150 J

7 m

3 m

ExampleA boy pushes a

lawnmower 20 meters across the yard If he pushed with a force of 200 N and the angle between the handle and the ground was 50 degrees how much work did he do

F

Displacement = 20 m

F cos

W = (F cos )dW = (200 cos 50) 20W = 2571 J

NOTE If while pushing an object it is moving at a constant velocity

the NET force must be zero

Sohellip Your applied force must be exactly equal to any resistant forces like friction

How much work do you do to carry a 30 kg cat from one side of the room to the other if the room is 10 meters long

ZERO because your Force is vertical but the displacement is horizontal

bull Energy and Work have no direction associated with them and are therefore scalar quantities not vectors

Power

The rate at which work is done

1 Power = Work divide time

Unit for power = J divide s

= Watt W

What is a Watt in ldquofundamental unitsrdquo

Since work is also the energy transferred or transformed ldquopowerrdquo is the rate at which energy is transferred or transformed

2 P = Energy divide time

This energy could be in ANY form heat light potential chemical nuclear

Since NET work = K

3 P = K divide t

And yet another approach

P = W divide t = (Fd) divide t = F middot (d divide t)

P = F middot v

Kinetic Energy

the energy of motion

K = frac12 mv2

Kinetic Energy

the energy of motion

K = frac12 mv2

Where does K = frac12 mv2 come from

Did your amazing teacher just arbitrarily make that equation up

Hmmmhellip

The ldquoWork- Kinetic Energy TheoremrdquoWorknet = K

Fnet bulld = K = frac12 mvf2 - frac12 mvo

2

You are supposed to be able touse Fnet = ma kinematics equations and Wnet = Fnet bulld to

derive the work-kinetic energy theoremhellip

Sohellip do it Then yoursquoll see where the equation we call ldquokinetic energyrdquo comes from

(Hint start with Fnet = ma and use the kinematics equation that doesnrsquot involve time)

Potential EnergyStored energy

It is called potential energy because it has the potential to do work

bull Example 1 Spring potential energy in the stretched string of a bow or spring or rubber band SPE = frac12 kx2

bull Example 2 Chemical potential energy in fuels- gasoline propane batteries food

bull Example 3 Gravitational potential energy- stored in an object due to its position from a chosen reference point

Gravitational potential energy

GPE = weight x height

GPE = mgh

bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative

bull The GPE only depends on the weight and the height not on the path that it took to get to that height

Work and EnergyOften some force must do work

to give an object potential or kinetic energy

You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy

Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy

Work = Force x distance = change in energy

Example

How much more distance is required to stop if a car is going twice as fast

Fd = frac12 mv2

The work done by the brakes = the change in the kinetic energy

With TWICE the speed the car has

FOUR times the kinetic energy

Therefore it takes FOUR times the stopping distance

The Work-Kinetic Energy Theorem

NET Work done = Kinetic Energy

Wnet = frac12 mv2f ndash frac12 mv2

o

Varying Forces

The rule ishellip ldquoIf the Force varies you must integraterdquo

If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done

If the force is a function of velocity you must integrate to find the power output

Examples

If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5

If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms

Examples of Integration

An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force

W F x dx x dx x 33

43 4

Examples of Integration

At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)

v t a t dt tdt t C t( ) ( )

1515

20 75 32 2

x t v t dt t dt t t C t t( ) ( ) ( )

0 75 30 75

23 0 375 3 02 2

Examples of IntegrationA particle of mass m moves along the y-axis as

y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force

P = Fmiddotv

We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )

P F v m at b at bty y ( )( )12 2 4 2 22 3

Examples of Definite Integration

An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s

x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2

62

2

63 2

2

6 3 2 3 22 8 62

3

8

26

2

36

8

26 6 6

2

32

8

22 6 2 45 33

Examples of Definite Integration

To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x

What work is done to stretch the spring 2 meters beyond its equilibrium position

W F ds x x ds x x Jmm

m ( ) ( ) ( ) 4

4

3

1

2

4

32

1

22

4

30

1

20 8 6672

0

23

0

22

0

2 3 2 2

Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under

the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m

P = Fmiddotv but what is the velocity at x = 3m

Hmmmhellip

W F x dx and W K so ( )

F x dx K mv mvx

x

f o

o

f

( ) 12

12

2 2 ( )6 4 3 12

2

0

3

32 x x dx K mv x

( ) ( )6 4 3 64

2

3

36 3

4

23

3

33 0 9 1

22

0

3

2 3

0

3 2 33

2 x x dx x x x J mv x

v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5

  • Slide 1
  • Work Energy and Power AP style
  • Energy
  • Slide 4
  • Slide 5
  • Work = Force x displacement
  • Slide 7
  • Slide 8
  • Slide 9
  • Slide 10
  • Slide 11
  • Example
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Power
  • Slide 18
  • Slide 19
  • Slide 20
  • Slide 21
  • The ldquoWork- Kinetic Energy Theoremrdquo
  • Potential Energy
  • Slide 24
  • Gravitational potential energy
  • Slide 26
  • Work and Energy
  • Slide 28
  • The Work-Kinetic Energy Theorem
  • Varying Forces
  • Slide 31
  • Examples of Integration
  • Slide 33
  • Slide 34
  • Examples of Definite Integration
  • Slide 36
  • Slide 37
Page 12: Work, Energy and Power AP style Energy Energy: the currency of the universe. Everything has to be “paid for” with energy. Energy can’t be created or.

ExampleA boy pushes a

lawnmower 20 meters across the yard If he pushed with a force of 200 N and the angle between the handle and the ground was 50 degrees how much work did he do

F

Displacement = 20 m

F cos

W = (F cos )dW = (200 cos 50) 20W = 2571 J

NOTE If while pushing an object it is moving at a constant velocity

the NET force must be zero

Sohellip Your applied force must be exactly equal to any resistant forces like friction

How much work do you do to carry a 30 kg cat from one side of the room to the other if the room is 10 meters long

ZERO because your Force is vertical but the displacement is horizontal

bull Energy and Work have no direction associated with them and are therefore scalar quantities not vectors

Power

The rate at which work is done

1 Power = Work divide time

Unit for power = J divide s

= Watt W

What is a Watt in ldquofundamental unitsrdquo

Since work is also the energy transferred or transformed ldquopowerrdquo is the rate at which energy is transferred or transformed

2 P = Energy divide time

This energy could be in ANY form heat light potential chemical nuclear

Since NET work = K

3 P = K divide t

And yet another approach

P = W divide t = (Fd) divide t = F middot (d divide t)

P = F middot v

Kinetic Energy

the energy of motion

K = frac12 mv2

Kinetic Energy

the energy of motion

K = frac12 mv2

Where does K = frac12 mv2 come from

Did your amazing teacher just arbitrarily make that equation up

Hmmmhellip

The ldquoWork- Kinetic Energy TheoremrdquoWorknet = K

Fnet bulld = K = frac12 mvf2 - frac12 mvo

2

You are supposed to be able touse Fnet = ma kinematics equations and Wnet = Fnet bulld to

derive the work-kinetic energy theoremhellip

Sohellip do it Then yoursquoll see where the equation we call ldquokinetic energyrdquo comes from

(Hint start with Fnet = ma and use the kinematics equation that doesnrsquot involve time)

Potential EnergyStored energy

It is called potential energy because it has the potential to do work

bull Example 1 Spring potential energy in the stretched string of a bow or spring or rubber band SPE = frac12 kx2

bull Example 2 Chemical potential energy in fuels- gasoline propane batteries food

bull Example 3 Gravitational potential energy- stored in an object due to its position from a chosen reference point

Gravitational potential energy

GPE = weight x height

GPE = mgh

bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative

bull The GPE only depends on the weight and the height not on the path that it took to get to that height

Work and EnergyOften some force must do work

to give an object potential or kinetic energy

You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy

Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy

Work = Force x distance = change in energy

Example

How much more distance is required to stop if a car is going twice as fast

Fd = frac12 mv2

The work done by the brakes = the change in the kinetic energy

With TWICE the speed the car has

FOUR times the kinetic energy

Therefore it takes FOUR times the stopping distance

The Work-Kinetic Energy Theorem

NET Work done = Kinetic Energy

Wnet = frac12 mv2f ndash frac12 mv2

o

Varying Forces

The rule ishellip ldquoIf the Force varies you must integraterdquo

If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done

If the force is a function of velocity you must integrate to find the power output

Examples

If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5

If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms

Examples of Integration

An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force

W F x dx x dx x 33

43 4

Examples of Integration

At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)

v t a t dt tdt t C t( ) ( )

1515

20 75 32 2

x t v t dt t dt t t C t t( ) ( ) ( )

0 75 30 75

23 0 375 3 02 2

Examples of IntegrationA particle of mass m moves along the y-axis as

y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force

P = Fmiddotv

We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )

P F v m at b at bty y ( )( )12 2 4 2 22 3

Examples of Definite Integration

An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s

x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2

62

2

63 2

2

6 3 2 3 22 8 62

3

8

26

2

36

8

26 6 6

2

32

8

22 6 2 45 33

Examples of Definite Integration

To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x

What work is done to stretch the spring 2 meters beyond its equilibrium position

W F ds x x ds x x Jmm

m ( ) ( ) ( ) 4

4

3

1

2

4

32

1

22

4

30

1

20 8 6672

0

23

0

22

0

2 3 2 2

Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under

the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m

P = Fmiddotv but what is the velocity at x = 3m

Hmmmhellip

W F x dx and W K so ( )

F x dx K mv mvx

x

f o

o

f

( ) 12

12

2 2 ( )6 4 3 12

2

0

3

32 x x dx K mv x

( ) ( )6 4 3 64

2

3

36 3

4

23

3

33 0 9 1

22

0

3

2 3

0

3 2 33

2 x x dx x x x J mv x

v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5

  • Slide 1
  • Work Energy and Power AP style
  • Energy
  • Slide 4
  • Slide 5
  • Work = Force x displacement
  • Slide 7
  • Slide 8
  • Slide 9
  • Slide 10
  • Slide 11
  • Example
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Power
  • Slide 18
  • Slide 19
  • Slide 20
  • Slide 21
  • The ldquoWork- Kinetic Energy Theoremrdquo
  • Potential Energy
  • Slide 24
  • Gravitational potential energy
  • Slide 26
  • Work and Energy
  • Slide 28
  • The Work-Kinetic Energy Theorem
  • Varying Forces
  • Slide 31
  • Examples of Integration
  • Slide 33
  • Slide 34
  • Examples of Definite Integration
  • Slide 36
  • Slide 37
Page 13: Work, Energy and Power AP style Energy Energy: the currency of the universe. Everything has to be “paid for” with energy. Energy can’t be created or.

NOTE If while pushing an object it is moving at a constant velocity

the NET force must be zero

Sohellip Your applied force must be exactly equal to any resistant forces like friction

How much work do you do to carry a 30 kg cat from one side of the room to the other if the room is 10 meters long

ZERO because your Force is vertical but the displacement is horizontal

bull Energy and Work have no direction associated with them and are therefore scalar quantities not vectors

Power

The rate at which work is done

1 Power = Work divide time

Unit for power = J divide s

= Watt W

What is a Watt in ldquofundamental unitsrdquo

Since work is also the energy transferred or transformed ldquopowerrdquo is the rate at which energy is transferred or transformed

2 P = Energy divide time

This energy could be in ANY form heat light potential chemical nuclear

Since NET work = K

3 P = K divide t

And yet another approach

P = W divide t = (Fd) divide t = F middot (d divide t)

P = F middot v

Kinetic Energy

the energy of motion

K = frac12 mv2

Kinetic Energy

the energy of motion

K = frac12 mv2

Where does K = frac12 mv2 come from

Did your amazing teacher just arbitrarily make that equation up

Hmmmhellip

The ldquoWork- Kinetic Energy TheoremrdquoWorknet = K

Fnet bulld = K = frac12 mvf2 - frac12 mvo

2

You are supposed to be able touse Fnet = ma kinematics equations and Wnet = Fnet bulld to

derive the work-kinetic energy theoremhellip

Sohellip do it Then yoursquoll see where the equation we call ldquokinetic energyrdquo comes from

(Hint start with Fnet = ma and use the kinematics equation that doesnrsquot involve time)

Potential EnergyStored energy

It is called potential energy because it has the potential to do work

bull Example 1 Spring potential energy in the stretched string of a bow or spring or rubber band SPE = frac12 kx2

bull Example 2 Chemical potential energy in fuels- gasoline propane batteries food

bull Example 3 Gravitational potential energy- stored in an object due to its position from a chosen reference point

Gravitational potential energy

GPE = weight x height

GPE = mgh

bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative

bull The GPE only depends on the weight and the height not on the path that it took to get to that height

Work and EnergyOften some force must do work

to give an object potential or kinetic energy

You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy

Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy

Work = Force x distance = change in energy

Example

How much more distance is required to stop if a car is going twice as fast

Fd = frac12 mv2

The work done by the brakes = the change in the kinetic energy

With TWICE the speed the car has

FOUR times the kinetic energy

Therefore it takes FOUR times the stopping distance

The Work-Kinetic Energy Theorem

NET Work done = Kinetic Energy

Wnet = frac12 mv2f ndash frac12 mv2

o

Varying Forces

The rule ishellip ldquoIf the Force varies you must integraterdquo

If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done

If the force is a function of velocity you must integrate to find the power output

Examples

If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5

If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms

Examples of Integration

An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force

W F x dx x dx x 33

43 4

Examples of Integration

At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)

v t a t dt tdt t C t( ) ( )

1515

20 75 32 2

x t v t dt t dt t t C t t( ) ( ) ( )

0 75 30 75

23 0 375 3 02 2

Examples of IntegrationA particle of mass m moves along the y-axis as

y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force

P = Fmiddotv

We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )

P F v m at b at bty y ( )( )12 2 4 2 22 3

Examples of Definite Integration

An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s

x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2

62

2

63 2

2

6 3 2 3 22 8 62

3

8

26

2

36

8

26 6 6

2

32

8

22 6 2 45 33

Examples of Definite Integration

To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x

What work is done to stretch the spring 2 meters beyond its equilibrium position

W F ds x x ds x x Jmm

m ( ) ( ) ( ) 4

4

3

1

2

4

32

1

22

4

30

1

20 8 6672

0

23

0

22

0

2 3 2 2

Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under

the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m

P = Fmiddotv but what is the velocity at x = 3m

Hmmmhellip

W F x dx and W K so ( )

F x dx K mv mvx

x

f o

o

f

( ) 12

12

2 2 ( )6 4 3 12

2

0

3

32 x x dx K mv x

( ) ( )6 4 3 64

2

3

36 3

4

23

3

33 0 9 1

22

0

3

2 3

0

3 2 33

2 x x dx x x x J mv x

v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5

  • Slide 1
  • Work Energy and Power AP style
  • Energy
  • Slide 4
  • Slide 5
  • Work = Force x displacement
  • Slide 7
  • Slide 8
  • Slide 9
  • Slide 10
  • Slide 11
  • Example
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Power
  • Slide 18
  • Slide 19
  • Slide 20
  • Slide 21
  • The ldquoWork- Kinetic Energy Theoremrdquo
  • Potential Energy
  • Slide 24
  • Gravitational potential energy
  • Slide 26
  • Work and Energy
  • Slide 28
  • The Work-Kinetic Energy Theorem
  • Varying Forces
  • Slide 31
  • Examples of Integration
  • Slide 33
  • Slide 34
  • Examples of Definite Integration
  • Slide 36
  • Slide 37
Page 14: Work, Energy and Power AP style Energy Energy: the currency of the universe. Everything has to be “paid for” with energy. Energy can’t be created or.

How much work do you do to carry a 30 kg cat from one side of the room to the other if the room is 10 meters long

ZERO because your Force is vertical but the displacement is horizontal

bull Energy and Work have no direction associated with them and are therefore scalar quantities not vectors

Power

The rate at which work is done

1 Power = Work divide time

Unit for power = J divide s

= Watt W

What is a Watt in ldquofundamental unitsrdquo

Since work is also the energy transferred or transformed ldquopowerrdquo is the rate at which energy is transferred or transformed

2 P = Energy divide time

This energy could be in ANY form heat light potential chemical nuclear

Since NET work = K

3 P = K divide t

And yet another approach

P = W divide t = (Fd) divide t = F middot (d divide t)

P = F middot v

Kinetic Energy

the energy of motion

K = frac12 mv2

Kinetic Energy

the energy of motion

K = frac12 mv2

Where does K = frac12 mv2 come from

Did your amazing teacher just arbitrarily make that equation up

Hmmmhellip

The ldquoWork- Kinetic Energy TheoremrdquoWorknet = K

Fnet bulld = K = frac12 mvf2 - frac12 mvo

2

You are supposed to be able touse Fnet = ma kinematics equations and Wnet = Fnet bulld to

derive the work-kinetic energy theoremhellip

Sohellip do it Then yoursquoll see where the equation we call ldquokinetic energyrdquo comes from

(Hint start with Fnet = ma and use the kinematics equation that doesnrsquot involve time)

Potential EnergyStored energy

It is called potential energy because it has the potential to do work

bull Example 1 Spring potential energy in the stretched string of a bow or spring or rubber band SPE = frac12 kx2

bull Example 2 Chemical potential energy in fuels- gasoline propane batteries food

bull Example 3 Gravitational potential energy- stored in an object due to its position from a chosen reference point

Gravitational potential energy

GPE = weight x height

GPE = mgh

bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative

bull The GPE only depends on the weight and the height not on the path that it took to get to that height

Work and EnergyOften some force must do work

to give an object potential or kinetic energy

You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy

Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy

Work = Force x distance = change in energy

Example

How much more distance is required to stop if a car is going twice as fast

Fd = frac12 mv2

The work done by the brakes = the change in the kinetic energy

With TWICE the speed the car has

FOUR times the kinetic energy

Therefore it takes FOUR times the stopping distance

The Work-Kinetic Energy Theorem

NET Work done = Kinetic Energy

Wnet = frac12 mv2f ndash frac12 mv2

o

Varying Forces

The rule ishellip ldquoIf the Force varies you must integraterdquo

If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done

If the force is a function of velocity you must integrate to find the power output

Examples

If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5

If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms

Examples of Integration

An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force

W F x dx x dx x 33

43 4

Examples of Integration

At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)

v t a t dt tdt t C t( ) ( )

1515

20 75 32 2

x t v t dt t dt t t C t t( ) ( ) ( )

0 75 30 75

23 0 375 3 02 2

Examples of IntegrationA particle of mass m moves along the y-axis as

y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force

P = Fmiddotv

We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )

P F v m at b at bty y ( )( )12 2 4 2 22 3

Examples of Definite Integration

An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s

x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2

62

2

63 2

2

6 3 2 3 22 8 62

3

8

26

2

36

8

26 6 6

2

32

8

22 6 2 45 33

Examples of Definite Integration

To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x

What work is done to stretch the spring 2 meters beyond its equilibrium position

W F ds x x ds x x Jmm

m ( ) ( ) ( ) 4

4

3

1

2

4

32

1

22

4

30

1

20 8 6672

0

23

0

22

0

2 3 2 2

Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under

the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m

P = Fmiddotv but what is the velocity at x = 3m

Hmmmhellip

W F x dx and W K so ( )

F x dx K mv mvx

x

f o

o

f

( ) 12

12

2 2 ( )6 4 3 12

2

0

3

32 x x dx K mv x

( ) ( )6 4 3 64

2

3

36 3

4

23

3

33 0 9 1

22

0

3

2 3

0

3 2 33

2 x x dx x x x J mv x

v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5

  • Slide 1
  • Work Energy and Power AP style
  • Energy
  • Slide 4
  • Slide 5
  • Work = Force x displacement
  • Slide 7
  • Slide 8
  • Slide 9
  • Slide 10
  • Slide 11
  • Example
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Power
  • Slide 18
  • Slide 19
  • Slide 20
  • Slide 21
  • The ldquoWork- Kinetic Energy Theoremrdquo
  • Potential Energy
  • Slide 24
  • Gravitational potential energy
  • Slide 26
  • Work and Energy
  • Slide 28
  • The Work-Kinetic Energy Theorem
  • Varying Forces
  • Slide 31
  • Examples of Integration
  • Slide 33
  • Slide 34
  • Examples of Definite Integration
  • Slide 36
  • Slide 37
Page 15: Work, Energy and Power AP style Energy Energy: the currency of the universe. Everything has to be “paid for” with energy. Energy can’t be created or.

bull Energy and Work have no direction associated with them and are therefore scalar quantities not vectors

Power

The rate at which work is done

1 Power = Work divide time

Unit for power = J divide s

= Watt W

What is a Watt in ldquofundamental unitsrdquo

Since work is also the energy transferred or transformed ldquopowerrdquo is the rate at which energy is transferred or transformed

2 P = Energy divide time

This energy could be in ANY form heat light potential chemical nuclear

Since NET work = K

3 P = K divide t

And yet another approach

P = W divide t = (Fd) divide t = F middot (d divide t)

P = F middot v

Kinetic Energy

the energy of motion

K = frac12 mv2

Kinetic Energy

the energy of motion

K = frac12 mv2

Where does K = frac12 mv2 come from

Did your amazing teacher just arbitrarily make that equation up

Hmmmhellip

The ldquoWork- Kinetic Energy TheoremrdquoWorknet = K

Fnet bulld = K = frac12 mvf2 - frac12 mvo

2

You are supposed to be able touse Fnet = ma kinematics equations and Wnet = Fnet bulld to

derive the work-kinetic energy theoremhellip

Sohellip do it Then yoursquoll see where the equation we call ldquokinetic energyrdquo comes from

(Hint start with Fnet = ma and use the kinematics equation that doesnrsquot involve time)

Potential EnergyStored energy

It is called potential energy because it has the potential to do work

bull Example 1 Spring potential energy in the stretched string of a bow or spring or rubber band SPE = frac12 kx2

bull Example 2 Chemical potential energy in fuels- gasoline propane batteries food

bull Example 3 Gravitational potential energy- stored in an object due to its position from a chosen reference point

Gravitational potential energy

GPE = weight x height

GPE = mgh

bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative

bull The GPE only depends on the weight and the height not on the path that it took to get to that height

Work and EnergyOften some force must do work

to give an object potential or kinetic energy

You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy

Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy

Work = Force x distance = change in energy

Example

How much more distance is required to stop if a car is going twice as fast

Fd = frac12 mv2

The work done by the brakes = the change in the kinetic energy

With TWICE the speed the car has

FOUR times the kinetic energy

Therefore it takes FOUR times the stopping distance

The Work-Kinetic Energy Theorem

NET Work done = Kinetic Energy

Wnet = frac12 mv2f ndash frac12 mv2

o

Varying Forces

The rule ishellip ldquoIf the Force varies you must integraterdquo

If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done

If the force is a function of velocity you must integrate to find the power output

Examples

If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5

If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms

Examples of Integration

An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force

W F x dx x dx x 33

43 4

Examples of Integration

At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)

v t a t dt tdt t C t( ) ( )

1515

20 75 32 2

x t v t dt t dt t t C t t( ) ( ) ( )

0 75 30 75

23 0 375 3 02 2

Examples of IntegrationA particle of mass m moves along the y-axis as

y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force

P = Fmiddotv

We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )

P F v m at b at bty y ( )( )12 2 4 2 22 3

Examples of Definite Integration

An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s

x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2

62

2

63 2

2

6 3 2 3 22 8 62

3

8

26

2

36

8

26 6 6

2

32

8

22 6 2 45 33

Examples of Definite Integration

To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x

What work is done to stretch the spring 2 meters beyond its equilibrium position

W F ds x x ds x x Jmm

m ( ) ( ) ( ) 4

4

3

1

2

4

32

1

22

4

30

1

20 8 6672

0

23

0

22

0

2 3 2 2

Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under

the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m

P = Fmiddotv but what is the velocity at x = 3m

Hmmmhellip

W F x dx and W K so ( )

F x dx K mv mvx

x

f o

o

f

( ) 12

12

2 2 ( )6 4 3 12

2

0

3

32 x x dx K mv x

( ) ( )6 4 3 64

2

3

36 3

4

23

3

33 0 9 1

22

0

3

2 3

0

3 2 33

2 x x dx x x x J mv x

v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5

  • Slide 1
  • Work Energy and Power AP style
  • Energy
  • Slide 4
  • Slide 5
  • Work = Force x displacement
  • Slide 7
  • Slide 8
  • Slide 9
  • Slide 10
  • Slide 11
  • Example
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Power
  • Slide 18
  • Slide 19
  • Slide 20
  • Slide 21
  • The ldquoWork- Kinetic Energy Theoremrdquo
  • Potential Energy
  • Slide 24
  • Gravitational potential energy
  • Slide 26
  • Work and Energy
  • Slide 28
  • The Work-Kinetic Energy Theorem
  • Varying Forces
  • Slide 31
  • Examples of Integration
  • Slide 33
  • Slide 34
  • Examples of Definite Integration
  • Slide 36
  • Slide 37
Page 16: Work, Energy and Power AP style Energy Energy: the currency of the universe. Everything has to be “paid for” with energy. Energy can’t be created or.

Power

The rate at which work is done

1 Power = Work divide time

Unit for power = J divide s

= Watt W

What is a Watt in ldquofundamental unitsrdquo

Since work is also the energy transferred or transformed ldquopowerrdquo is the rate at which energy is transferred or transformed

2 P = Energy divide time

This energy could be in ANY form heat light potential chemical nuclear

Since NET work = K

3 P = K divide t

And yet another approach

P = W divide t = (Fd) divide t = F middot (d divide t)

P = F middot v

Kinetic Energy

the energy of motion

K = frac12 mv2

Kinetic Energy

the energy of motion

K = frac12 mv2

Where does K = frac12 mv2 come from

Did your amazing teacher just arbitrarily make that equation up

Hmmmhellip

The ldquoWork- Kinetic Energy TheoremrdquoWorknet = K

Fnet bulld = K = frac12 mvf2 - frac12 mvo

2

You are supposed to be able touse Fnet = ma kinematics equations and Wnet = Fnet bulld to

derive the work-kinetic energy theoremhellip

Sohellip do it Then yoursquoll see where the equation we call ldquokinetic energyrdquo comes from

(Hint start with Fnet = ma and use the kinematics equation that doesnrsquot involve time)

Potential EnergyStored energy

It is called potential energy because it has the potential to do work

bull Example 1 Spring potential energy in the stretched string of a bow or spring or rubber band SPE = frac12 kx2

bull Example 2 Chemical potential energy in fuels- gasoline propane batteries food

bull Example 3 Gravitational potential energy- stored in an object due to its position from a chosen reference point

Gravitational potential energy

GPE = weight x height

GPE = mgh

bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative

bull The GPE only depends on the weight and the height not on the path that it took to get to that height

Work and EnergyOften some force must do work

to give an object potential or kinetic energy

You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy

Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy

Work = Force x distance = change in energy

Example

How much more distance is required to stop if a car is going twice as fast

Fd = frac12 mv2

The work done by the brakes = the change in the kinetic energy

With TWICE the speed the car has

FOUR times the kinetic energy

Therefore it takes FOUR times the stopping distance

The Work-Kinetic Energy Theorem

NET Work done = Kinetic Energy

Wnet = frac12 mv2f ndash frac12 mv2

o

Varying Forces

The rule ishellip ldquoIf the Force varies you must integraterdquo

If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done

If the force is a function of velocity you must integrate to find the power output

Examples

If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5

If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms

Examples of Integration

An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force

W F x dx x dx x 33

43 4

Examples of Integration

At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)

v t a t dt tdt t C t( ) ( )

1515

20 75 32 2

x t v t dt t dt t t C t t( ) ( ) ( )

0 75 30 75

23 0 375 3 02 2

Examples of IntegrationA particle of mass m moves along the y-axis as

y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force

P = Fmiddotv

We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )

P F v m at b at bty y ( )( )12 2 4 2 22 3

Examples of Definite Integration

An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s

x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2

62

2

63 2

2

6 3 2 3 22 8 62

3

8

26

2

36

8

26 6 6

2

32

8

22 6 2 45 33

Examples of Definite Integration

To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x

What work is done to stretch the spring 2 meters beyond its equilibrium position

W F ds x x ds x x Jmm

m ( ) ( ) ( ) 4

4

3

1

2

4

32

1

22

4

30

1

20 8 6672

0

23

0

22

0

2 3 2 2

Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under

the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m

P = Fmiddotv but what is the velocity at x = 3m

Hmmmhellip

W F x dx and W K so ( )

F x dx K mv mvx

x

f o

o

f

( ) 12

12

2 2 ( )6 4 3 12

2

0

3

32 x x dx K mv x

( ) ( )6 4 3 64

2

3

36 3

4

23

3

33 0 9 1

22

0

3

2 3

0

3 2 33

2 x x dx x x x J mv x

v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5

  • Slide 1
  • Work Energy and Power AP style
  • Energy
  • Slide 4
  • Slide 5
  • Work = Force x displacement
  • Slide 7
  • Slide 8
  • Slide 9
  • Slide 10
  • Slide 11
  • Example
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Power
  • Slide 18
  • Slide 19
  • Slide 20
  • Slide 21
  • The ldquoWork- Kinetic Energy Theoremrdquo
  • Potential Energy
  • Slide 24
  • Gravitational potential energy
  • Slide 26
  • Work and Energy
  • Slide 28
  • The Work-Kinetic Energy Theorem
  • Varying Forces
  • Slide 31
  • Examples of Integration
  • Slide 33
  • Slide 34
  • Examples of Definite Integration
  • Slide 36
  • Slide 37
Page 17: Work, Energy and Power AP style Energy Energy: the currency of the universe. Everything has to be “paid for” with energy. Energy can’t be created or.

Since work is also the energy transferred or transformed ldquopowerrdquo is the rate at which energy is transferred or transformed

2 P = Energy divide time

This energy could be in ANY form heat light potential chemical nuclear

Since NET work = K

3 P = K divide t

And yet another approach

P = W divide t = (Fd) divide t = F middot (d divide t)

P = F middot v

Kinetic Energy

the energy of motion

K = frac12 mv2

Kinetic Energy

the energy of motion

K = frac12 mv2

Where does K = frac12 mv2 come from

Did your amazing teacher just arbitrarily make that equation up

Hmmmhellip

The ldquoWork- Kinetic Energy TheoremrdquoWorknet = K

Fnet bulld = K = frac12 mvf2 - frac12 mvo

2

You are supposed to be able touse Fnet = ma kinematics equations and Wnet = Fnet bulld to

derive the work-kinetic energy theoremhellip

Sohellip do it Then yoursquoll see where the equation we call ldquokinetic energyrdquo comes from

(Hint start with Fnet = ma and use the kinematics equation that doesnrsquot involve time)

Potential EnergyStored energy

It is called potential energy because it has the potential to do work

bull Example 1 Spring potential energy in the stretched string of a bow or spring or rubber band SPE = frac12 kx2

bull Example 2 Chemical potential energy in fuels- gasoline propane batteries food

bull Example 3 Gravitational potential energy- stored in an object due to its position from a chosen reference point

Gravitational potential energy

GPE = weight x height

GPE = mgh

bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative

bull The GPE only depends on the weight and the height not on the path that it took to get to that height

Work and EnergyOften some force must do work

to give an object potential or kinetic energy

You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy

Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy

Work = Force x distance = change in energy

Example

How much more distance is required to stop if a car is going twice as fast

Fd = frac12 mv2

The work done by the brakes = the change in the kinetic energy

With TWICE the speed the car has

FOUR times the kinetic energy

Therefore it takes FOUR times the stopping distance

The Work-Kinetic Energy Theorem

NET Work done = Kinetic Energy

Wnet = frac12 mv2f ndash frac12 mv2

o

Varying Forces

The rule ishellip ldquoIf the Force varies you must integraterdquo

If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done

If the force is a function of velocity you must integrate to find the power output

Examples

If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5

If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms

Examples of Integration

An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force

W F x dx x dx x 33

43 4

Examples of Integration

At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)

v t a t dt tdt t C t( ) ( )

1515

20 75 32 2

x t v t dt t dt t t C t t( ) ( ) ( )

0 75 30 75

23 0 375 3 02 2

Examples of IntegrationA particle of mass m moves along the y-axis as

y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force

P = Fmiddotv

We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )

P F v m at b at bty y ( )( )12 2 4 2 22 3

Examples of Definite Integration

An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s

x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2

62

2

63 2

2

6 3 2 3 22 8 62

3

8

26

2

36

8

26 6 6

2

32

8

22 6 2 45 33

Examples of Definite Integration

To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x

What work is done to stretch the spring 2 meters beyond its equilibrium position

W F ds x x ds x x Jmm

m ( ) ( ) ( ) 4

4

3

1

2

4

32

1

22

4

30

1

20 8 6672

0

23

0

22

0

2 3 2 2

Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under

the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m

P = Fmiddotv but what is the velocity at x = 3m

Hmmmhellip

W F x dx and W K so ( )

F x dx K mv mvx

x

f o

o

f

( ) 12

12

2 2 ( )6 4 3 12

2

0

3

32 x x dx K mv x

( ) ( )6 4 3 64

2

3

36 3

4

23

3

33 0 9 1

22

0

3

2 3

0

3 2 33

2 x x dx x x x J mv x

v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5

  • Slide 1
  • Work Energy and Power AP style
  • Energy
  • Slide 4
  • Slide 5
  • Work = Force x displacement
  • Slide 7
  • Slide 8
  • Slide 9
  • Slide 10
  • Slide 11
  • Example
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Power
  • Slide 18
  • Slide 19
  • Slide 20
  • Slide 21
  • The ldquoWork- Kinetic Energy Theoremrdquo
  • Potential Energy
  • Slide 24
  • Gravitational potential energy
  • Slide 26
  • Work and Energy
  • Slide 28
  • The Work-Kinetic Energy Theorem
  • Varying Forces
  • Slide 31
  • Examples of Integration
  • Slide 33
  • Slide 34
  • Examples of Definite Integration
  • Slide 36
  • Slide 37
Page 18: Work, Energy and Power AP style Energy Energy: the currency of the universe. Everything has to be “paid for” with energy. Energy can’t be created or.

Kinetic Energy

the energy of motion

K = frac12 mv2

Kinetic Energy

the energy of motion

K = frac12 mv2

Where does K = frac12 mv2 come from

Did your amazing teacher just arbitrarily make that equation up

Hmmmhellip

The ldquoWork- Kinetic Energy TheoremrdquoWorknet = K

Fnet bulld = K = frac12 mvf2 - frac12 mvo

2

You are supposed to be able touse Fnet = ma kinematics equations and Wnet = Fnet bulld to

derive the work-kinetic energy theoremhellip

Sohellip do it Then yoursquoll see where the equation we call ldquokinetic energyrdquo comes from

(Hint start with Fnet = ma and use the kinematics equation that doesnrsquot involve time)

Potential EnergyStored energy

It is called potential energy because it has the potential to do work

bull Example 1 Spring potential energy in the stretched string of a bow or spring or rubber band SPE = frac12 kx2

bull Example 2 Chemical potential energy in fuels- gasoline propane batteries food

bull Example 3 Gravitational potential energy- stored in an object due to its position from a chosen reference point

Gravitational potential energy

GPE = weight x height

GPE = mgh

bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative

bull The GPE only depends on the weight and the height not on the path that it took to get to that height

Work and EnergyOften some force must do work

to give an object potential or kinetic energy

You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy

Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy

Work = Force x distance = change in energy

Example

How much more distance is required to stop if a car is going twice as fast

Fd = frac12 mv2

The work done by the brakes = the change in the kinetic energy

With TWICE the speed the car has

FOUR times the kinetic energy

Therefore it takes FOUR times the stopping distance

The Work-Kinetic Energy Theorem

NET Work done = Kinetic Energy

Wnet = frac12 mv2f ndash frac12 mv2

o

Varying Forces

The rule ishellip ldquoIf the Force varies you must integraterdquo

If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done

If the force is a function of velocity you must integrate to find the power output

Examples

If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5

If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms

Examples of Integration

An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force

W F x dx x dx x 33

43 4

Examples of Integration

At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)

v t a t dt tdt t C t( ) ( )

1515

20 75 32 2

x t v t dt t dt t t C t t( ) ( ) ( )

0 75 30 75

23 0 375 3 02 2

Examples of IntegrationA particle of mass m moves along the y-axis as

y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force

P = Fmiddotv

We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )

P F v m at b at bty y ( )( )12 2 4 2 22 3

Examples of Definite Integration

An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s

x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2

62

2

63 2

2

6 3 2 3 22 8 62

3

8

26

2

36

8

26 6 6

2

32

8

22 6 2 45 33

Examples of Definite Integration

To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x

What work is done to stretch the spring 2 meters beyond its equilibrium position

W F ds x x ds x x Jmm

m ( ) ( ) ( ) 4

4

3

1

2

4

32

1

22

4

30

1

20 8 6672

0

23

0

22

0

2 3 2 2

Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under

the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m

P = Fmiddotv but what is the velocity at x = 3m

Hmmmhellip

W F x dx and W K so ( )

F x dx K mv mvx

x

f o

o

f

( ) 12

12

2 2 ( )6 4 3 12

2

0

3

32 x x dx K mv x

( ) ( )6 4 3 64

2

3

36 3

4

23

3

33 0 9 1

22

0

3

2 3

0

3 2 33

2 x x dx x x x J mv x

v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5

  • Slide 1
  • Work Energy and Power AP style
  • Energy
  • Slide 4
  • Slide 5
  • Work = Force x displacement
  • Slide 7
  • Slide 8
  • Slide 9
  • Slide 10
  • Slide 11
  • Example
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Power
  • Slide 18
  • Slide 19
  • Slide 20
  • Slide 21
  • The ldquoWork- Kinetic Energy Theoremrdquo
  • Potential Energy
  • Slide 24
  • Gravitational potential energy
  • Slide 26
  • Work and Energy
  • Slide 28
  • The Work-Kinetic Energy Theorem
  • Varying Forces
  • Slide 31
  • Examples of Integration
  • Slide 33
  • Slide 34
  • Examples of Definite Integration
  • Slide 36
  • Slide 37
Page 19: Work, Energy and Power AP style Energy Energy: the currency of the universe. Everything has to be “paid for” with energy. Energy can’t be created or.

Where does K = frac12 mv2 come from

Did your amazing teacher just arbitrarily make that equation up

Hmmmhellip

The ldquoWork- Kinetic Energy TheoremrdquoWorknet = K

Fnet bulld = K = frac12 mvf2 - frac12 mvo

2

You are supposed to be able touse Fnet = ma kinematics equations and Wnet = Fnet bulld to

derive the work-kinetic energy theoremhellip

Sohellip do it Then yoursquoll see where the equation we call ldquokinetic energyrdquo comes from

(Hint start with Fnet = ma and use the kinematics equation that doesnrsquot involve time)

Potential EnergyStored energy

It is called potential energy because it has the potential to do work

bull Example 1 Spring potential energy in the stretched string of a bow or spring or rubber band SPE = frac12 kx2

bull Example 2 Chemical potential energy in fuels- gasoline propane batteries food

bull Example 3 Gravitational potential energy- stored in an object due to its position from a chosen reference point

Gravitational potential energy

GPE = weight x height

GPE = mgh

bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative

bull The GPE only depends on the weight and the height not on the path that it took to get to that height

Work and EnergyOften some force must do work

to give an object potential or kinetic energy

You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy

Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy

Work = Force x distance = change in energy

Example

How much more distance is required to stop if a car is going twice as fast

Fd = frac12 mv2

The work done by the brakes = the change in the kinetic energy

With TWICE the speed the car has

FOUR times the kinetic energy

Therefore it takes FOUR times the stopping distance

The Work-Kinetic Energy Theorem

NET Work done = Kinetic Energy

Wnet = frac12 mv2f ndash frac12 mv2

o

Varying Forces

The rule ishellip ldquoIf the Force varies you must integraterdquo

If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done

If the force is a function of velocity you must integrate to find the power output

Examples

If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5

If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms

Examples of Integration

An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force

W F x dx x dx x 33

43 4

Examples of Integration

At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)

v t a t dt tdt t C t( ) ( )

1515

20 75 32 2

x t v t dt t dt t t C t t( ) ( ) ( )

0 75 30 75

23 0 375 3 02 2

Examples of IntegrationA particle of mass m moves along the y-axis as

y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force

P = Fmiddotv

We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )

P F v m at b at bty y ( )( )12 2 4 2 22 3

Examples of Definite Integration

An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s

x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2

62

2

63 2

2

6 3 2 3 22 8 62

3

8

26

2

36

8

26 6 6

2

32

8

22 6 2 45 33

Examples of Definite Integration

To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x

What work is done to stretch the spring 2 meters beyond its equilibrium position

W F ds x x ds x x Jmm

m ( ) ( ) ( ) 4

4

3

1

2

4

32

1

22

4

30

1

20 8 6672

0

23

0

22

0

2 3 2 2

Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under

the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m

P = Fmiddotv but what is the velocity at x = 3m

Hmmmhellip

W F x dx and W K so ( )

F x dx K mv mvx

x

f o

o

f

( ) 12

12

2 2 ( )6 4 3 12

2

0

3

32 x x dx K mv x

( ) ( )6 4 3 64

2

3

36 3

4

23

3

33 0 9 1

22

0

3

2 3

0

3 2 33

2 x x dx x x x J mv x

v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5

  • Slide 1
  • Work Energy and Power AP style
  • Energy
  • Slide 4
  • Slide 5
  • Work = Force x displacement
  • Slide 7
  • Slide 8
  • Slide 9
  • Slide 10
  • Slide 11
  • Example
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Power
  • Slide 18
  • Slide 19
  • Slide 20
  • Slide 21
  • The ldquoWork- Kinetic Energy Theoremrdquo
  • Potential Energy
  • Slide 24
  • Gravitational potential energy
  • Slide 26
  • Work and Energy
  • Slide 28
  • The Work-Kinetic Energy Theorem
  • Varying Forces
  • Slide 31
  • Examples of Integration
  • Slide 33
  • Slide 34
  • Examples of Definite Integration
  • Slide 36
  • Slide 37
Page 20: Work, Energy and Power AP style Energy Energy: the currency of the universe. Everything has to be “paid for” with energy. Energy can’t be created or.

The ldquoWork- Kinetic Energy TheoremrdquoWorknet = K

Fnet bulld = K = frac12 mvf2 - frac12 mvo

2

You are supposed to be able touse Fnet = ma kinematics equations and Wnet = Fnet bulld to

derive the work-kinetic energy theoremhellip

Sohellip do it Then yoursquoll see where the equation we call ldquokinetic energyrdquo comes from

(Hint start with Fnet = ma and use the kinematics equation that doesnrsquot involve time)

Potential EnergyStored energy

It is called potential energy because it has the potential to do work

bull Example 1 Spring potential energy in the stretched string of a bow or spring or rubber band SPE = frac12 kx2

bull Example 2 Chemical potential energy in fuels- gasoline propane batteries food

bull Example 3 Gravitational potential energy- stored in an object due to its position from a chosen reference point

Gravitational potential energy

GPE = weight x height

GPE = mgh

bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative

bull The GPE only depends on the weight and the height not on the path that it took to get to that height

Work and EnergyOften some force must do work

to give an object potential or kinetic energy

You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy

Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy

Work = Force x distance = change in energy

Example

How much more distance is required to stop if a car is going twice as fast

Fd = frac12 mv2

The work done by the brakes = the change in the kinetic energy

With TWICE the speed the car has

FOUR times the kinetic energy

Therefore it takes FOUR times the stopping distance

The Work-Kinetic Energy Theorem

NET Work done = Kinetic Energy

Wnet = frac12 mv2f ndash frac12 mv2

o

Varying Forces

The rule ishellip ldquoIf the Force varies you must integraterdquo

If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done

If the force is a function of velocity you must integrate to find the power output

Examples

If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5

If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms

Examples of Integration

An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force

W F x dx x dx x 33

43 4

Examples of Integration

At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)

v t a t dt tdt t C t( ) ( )

1515

20 75 32 2

x t v t dt t dt t t C t t( ) ( ) ( )

0 75 30 75

23 0 375 3 02 2

Examples of IntegrationA particle of mass m moves along the y-axis as

y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force

P = Fmiddotv

We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )

P F v m at b at bty y ( )( )12 2 4 2 22 3

Examples of Definite Integration

An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s

x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2

62

2

63 2

2

6 3 2 3 22 8 62

3

8

26

2

36

8

26 6 6

2

32

8

22 6 2 45 33

Examples of Definite Integration

To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x

What work is done to stretch the spring 2 meters beyond its equilibrium position

W F ds x x ds x x Jmm

m ( ) ( ) ( ) 4

4

3

1

2

4

32

1

22

4

30

1

20 8 6672

0

23

0

22

0

2 3 2 2

Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under

the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m

P = Fmiddotv but what is the velocity at x = 3m

Hmmmhellip

W F x dx and W K so ( )

F x dx K mv mvx

x

f o

o

f

( ) 12

12

2 2 ( )6 4 3 12

2

0

3

32 x x dx K mv x

( ) ( )6 4 3 64

2

3

36 3

4

23

3

33 0 9 1

22

0

3

2 3

0

3 2 33

2 x x dx x x x J mv x

v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5

  • Slide 1
  • Work Energy and Power AP style
  • Energy
  • Slide 4
  • Slide 5
  • Work = Force x displacement
  • Slide 7
  • Slide 8
  • Slide 9
  • Slide 10
  • Slide 11
  • Example
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Power
  • Slide 18
  • Slide 19
  • Slide 20
  • Slide 21
  • The ldquoWork- Kinetic Energy Theoremrdquo
  • Potential Energy
  • Slide 24
  • Gravitational potential energy
  • Slide 26
  • Work and Energy
  • Slide 28
  • The Work-Kinetic Energy Theorem
  • Varying Forces
  • Slide 31
  • Examples of Integration
  • Slide 33
  • Slide 34
  • Examples of Definite Integration
  • Slide 36
  • Slide 37
Page 21: Work, Energy and Power AP style Energy Energy: the currency of the universe. Everything has to be “paid for” with energy. Energy can’t be created or.

Potential EnergyStored energy

It is called potential energy because it has the potential to do work

bull Example 1 Spring potential energy in the stretched string of a bow or spring or rubber band SPE = frac12 kx2

bull Example 2 Chemical potential energy in fuels- gasoline propane batteries food

bull Example 3 Gravitational potential energy- stored in an object due to its position from a chosen reference point

Gravitational potential energy

GPE = weight x height

GPE = mgh

bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative

bull The GPE only depends on the weight and the height not on the path that it took to get to that height

Work and EnergyOften some force must do work

to give an object potential or kinetic energy

You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy

Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy

Work = Force x distance = change in energy

Example

How much more distance is required to stop if a car is going twice as fast

Fd = frac12 mv2

The work done by the brakes = the change in the kinetic energy

With TWICE the speed the car has

FOUR times the kinetic energy

Therefore it takes FOUR times the stopping distance

The Work-Kinetic Energy Theorem

NET Work done = Kinetic Energy

Wnet = frac12 mv2f ndash frac12 mv2

o

Varying Forces

The rule ishellip ldquoIf the Force varies you must integraterdquo

If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done

If the force is a function of velocity you must integrate to find the power output

Examples

If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5

If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms

Examples of Integration

An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force

W F x dx x dx x 33

43 4

Examples of Integration

At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)

v t a t dt tdt t C t( ) ( )

1515

20 75 32 2

x t v t dt t dt t t C t t( ) ( ) ( )

0 75 30 75

23 0 375 3 02 2

Examples of IntegrationA particle of mass m moves along the y-axis as

y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force

P = Fmiddotv

We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )

P F v m at b at bty y ( )( )12 2 4 2 22 3

Examples of Definite Integration

An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s

x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2

62

2

63 2

2

6 3 2 3 22 8 62

3

8

26

2

36

8

26 6 6

2

32

8

22 6 2 45 33

Examples of Definite Integration

To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x

What work is done to stretch the spring 2 meters beyond its equilibrium position

W F ds x x ds x x Jmm

m ( ) ( ) ( ) 4

4

3

1

2

4

32

1

22

4

30

1

20 8 6672

0

23

0

22

0

2 3 2 2

Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under

the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m

P = Fmiddotv but what is the velocity at x = 3m

Hmmmhellip

W F x dx and W K so ( )

F x dx K mv mvx

x

f o

o

f

( ) 12

12

2 2 ( )6 4 3 12

2

0

3

32 x x dx K mv x

( ) ( )6 4 3 64

2

3

36 3

4

23

3

33 0 9 1

22

0

3

2 3

0

3 2 33

2 x x dx x x x J mv x

v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5

  • Slide 1
  • Work Energy and Power AP style
  • Energy
  • Slide 4
  • Slide 5
  • Work = Force x displacement
  • Slide 7
  • Slide 8
  • Slide 9
  • Slide 10
  • Slide 11
  • Example
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Power
  • Slide 18
  • Slide 19
  • Slide 20
  • Slide 21
  • The ldquoWork- Kinetic Energy Theoremrdquo
  • Potential Energy
  • Slide 24
  • Gravitational potential energy
  • Slide 26
  • Work and Energy
  • Slide 28
  • The Work-Kinetic Energy Theorem
  • Varying Forces
  • Slide 31
  • Examples of Integration
  • Slide 33
  • Slide 34
  • Examples of Definite Integration
  • Slide 36
  • Slide 37
Page 22: Work, Energy and Power AP style Energy Energy: the currency of the universe. Everything has to be “paid for” with energy. Energy can’t be created or.

bull Example 1 Spring potential energy in the stretched string of a bow or spring or rubber band SPE = frac12 kx2

bull Example 2 Chemical potential energy in fuels- gasoline propane batteries food

bull Example 3 Gravitational potential energy- stored in an object due to its position from a chosen reference point

Gravitational potential energy

GPE = weight x height

GPE = mgh

bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative

bull The GPE only depends on the weight and the height not on the path that it took to get to that height

Work and EnergyOften some force must do work

to give an object potential or kinetic energy

You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy

Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy

Work = Force x distance = change in energy

Example

How much more distance is required to stop if a car is going twice as fast

Fd = frac12 mv2

The work done by the brakes = the change in the kinetic energy

With TWICE the speed the car has

FOUR times the kinetic energy

Therefore it takes FOUR times the stopping distance

The Work-Kinetic Energy Theorem

NET Work done = Kinetic Energy

Wnet = frac12 mv2f ndash frac12 mv2

o

Varying Forces

The rule ishellip ldquoIf the Force varies you must integraterdquo

If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done

If the force is a function of velocity you must integrate to find the power output

Examples

If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5

If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms

Examples of Integration

An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force

W F x dx x dx x 33

43 4

Examples of Integration

At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)

v t a t dt tdt t C t( ) ( )

1515

20 75 32 2

x t v t dt t dt t t C t t( ) ( ) ( )

0 75 30 75

23 0 375 3 02 2

Examples of IntegrationA particle of mass m moves along the y-axis as

y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force

P = Fmiddotv

We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )

P F v m at b at bty y ( )( )12 2 4 2 22 3

Examples of Definite Integration

An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s

x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2

62

2

63 2

2

6 3 2 3 22 8 62

3

8

26

2

36

8

26 6 6

2

32

8

22 6 2 45 33

Examples of Definite Integration

To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x

What work is done to stretch the spring 2 meters beyond its equilibrium position

W F ds x x ds x x Jmm

m ( ) ( ) ( ) 4

4

3

1

2

4

32

1

22

4

30

1

20 8 6672

0

23

0

22

0

2 3 2 2

Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under

the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m

P = Fmiddotv but what is the velocity at x = 3m

Hmmmhellip

W F x dx and W K so ( )

F x dx K mv mvx

x

f o

o

f

( ) 12

12

2 2 ( )6 4 3 12

2

0

3

32 x x dx K mv x

( ) ( )6 4 3 64

2

3

36 3

4

23

3

33 0 9 1

22

0

3

2 3

0

3 2 33

2 x x dx x x x J mv x

v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5

  • Slide 1
  • Work Energy and Power AP style
  • Energy
  • Slide 4
  • Slide 5
  • Work = Force x displacement
  • Slide 7
  • Slide 8
  • Slide 9
  • Slide 10
  • Slide 11
  • Example
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Power
  • Slide 18
  • Slide 19
  • Slide 20
  • Slide 21
  • The ldquoWork- Kinetic Energy Theoremrdquo
  • Potential Energy
  • Slide 24
  • Gravitational potential energy
  • Slide 26
  • Work and Energy
  • Slide 28
  • The Work-Kinetic Energy Theorem
  • Varying Forces
  • Slide 31
  • Examples of Integration
  • Slide 33
  • Slide 34
  • Examples of Definite Integration
  • Slide 36
  • Slide 37
Page 23: Work, Energy and Power AP style Energy Energy: the currency of the universe. Everything has to be “paid for” with energy. Energy can’t be created or.

Gravitational potential energy

GPE = weight x height

GPE = mgh

bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative

bull The GPE only depends on the weight and the height not on the path that it took to get to that height

Work and EnergyOften some force must do work

to give an object potential or kinetic energy

You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy

Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy

Work = Force x distance = change in energy

Example

How much more distance is required to stop if a car is going twice as fast

Fd = frac12 mv2

The work done by the brakes = the change in the kinetic energy

With TWICE the speed the car has

FOUR times the kinetic energy

Therefore it takes FOUR times the stopping distance

The Work-Kinetic Energy Theorem

NET Work done = Kinetic Energy

Wnet = frac12 mv2f ndash frac12 mv2

o

Varying Forces

The rule ishellip ldquoIf the Force varies you must integraterdquo

If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done

If the force is a function of velocity you must integrate to find the power output

Examples

If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5

If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms

Examples of Integration

An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force

W F x dx x dx x 33

43 4

Examples of Integration

At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)

v t a t dt tdt t C t( ) ( )

1515

20 75 32 2

x t v t dt t dt t t C t t( ) ( ) ( )

0 75 30 75

23 0 375 3 02 2

Examples of IntegrationA particle of mass m moves along the y-axis as

y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force

P = Fmiddotv

We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )

P F v m at b at bty y ( )( )12 2 4 2 22 3

Examples of Definite Integration

An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s

x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2

62

2

63 2

2

6 3 2 3 22 8 62

3

8

26

2

36

8

26 6 6

2

32

8

22 6 2 45 33

Examples of Definite Integration

To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x

What work is done to stretch the spring 2 meters beyond its equilibrium position

W F ds x x ds x x Jmm

m ( ) ( ) ( ) 4

4

3

1

2

4

32

1

22

4

30

1

20 8 6672

0

23

0

22

0

2 3 2 2

Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under

the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m

P = Fmiddotv but what is the velocity at x = 3m

Hmmmhellip

W F x dx and W K so ( )

F x dx K mv mvx

x

f o

o

f

( ) 12

12

2 2 ( )6 4 3 12

2

0

3

32 x x dx K mv x

( ) ( )6 4 3 64

2

3

36 3

4

23

3

33 0 9 1

22

0

3

2 3

0

3 2 33

2 x x dx x x x J mv x

v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5

  • Slide 1
  • Work Energy and Power AP style
  • Energy
  • Slide 4
  • Slide 5
  • Work = Force x displacement
  • Slide 7
  • Slide 8
  • Slide 9
  • Slide 10
  • Slide 11
  • Example
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Power
  • Slide 18
  • Slide 19
  • Slide 20
  • Slide 21
  • The ldquoWork- Kinetic Energy Theoremrdquo
  • Potential Energy
  • Slide 24
  • Gravitational potential energy
  • Slide 26
  • Work and Energy
  • Slide 28
  • The Work-Kinetic Energy Theorem
  • Varying Forces
  • Slide 31
  • Examples of Integration
  • Slide 33
  • Slide 34
  • Examples of Definite Integration
  • Slide 36
  • Slide 37
Page 24: Work, Energy and Power AP style Energy Energy: the currency of the universe. Everything has to be “paid for” with energy. Energy can’t be created or.

bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative

bull The GPE only depends on the weight and the height not on the path that it took to get to that height

Work and EnergyOften some force must do work

to give an object potential or kinetic energy

You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy

Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy

Work = Force x distance = change in energy

Example

How much more distance is required to stop if a car is going twice as fast

Fd = frac12 mv2

The work done by the brakes = the change in the kinetic energy

With TWICE the speed the car has

FOUR times the kinetic energy

Therefore it takes FOUR times the stopping distance

The Work-Kinetic Energy Theorem

NET Work done = Kinetic Energy

Wnet = frac12 mv2f ndash frac12 mv2

o

Varying Forces

The rule ishellip ldquoIf the Force varies you must integraterdquo

If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done

If the force is a function of velocity you must integrate to find the power output

Examples

If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5

If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms

Examples of Integration

An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force

W F x dx x dx x 33

43 4

Examples of Integration

At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)

v t a t dt tdt t C t( ) ( )

1515

20 75 32 2

x t v t dt t dt t t C t t( ) ( ) ( )

0 75 30 75

23 0 375 3 02 2

Examples of IntegrationA particle of mass m moves along the y-axis as

y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force

P = Fmiddotv

We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )

P F v m at b at bty y ( )( )12 2 4 2 22 3

Examples of Definite Integration

An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s

x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2

62

2

63 2

2

6 3 2 3 22 8 62

3

8

26

2

36

8

26 6 6

2

32

8

22 6 2 45 33

Examples of Definite Integration

To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x

What work is done to stretch the spring 2 meters beyond its equilibrium position

W F ds x x ds x x Jmm

m ( ) ( ) ( ) 4

4

3

1

2

4

32

1

22

4

30

1

20 8 6672

0

23

0

22

0

2 3 2 2

Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under

the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m

P = Fmiddotv but what is the velocity at x = 3m

Hmmmhellip

W F x dx and W K so ( )

F x dx K mv mvx

x

f o

o

f

( ) 12

12

2 2 ( )6 4 3 12

2

0

3

32 x x dx K mv x

( ) ( )6 4 3 64

2

3

36 3

4

23

3

33 0 9 1

22

0

3

2 3

0

3 2 33

2 x x dx x x x J mv x

v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5

  • Slide 1
  • Work Energy and Power AP style
  • Energy
  • Slide 4
  • Slide 5
  • Work = Force x displacement
  • Slide 7
  • Slide 8
  • Slide 9
  • Slide 10
  • Slide 11
  • Example
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Power
  • Slide 18
  • Slide 19
  • Slide 20
  • Slide 21
  • The ldquoWork- Kinetic Energy Theoremrdquo
  • Potential Energy
  • Slide 24
  • Gravitational potential energy
  • Slide 26
  • Work and Energy
  • Slide 28
  • The Work-Kinetic Energy Theorem
  • Varying Forces
  • Slide 31
  • Examples of Integration
  • Slide 33
  • Slide 34
  • Examples of Definite Integration
  • Slide 36
  • Slide 37
Page 25: Work, Energy and Power AP style Energy Energy: the currency of the universe. Everything has to be “paid for” with energy. Energy can’t be created or.

Work and EnergyOften some force must do work

to give an object potential or kinetic energy

You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy

Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy

Work = Force x distance = change in energy

Example

How much more distance is required to stop if a car is going twice as fast

Fd = frac12 mv2

The work done by the brakes = the change in the kinetic energy

With TWICE the speed the car has

FOUR times the kinetic energy

Therefore it takes FOUR times the stopping distance

The Work-Kinetic Energy Theorem

NET Work done = Kinetic Energy

Wnet = frac12 mv2f ndash frac12 mv2

o

Varying Forces

The rule ishellip ldquoIf the Force varies you must integraterdquo

If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done

If the force is a function of velocity you must integrate to find the power output

Examples

If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5

If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms

Examples of Integration

An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force

W F x dx x dx x 33

43 4

Examples of Integration

At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)

v t a t dt tdt t C t( ) ( )

1515

20 75 32 2

x t v t dt t dt t t C t t( ) ( ) ( )

0 75 30 75

23 0 375 3 02 2

Examples of IntegrationA particle of mass m moves along the y-axis as

y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force

P = Fmiddotv

We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )

P F v m at b at bty y ( )( )12 2 4 2 22 3

Examples of Definite Integration

An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s

x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2

62

2

63 2

2

6 3 2 3 22 8 62

3

8

26

2

36

8

26 6 6

2

32

8

22 6 2 45 33

Examples of Definite Integration

To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x

What work is done to stretch the spring 2 meters beyond its equilibrium position

W F ds x x ds x x Jmm

m ( ) ( ) ( ) 4

4

3

1

2

4

32

1

22

4

30

1

20 8 6672

0

23

0

22

0

2 3 2 2

Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under

the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m

P = Fmiddotv but what is the velocity at x = 3m

Hmmmhellip

W F x dx and W K so ( )

F x dx K mv mvx

x

f o

o

f

( ) 12

12

2 2 ( )6 4 3 12

2

0

3

32 x x dx K mv x

( ) ( )6 4 3 64

2

3

36 3

4

23

3

33 0 9 1

22

0

3

2 3

0

3 2 33

2 x x dx x x x J mv x

v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5

  • Slide 1
  • Work Energy and Power AP style
  • Energy
  • Slide 4
  • Slide 5
  • Work = Force x displacement
  • Slide 7
  • Slide 8
  • Slide 9
  • Slide 10
  • Slide 11
  • Example
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Power
  • Slide 18
  • Slide 19
  • Slide 20
  • Slide 21
  • The ldquoWork- Kinetic Energy Theoremrdquo
  • Potential Energy
  • Slide 24
  • Gravitational potential energy
  • Slide 26
  • Work and Energy
  • Slide 28
  • The Work-Kinetic Energy Theorem
  • Varying Forces
  • Slide 31
  • Examples of Integration
  • Slide 33
  • Slide 34
  • Examples of Definite Integration
  • Slide 36
  • Slide 37
Page 26: Work, Energy and Power AP style Energy Energy: the currency of the universe. Everything has to be “paid for” with energy. Energy can’t be created or.

Example

How much more distance is required to stop if a car is going twice as fast

Fd = frac12 mv2

The work done by the brakes = the change in the kinetic energy

With TWICE the speed the car has

FOUR times the kinetic energy

Therefore it takes FOUR times the stopping distance

The Work-Kinetic Energy Theorem

NET Work done = Kinetic Energy

Wnet = frac12 mv2f ndash frac12 mv2

o

Varying Forces

The rule ishellip ldquoIf the Force varies you must integraterdquo

If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done

If the force is a function of velocity you must integrate to find the power output

Examples

If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5

If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms

Examples of Integration

An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force

W F x dx x dx x 33

43 4

Examples of Integration

At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)

v t a t dt tdt t C t( ) ( )

1515

20 75 32 2

x t v t dt t dt t t C t t( ) ( ) ( )

0 75 30 75

23 0 375 3 02 2

Examples of IntegrationA particle of mass m moves along the y-axis as

y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force

P = Fmiddotv

We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )

P F v m at b at bty y ( )( )12 2 4 2 22 3

Examples of Definite Integration

An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s

x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2

62

2

63 2

2

6 3 2 3 22 8 62

3

8

26

2

36

8

26 6 6

2

32

8

22 6 2 45 33

Examples of Definite Integration

To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x

What work is done to stretch the spring 2 meters beyond its equilibrium position

W F ds x x ds x x Jmm

m ( ) ( ) ( ) 4

4

3

1

2

4

32

1

22

4

30

1

20 8 6672

0

23

0

22

0

2 3 2 2

Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under

the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m

P = Fmiddotv but what is the velocity at x = 3m

Hmmmhellip

W F x dx and W K so ( )

F x dx K mv mvx

x

f o

o

f

( ) 12

12

2 2 ( )6 4 3 12

2

0

3

32 x x dx K mv x

( ) ( )6 4 3 64

2

3

36 3

4

23

3

33 0 9 1

22

0

3

2 3

0

3 2 33

2 x x dx x x x J mv x

v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5

  • Slide 1
  • Work Energy and Power AP style
  • Energy
  • Slide 4
  • Slide 5
  • Work = Force x displacement
  • Slide 7
  • Slide 8
  • Slide 9
  • Slide 10
  • Slide 11
  • Example
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Power
  • Slide 18
  • Slide 19
  • Slide 20
  • Slide 21
  • The ldquoWork- Kinetic Energy Theoremrdquo
  • Potential Energy
  • Slide 24
  • Gravitational potential energy
  • Slide 26
  • Work and Energy
  • Slide 28
  • The Work-Kinetic Energy Theorem
  • Varying Forces
  • Slide 31
  • Examples of Integration
  • Slide 33
  • Slide 34
  • Examples of Definite Integration
  • Slide 36
  • Slide 37
Page 27: Work, Energy and Power AP style Energy Energy: the currency of the universe. Everything has to be “paid for” with energy. Energy can’t be created or.

The Work-Kinetic Energy Theorem

NET Work done = Kinetic Energy

Wnet = frac12 mv2f ndash frac12 mv2

o

Varying Forces

The rule ishellip ldquoIf the Force varies you must integraterdquo

If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done

If the force is a function of velocity you must integrate to find the power output

Examples

If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5

If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms

Examples of Integration

An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force

W F x dx x dx x 33

43 4

Examples of Integration

At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)

v t a t dt tdt t C t( ) ( )

1515

20 75 32 2

x t v t dt t dt t t C t t( ) ( ) ( )

0 75 30 75

23 0 375 3 02 2

Examples of IntegrationA particle of mass m moves along the y-axis as

y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force

P = Fmiddotv

We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )

P F v m at b at bty y ( )( )12 2 4 2 22 3

Examples of Definite Integration

An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s

x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2

62

2

63 2

2

6 3 2 3 22 8 62

3

8

26

2

36

8

26 6 6

2

32

8

22 6 2 45 33

Examples of Definite Integration

To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x

What work is done to stretch the spring 2 meters beyond its equilibrium position

W F ds x x ds x x Jmm

m ( ) ( ) ( ) 4

4

3

1

2

4

32

1

22

4

30

1

20 8 6672

0

23

0

22

0

2 3 2 2

Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under

the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m

P = Fmiddotv but what is the velocity at x = 3m

Hmmmhellip

W F x dx and W K so ( )

F x dx K mv mvx

x

f o

o

f

( ) 12

12

2 2 ( )6 4 3 12

2

0

3

32 x x dx K mv x

( ) ( )6 4 3 64

2

3

36 3

4

23

3

33 0 9 1

22

0

3

2 3

0

3 2 33

2 x x dx x x x J mv x

v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5

  • Slide 1
  • Work Energy and Power AP style
  • Energy
  • Slide 4
  • Slide 5
  • Work = Force x displacement
  • Slide 7
  • Slide 8
  • Slide 9
  • Slide 10
  • Slide 11
  • Example
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Power
  • Slide 18
  • Slide 19
  • Slide 20
  • Slide 21
  • The ldquoWork- Kinetic Energy Theoremrdquo
  • Potential Energy
  • Slide 24
  • Gravitational potential energy
  • Slide 26
  • Work and Energy
  • Slide 28
  • The Work-Kinetic Energy Theorem
  • Varying Forces
  • Slide 31
  • Examples of Integration
  • Slide 33
  • Slide 34
  • Examples of Definite Integration
  • Slide 36
  • Slide 37
Page 28: Work, Energy and Power AP style Energy Energy: the currency of the universe. Everything has to be “paid for” with energy. Energy can’t be created or.

Varying Forces

The rule ishellip ldquoIf the Force varies you must integraterdquo

If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done

If the force is a function of velocity you must integrate to find the power output

Examples

If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5

If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms

Examples of Integration

An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force

W F x dx x dx x 33

43 4

Examples of Integration

At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)

v t a t dt tdt t C t( ) ( )

1515

20 75 32 2

x t v t dt t dt t t C t t( ) ( ) ( )

0 75 30 75

23 0 375 3 02 2

Examples of IntegrationA particle of mass m moves along the y-axis as

y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force

P = Fmiddotv

We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )

P F v m at b at bty y ( )( )12 2 4 2 22 3

Examples of Definite Integration

An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s

x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2

62

2

63 2

2

6 3 2 3 22 8 62

3

8

26

2

36

8

26 6 6

2

32

8

22 6 2 45 33

Examples of Definite Integration

To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x

What work is done to stretch the spring 2 meters beyond its equilibrium position

W F ds x x ds x x Jmm

m ( ) ( ) ( ) 4

4

3

1

2

4

32

1

22

4

30

1

20 8 6672

0

23

0

22

0

2 3 2 2

Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under

the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m

P = Fmiddotv but what is the velocity at x = 3m

Hmmmhellip

W F x dx and W K so ( )

F x dx K mv mvx

x

f o

o

f

( ) 12

12

2 2 ( )6 4 3 12

2

0

3

32 x x dx K mv x

( ) ( )6 4 3 64

2

3

36 3

4

23

3

33 0 9 1

22

0

3

2 3

0

3 2 33

2 x x dx x x x J mv x

v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5

  • Slide 1
  • Work Energy and Power AP style
  • Energy
  • Slide 4
  • Slide 5
  • Work = Force x displacement
  • Slide 7
  • Slide 8
  • Slide 9
  • Slide 10
  • Slide 11
  • Example
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Power
  • Slide 18
  • Slide 19
  • Slide 20
  • Slide 21
  • The ldquoWork- Kinetic Energy Theoremrdquo
  • Potential Energy
  • Slide 24
  • Gravitational potential energy
  • Slide 26
  • Work and Energy
  • Slide 28
  • The Work-Kinetic Energy Theorem
  • Varying Forces
  • Slide 31
  • Examples of Integration
  • Slide 33
  • Slide 34
  • Examples of Definite Integration
  • Slide 36
  • Slide 37
Page 29: Work, Energy and Power AP style Energy Energy: the currency of the universe. Everything has to be “paid for” with energy. Energy can’t be created or.

Examples

If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5

If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms

Examples of Integration

An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force

W F x dx x dx x 33

43 4

Examples of Integration

At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)

v t a t dt tdt t C t( ) ( )

1515

20 75 32 2

x t v t dt t dt t t C t t( ) ( ) ( )

0 75 30 75

23 0 375 3 02 2

Examples of IntegrationA particle of mass m moves along the y-axis as

y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force

P = Fmiddotv

We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )

P F v m at b at bty y ( )( )12 2 4 2 22 3

Examples of Definite Integration

An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s

x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2

62

2

63 2

2

6 3 2 3 22 8 62

3

8

26

2

36

8

26 6 6

2

32

8

22 6 2 45 33

Examples of Definite Integration

To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x

What work is done to stretch the spring 2 meters beyond its equilibrium position

W F ds x x ds x x Jmm

m ( ) ( ) ( ) 4

4

3

1

2

4

32

1

22

4

30

1

20 8 6672

0

23

0

22

0

2 3 2 2

Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under

the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m

P = Fmiddotv but what is the velocity at x = 3m

Hmmmhellip

W F x dx and W K so ( )

F x dx K mv mvx

x

f o

o

f

( ) 12

12

2 2 ( )6 4 3 12

2

0

3

32 x x dx K mv x

( ) ( )6 4 3 64

2

3

36 3

4

23

3

33 0 9 1

22

0

3

2 3

0

3 2 33

2 x x dx x x x J mv x

v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5

  • Slide 1
  • Work Energy and Power AP style
  • Energy
  • Slide 4
  • Slide 5
  • Work = Force x displacement
  • Slide 7
  • Slide 8
  • Slide 9
  • Slide 10
  • Slide 11
  • Example
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Power
  • Slide 18
  • Slide 19
  • Slide 20
  • Slide 21
  • The ldquoWork- Kinetic Energy Theoremrdquo
  • Potential Energy
  • Slide 24
  • Gravitational potential energy
  • Slide 26
  • Work and Energy
  • Slide 28
  • The Work-Kinetic Energy Theorem
  • Varying Forces
  • Slide 31
  • Examples of Integration
  • Slide 33
  • Slide 34
  • Examples of Definite Integration
  • Slide 36
  • Slide 37
Page 30: Work, Energy and Power AP style Energy Energy: the currency of the universe. Everything has to be “paid for” with energy. Energy can’t be created or.

Examples of Integration

An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force

W F x dx x dx x 33

43 4

Examples of Integration

At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)

v t a t dt tdt t C t( ) ( )

1515

20 75 32 2

x t v t dt t dt t t C t t( ) ( ) ( )

0 75 30 75

23 0 375 3 02 2

Examples of IntegrationA particle of mass m moves along the y-axis as

y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force

P = Fmiddotv

We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )

P F v m at b at bty y ( )( )12 2 4 2 22 3

Examples of Definite Integration

An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s

x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2

62

2

63 2

2

6 3 2 3 22 8 62

3

8

26

2

36

8

26 6 6

2

32

8

22 6 2 45 33

Examples of Definite Integration

To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x

What work is done to stretch the spring 2 meters beyond its equilibrium position

W F ds x x ds x x Jmm

m ( ) ( ) ( ) 4

4

3

1

2

4

32

1

22

4

30

1

20 8 6672

0

23

0

22

0

2 3 2 2

Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under

the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m

P = Fmiddotv but what is the velocity at x = 3m

Hmmmhellip

W F x dx and W K so ( )

F x dx K mv mvx

x

f o

o

f

( ) 12

12

2 2 ( )6 4 3 12

2

0

3

32 x x dx K mv x

( ) ( )6 4 3 64

2

3

36 3

4

23

3

33 0 9 1

22

0

3

2 3

0

3 2 33

2 x x dx x x x J mv x

v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5

  • Slide 1
  • Work Energy and Power AP style
  • Energy
  • Slide 4
  • Slide 5
  • Work = Force x displacement
  • Slide 7
  • Slide 8
  • Slide 9
  • Slide 10
  • Slide 11
  • Example
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Power
  • Slide 18
  • Slide 19
  • Slide 20
  • Slide 21
  • The ldquoWork- Kinetic Energy Theoremrdquo
  • Potential Energy
  • Slide 24
  • Gravitational potential energy
  • Slide 26
  • Work and Energy
  • Slide 28
  • The Work-Kinetic Energy Theorem
  • Varying Forces
  • Slide 31
  • Examples of Integration
  • Slide 33
  • Slide 34
  • Examples of Definite Integration
  • Slide 36
  • Slide 37
Page 31: Work, Energy and Power AP style Energy Energy: the currency of the universe. Everything has to be “paid for” with energy. Energy can’t be created or.

Examples of Integration

At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)

v t a t dt tdt t C t( ) ( )

1515

20 75 32 2

x t v t dt t dt t t C t t( ) ( ) ( )

0 75 30 75

23 0 375 3 02 2

Examples of IntegrationA particle of mass m moves along the y-axis as

y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force

P = Fmiddotv

We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )

P F v m at b at bty y ( )( )12 2 4 2 22 3

Examples of Definite Integration

An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s

x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2

62

2

63 2

2

6 3 2 3 22 8 62

3

8

26

2

36

8

26 6 6

2

32

8

22 6 2 45 33

Examples of Definite Integration

To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x

What work is done to stretch the spring 2 meters beyond its equilibrium position

W F ds x x ds x x Jmm

m ( ) ( ) ( ) 4

4

3

1

2

4

32

1

22

4

30

1

20 8 6672

0

23

0

22

0

2 3 2 2

Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under

the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m

P = Fmiddotv but what is the velocity at x = 3m

Hmmmhellip

W F x dx and W K so ( )

F x dx K mv mvx

x

f o

o

f

( ) 12

12

2 2 ( )6 4 3 12

2

0

3

32 x x dx K mv x

( ) ( )6 4 3 64

2

3

36 3

4

23

3

33 0 9 1

22

0

3

2 3

0

3 2 33

2 x x dx x x x J mv x

v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5

  • Slide 1
  • Work Energy and Power AP style
  • Energy
  • Slide 4
  • Slide 5
  • Work = Force x displacement
  • Slide 7
  • Slide 8
  • Slide 9
  • Slide 10
  • Slide 11
  • Example
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Power
  • Slide 18
  • Slide 19
  • Slide 20
  • Slide 21
  • The ldquoWork- Kinetic Energy Theoremrdquo
  • Potential Energy
  • Slide 24
  • Gravitational potential energy
  • Slide 26
  • Work and Energy
  • Slide 28
  • The Work-Kinetic Energy Theorem
  • Varying Forces
  • Slide 31
  • Examples of Integration
  • Slide 33
  • Slide 34
  • Examples of Definite Integration
  • Slide 36
  • Slide 37
Page 32: Work, Energy and Power AP style Energy Energy: the currency of the universe. Everything has to be “paid for” with energy. Energy can’t be created or.

Examples of IntegrationA particle of mass m moves along the y-axis as

y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force

P = Fmiddotv

We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )

P F v m at b at bty y ( )( )12 2 4 2 22 3

Examples of Definite Integration

An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s

x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2

62

2

63 2

2

6 3 2 3 22 8 62

3

8

26

2

36

8

26 6 6

2

32

8

22 6 2 45 33

Examples of Definite Integration

To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x

What work is done to stretch the spring 2 meters beyond its equilibrium position

W F ds x x ds x x Jmm

m ( ) ( ) ( ) 4

4

3

1

2

4

32

1

22

4

30

1

20 8 6672

0

23

0

22

0

2 3 2 2

Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under

the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m

P = Fmiddotv but what is the velocity at x = 3m

Hmmmhellip

W F x dx and W K so ( )

F x dx K mv mvx

x

f o

o

f

( ) 12

12

2 2 ( )6 4 3 12

2

0

3

32 x x dx K mv x

( ) ( )6 4 3 64

2

3

36 3

4

23

3

33 0 9 1

22

0

3

2 3

0

3 2 33

2 x x dx x x x J mv x

v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5

  • Slide 1
  • Work Energy and Power AP style
  • Energy
  • Slide 4
  • Slide 5
  • Work = Force x displacement
  • Slide 7
  • Slide 8
  • Slide 9
  • Slide 10
  • Slide 11
  • Example
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Power
  • Slide 18
  • Slide 19
  • Slide 20
  • Slide 21
  • The ldquoWork- Kinetic Energy Theoremrdquo
  • Potential Energy
  • Slide 24
  • Gravitational potential energy
  • Slide 26
  • Work and Energy
  • Slide 28
  • The Work-Kinetic Energy Theorem
  • Varying Forces
  • Slide 31
  • Examples of Integration
  • Slide 33
  • Slide 34
  • Examples of Definite Integration
  • Slide 36
  • Slide 37
Page 33: Work, Energy and Power AP style Energy Energy: the currency of the universe. Everything has to be “paid for” with energy. Energy can’t be created or.

Examples of Definite Integration

An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s

x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2

62

2

63 2

2

6 3 2 3 22 8 62

3

8

26

2

36

8

26 6 6

2

32

8

22 6 2 45 33

Examples of Definite Integration

To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x

What work is done to stretch the spring 2 meters beyond its equilibrium position

W F ds x x ds x x Jmm

m ( ) ( ) ( ) 4

4

3

1

2

4

32

1

22

4

30

1

20 8 6672

0

23

0

22

0

2 3 2 2

Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under

the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m

P = Fmiddotv but what is the velocity at x = 3m

Hmmmhellip

W F x dx and W K so ( )

F x dx K mv mvx

x

f o

o

f

( ) 12

12

2 2 ( )6 4 3 12

2

0

3

32 x x dx K mv x

( ) ( )6 4 3 64

2

3

36 3

4

23

3

33 0 9 1

22

0

3

2 3

0

3 2 33

2 x x dx x x x J mv x

v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5

  • Slide 1
  • Work Energy and Power AP style
  • Energy
  • Slide 4
  • Slide 5
  • Work = Force x displacement
  • Slide 7
  • Slide 8
  • Slide 9
  • Slide 10
  • Slide 11
  • Example
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Power
  • Slide 18
  • Slide 19
  • Slide 20
  • Slide 21
  • The ldquoWork- Kinetic Energy Theoremrdquo
  • Potential Energy
  • Slide 24
  • Gravitational potential energy
  • Slide 26
  • Work and Energy
  • Slide 28
  • The Work-Kinetic Energy Theorem
  • Varying Forces
  • Slide 31
  • Examples of Integration
  • Slide 33
  • Slide 34
  • Examples of Definite Integration
  • Slide 36
  • Slide 37
Page 34: Work, Energy and Power AP style Energy Energy: the currency of the universe. Everything has to be “paid for” with energy. Energy can’t be created or.

Examples of Definite Integration

To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x

What work is done to stretch the spring 2 meters beyond its equilibrium position

W F ds x x ds x x Jmm

m ( ) ( ) ( ) 4

4

3

1

2

4

32

1

22

4

30

1

20 8 6672

0

23

0

22

0

2 3 2 2

Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under

the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m

P = Fmiddotv but what is the velocity at x = 3m

Hmmmhellip

W F x dx and W K so ( )

F x dx K mv mvx

x

f o

o

f

( ) 12

12

2 2 ( )6 4 3 12

2

0

3

32 x x dx K mv x

( ) ( )6 4 3 64

2

3

36 3

4

23

3

33 0 9 1

22

0

3

2 3

0

3 2 33

2 x x dx x x x J mv x

v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5

  • Slide 1
  • Work Energy and Power AP style
  • Energy
  • Slide 4
  • Slide 5
  • Work = Force x displacement
  • Slide 7
  • Slide 8
  • Slide 9
  • Slide 10
  • Slide 11
  • Example
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Power
  • Slide 18
  • Slide 19
  • Slide 20
  • Slide 21
  • The ldquoWork- Kinetic Energy Theoremrdquo
  • Potential Energy
  • Slide 24
  • Gravitational potential energy
  • Slide 26
  • Work and Energy
  • Slide 28
  • The Work-Kinetic Energy Theorem
  • Varying Forces
  • Slide 31
  • Examples of Integration
  • Slide 33
  • Slide 34
  • Examples of Definite Integration
  • Slide 36
  • Slide 37
Page 35: Work, Energy and Power AP style Energy Energy: the currency of the universe. Everything has to be “paid for” with energy. Energy can’t be created or.

Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under

the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m

P = Fmiddotv but what is the velocity at x = 3m

Hmmmhellip

W F x dx and W K so ( )

F x dx K mv mvx

x

f o

o

f

( ) 12

12

2 2 ( )6 4 3 12

2

0

3

32 x x dx K mv x

( ) ( )6 4 3 64

2

3

36 3

4

23

3

33 0 9 1

22

0

3

2 3

0

3 2 33

2 x x dx x x x J mv x

v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5

  • Slide 1
  • Work Energy and Power AP style
  • Energy
  • Slide 4
  • Slide 5
  • Work = Force x displacement
  • Slide 7
  • Slide 8
  • Slide 9
  • Slide 10
  • Slide 11
  • Example
  • Slide 13
  • Slide 14
  • Slide 15
  • Slide 16
  • Power
  • Slide 18
  • Slide 19
  • Slide 20
  • Slide 21
  • The ldquoWork- Kinetic Energy Theoremrdquo
  • Potential Energy
  • Slide 24
  • Gravitational potential energy
  • Slide 26
  • Work and Energy
  • Slide 28
  • The Work-Kinetic Energy Theorem
  • Varying Forces
  • Slide 31
  • Examples of Integration
  • Slide 33
  • Slide 34
  • Examples of Definite Integration
  • Slide 36
  • Slide 37

Kingdom of Cambodia: Rural Energy Project (Proposed ...CURRENCY EQUIVALENTS (13 February 2012) Currency unit - Riel (KHR) KHR 1.00 - $0.00025 . $1.00 - KHR 4,043 . ABBREVIATIONS .

Kingdom of Cambodia: Rural Energy Project (Proposed ...CURRENCY EQUIVALENTS (13 February 2012) Currency unit - Riel (KHR) KHR 1.00 - $0.00025 . $1.00 - KHR 4,043 . ABBREVIATIONS .

Currency Derivatives...Currency Derivatives: Currency Futures Currency Futures • Currency Futures are contracts specifying a standard volume of a particular currency to be exchanged

Currency Derivatives...Currency Derivatives: Currency Futures Currency Futures • Currency Futures are contracts specifying a standard volume of a particular currency to be exchanged

ATP The Energy Currency of Living Organisms. Objectives Understand the molecule ATP.

ATP The Energy Currency of Living Organisms. Objectives Understand the molecule ATP.

ATP- The universal energy currency of cell

ATP- The universal energy currency of cell

Meeting the Needs. Plants can’t eat… …what do they release energy from?

Meeting the Needs. Plants can’t eat… …what do they release energy from?

Chapter 4: Energy, Chemistry, and Society. ENERGY? Like the energy of a crowd, You can’t see it, Can’t measure it, But you know it is there. What do you.

Chapter 4: Energy, Chemistry, and Society. ENERGY? Like the energy of a crowd, You can’t see it, Can’t measure it, But you know it is there. What do you.

Constitution #7 The Shrinking 10 th Amendment. Ch 2 EQ Answers 1.AofC: strengths: military, treaty, unify –Weak: can’t tax, regulate trade, currency,

Constitution #7 The Shrinking 10 th Amendment. Ch 2 EQ Answers 1.AofC: strengths: military, treaty, unify –Weak: can’t tax, regulate trade, currency,

Social Media and Social Consent: The New Market Currency for the Energy Sector

Social Media and Social Consent: The New Market Currency for the Energy Sector

Economy as Biophysical system Laws of physics Can’t make something from nothing, or vice versa Can’t do work without energy Entropy increases.

Economy as Biophysical system Laws of physics Can’t make something from nothing, or vice versa Can’t do work without energy Entropy increases.

Biol221 24a energy currency to be taught

Biol221 24a energy currency to be taught

Plasma Based Energy Researchrexresearch.com/kingmb/MorayKing_TeslaTech_July201… ·  · 2017-11-01• What is the energy source? –Cold Fusion, LENR ? –Hydrinos ? Can’t Happen!

Plasma Based Energy Researchrexresearch.com/kingmb/MorayKing_TeslaTech_July201… ·  · 2017-11-01• What is the energy source? –Cold Fusion, LENR ? –Hydrinos ? Can’t Happen!

Things to remember You can’t move from Standard back to Starter You can’t change your billing currency You can’t change your time zone information.

Things to remember You can’t move from Standard back to Starter You can’t change your billing currency You can’t change your time zone information.

Currency NOTES Currency NOTES

Currency NOTES Currency NOTES

King Versus Assembly of Notables Archaic banking system and a lack of paper money limited economic options –Can’t inflate currency –French gov. owes money.

King Versus Assembly of Notables Archaic banking system and a lack of paper money limited economic options –Can’t inflate currency –French gov. owes money.

Mitochondria III: ATP - the energy Currency of the Cell

Mitochondria III: ATP - the energy Currency of the Cell

Lecture 9 Generating Energy. Adenosine Triphosphate (ATP) The energy currency or coin of the cell.The energy currency or coin of the cell. Transfers energy.

Lecture 9 Generating Energy. Adenosine Triphosphate (ATP) The energy currency or coin of the cell.The energy currency or coin of the cell. Transfers energy.

ATP: Energy Currency of the Cell 3 - …docshare01.docshare.tips/files/23990/239906969.pdfnEL 3.2 atP: Energy Currency of the Cell 143 8159_Bio_Ch03.indd 143 4/9/12 6:23 PM. ATP is

ATP: Energy Currency of the Cell 3 - …docshare01.docshare.tips/files/23990/239906969.pdfnEL 3.2 atP: Energy Currency of the Cell 143 8159_Bio_Ch03.indd 143 4/9/12 6:23 PM. ATP is

1 S WISS A SIA C APITAL SAC GLOBAL ENERGY & MINING FUND FUNDAMENTALS CAN’T GET ANY BETTER SAC GLOBAL ENERGY & MINING FUND FUNDAMENTALS CAN’T GET ANY BETTER.

1 S WISS A SIA C APITAL SAC GLOBAL ENERGY & MINING FUND FUNDAMENTALS CAN’T GET ANY BETTER SAC GLOBAL ENERGY & MINING FUND FUNDAMENTALS CAN’T GET ANY BETTER.

Currency Hedging - Nebraska · Advantages Disadvantages Currency hedging is a cost-effective way to manage currency exposure The hedge ratio is static, so the manager can’t exploit

Currency Hedging - Nebraska · Advantages Disadvantages Currency hedging is a cost-effective way to manage currency exposure The hedge ratio is static, so the manager can’t exploit

Energy Transfer. ATP z“Energy Currency” zPotential energy in ATP used for all energy requiring processes of cells.

Energy Transfer. ATP z“Energy Currency” zPotential energy in ATP used for all energy requiring processes of cells.