Uniform Circular Motion, Acceleration

A particle moves with a constant speed in a circular path of radius r with an acceleration:

The centripetal acceleration, is directed toward the center of the circle

The centripetal acceleration is always perpendicular to the velocity

2

c

va

r

ca

Uniform Circular Motion, Force

A force, , is associated with the centripetal acceleration

The force is also directed toward the center of the circle

Applying Newton’s Second Law along the radial direction gives

2

c

vF ma m

r

rF

Uniform Circular Motion, cont

A force causing a centripetal acceleration acts toward the center of the circle

It causes a change in the direction of the velocity vector

If the force vanishes, the object would move in a straight-line path tangent to the circle See various release points in

the active figure

Motion in a Horizontal Circle

The speed at which the object moves depends on the mass of the object and the tension in the cord

The centripetal force is supplied by the tension

T=mv2/r hence Trv

m

Motion in Accelerated Frames

A fictitious force results from an accelerated frame of reference A fictitious force appears to act on an object in the

same way as a real force, but you cannot identify a second object for the fictitious force Remember that real forces are always interactions

between two objects

“Centrifugal” Force From the frame of the passenger (b), a

force appears to push her toward the door From the frame of the Earth, the car

applies a leftward force on the passenger The outward force is often called a

centrifugal force It is a fictitious force due to the centripetal

acceleration associated with the car’s change in direction

In actuality, friction supplies the force to allow the passenger to move with the car If the frictional force is not large enough,

the passenger continues on her initial path according to Newton’s First Law

“Coriolis Force” This is an apparent

force caused by changing the radial position of an object in a rotating coordinate system

The result of the rotation is the curved path of objectBall in figure to the right, winds, rivers and currents onearth. For winds we get the prevailing wind pattern below.

Fictitious Forces, examples

Although fictitious forces are not real forces, they can have real effects

Examples: Objects in the car do slide You feel pushed to the outside of a rotating

platform The Coriolis force is responsible for the rotation of

weather systems, including hurricanes, and ocean currents

Introduction to Energy

The concept of energy is one of the most important topics in science and engineering

Every physical process that occurs in the Universe involves energy and energy transfers or transformations

Energy is not easily defined

Work

The work, W, done on a system by an agent exerting a constant force on the system is the product of the magnitude F of the force, the magnitude r of the displacement of the point of application of the force, and cos where is the angle between the force and the displacement vectors

Work, cont.

W = F r cos F. r The displacement is that

of the point of application of the force

A force does no work on the object if the force does not move through a displacement

The work done by a force on a moving object is zero when the force applied is perpendicular to the displacement of its point of application

Work Example

The normal force and the gravitational force do no work on the object cos = cos 90° = 0

The force is the only force that does work on the object

F

Units of Work

Work is a scalar quantity The unit of work is a joule (J)

1 joule = 1 newton . 1 meter J = N · m ( Fr)

The sign of the work depends on the direction of the force relative to the displacement Work is positive when projection of onto is in the

same direction as the displacement Work is negative when the projection is in the opposite

direction

Work Done by a Varying Force

Assume that during a very small displacement, x, F is constant

For that displacement, W ~ F x

For all of the intervals,

f

i

x

xx

W F x

Work Done by a Varying Force, cont

Therefore,

The work done is equal to the area under the curve between xi and xf

lim0

ff

ii

xx

x x xxx

F x F dx

f

i

x

xxW F dx

Work Done By A Spring

A model of a common physical system for which the force varies with position

The block is on a horizontal, frictionless surface

Observe the motion of the block with various values of the spring constant

Hooke’s Law

The force exerted by the spring is

Fs = - kx x is the position of the block with respect to the equilibrium position (x =

0) k is called the spring constant or force constant and measures the

stiffness of the spring This is called Hooke’s Law

Hooke’s Law, cont.

When x is positive (spring is stretched), F is negative

When x is 0 (at the equilibrium position), F is 0

When x is negative (spring is compressed), F is positive

Hooke’s Law, final

The force exerted by the spring is always directed opposite to the displacement from equilibrium

The spring force is sometimes called the restoring force

If the block is released it will oscillate back and forth between –x and x

Hooke’s Law consider the spring When x is positive (spring is

stretched), Fs is negative When x is 0 (at the

equilibrium position), Fs is 0 When x is negative (spring is

compressed), Fs is positive Hence the restoring force Fs =Fs = -kx

Work Done by a Spring Identify the block as the system and see figure below The work as the block moves from xi = - xmax to xf = 0 is ½ kx2

Note: The total work done by the spring as the block moves from –xmax to xmax is zero see figure also

Ie. From the General definition

Or

max

0 2max

1

2f

i

x

s xx xW F dx kx dx kx

rF dWW net )(

0

xi

max-

dxkxdxkxdW ss )((fx

ixirF

)1/(. 1 nxdxxie nn

Work Done by a Spring,in general Assume the block undergoes an arbitrary

displacement from x = xi to x = xf The work done by the spring on the block is

If the motion ends where it begins, W = 0 NOTE the work is a change in the expression 1/2kx2 We say a change in elastic potential

energy..in general a energy expression is defined for various forces and the work done changes that energy.

2 21 1

2 2f

i

x

s i fxW kx dx kx kx

Kinetic Energy and Work-Kinetic Energy Theorem

Kinetic Energy is the energy of a particle due to its motion K = ½ mv2

K is the kinetic energy m is the mass of the particle v is the speed of the particle

A change in kinetic energy is one possible result of doing work to transfer energy into a system

Kinetic Energy Calculating the work:

2 21 1

2 2

f f

i i

f

i

x x

x x

v

v

f i

net f i

W F dx ma dx

W mv dv

W mv mv

W K K K

IE. a=dv/dtadx=dv/dt dx=dv dx/dt=vdv

Hence K=1/2 mv2 is a a natural for energy expression..And the last equation is called the Work-Kinetic Energy Theorem Again we note that the work done changes an energy expression …in this case a change in Kinetic energyThe speed of the system increases if the work done on it is positive

The speed of the system decreases if the net work is negativeAlso valid for changes in rotational speed

The Work-Kinetic Energy Theorem states W = Kf – Ki = K

Potential Energy in general

Potential energy is energy related to the configuration of a system in which the components of the system interact by forces The forces are internal to the system Can be associated with only specific types of

forces acting between members of a system

Gravitational Potential EnergyNEAR SURFACE OF EARTH ONLY

The system is the Earth and the book

Do work on the book by lifting it slowly through a vertical displacement

The work done on the system must appear as an increase in the energy of the system

ˆy r j

Gravitational Potential Energy, cont

There is no change in kinetic energy since the book starts and ends at rest

Gravitational potential energy is the energy associated with an object at a given location above the surface of the Earth

app

ˆ ˆ( ) f i

f i

W

W mg y y

W mgy mgy

F r

j j

Gravitational Potential Energy, final

The quantity mgy is identified as the gravitational potential energy, Ug

Ug = mgy THIS IS ONLY NEAR THE EARTH’s surface ……………

WHY???????

Units are joules (J) Is a scalar Work may change the gravitational potential energy

of the system Wnet = Ug

Conservative Forces and Potential Energy

Define a potential energy function, U, such that the work done by a conservative force equals the decrease in the potential energy of the system

The work done by such a force, F, is

U is negative when F and x are in the same direction

f

i

x

C xxW F dx U

Conservative Forces and Potential Energy

The conservative force is related to the potential energy function through

The x component of a conservative force acting on an object within a system equals the negative of the potential energy of the system with respect to x Can be extended to three dimensions

x

dUF

dx

Conservative Forces and Potential Energy – Check

Look at the case of a deformed spring

This is Hooke’s Law and confirms the equation for U

U is an important function because a conservative force can be derived from it

21

2s

s

dU dF kx kx

dx dx

Energy Diagrams and Equilibrium

Motion in a system can be observed in terms of a graph of its position and energy

In a spring-mass system example, the block oscillates between the turning points, x = ±xmax

The block will always accelerate back toward x = 0

Energy Diagrams and Stable Equilibrium

The x = 0 position is one of stable equilibrium

Configurations of stable equilibrium correspond to those for which U(x) is a minimum

x = xmax and x = -xmax are called the turning points

Energy Diagrams and Unstable Equilibrium Fx = 0 at x = 0, so the

particle is in equilibrium For any other value of x, the

particle moves away from the equilibrium position

This is an example of unstable equilibrium

Configurations of unstable equilibrium correspond to those for which U(x) is a maximum

Neutral Equilibrium

Neutral equilibrium occurs in a configuration when U is constant over some region

A small displacement from a position in this region will produce neither restoring nor disrupting forces

Ways to Transfer Energy Into or Out of A System

Work – transfers by applying a force and causing a displacement of the point of application of the force

Mechanical Waves – allow a disturbance to propagate through a medium

Heat – is driven by a temperature difference between two regions in space

A word from our sponsors: CONDUCTION, CONVECTION, RADIATION

More Ways to Transfer Energy Into or Out of A System

Matter Transfer – matter physically crosses the boundary of the system, carrying energy with it

Electrical Transmission – transfer is by electric current

Electromagnetic Radiation – energy is transferred by electromagnetic waves

Two New important Potential Energies

In the universe at large Gravitational force as defined by Newton prevails

Ie.. F = -Gm1m2 /r2 m the masses G a universal constant and r distance between the masses (negative is attractive force)

In the atomic world the electric force dominates defined as F=kq1q2 /r2 here r is the distance between the electric charges represented by q and k a universal constant

Charges can be + or - The Constant values.G,k depend upon units

used

Gravitational and Electric Potential energies (3D)

f

i

x

C xxW F dx U

With r replacing x we get and using the gravitational and electric forces equations and for integration from point initial to final

W = FGdr = - Gm1m2 = 1/r2 dr = -Gm1m2 (1/rf -1/ri)

W = Fedr = kq1q2 = 1/r2 dr = kq1q2 (1/rf -1/ri)Or potential energies for these forces go as 1/r

Note from above that F = -dU/dr with UG = Gm1m2 /r Ue = kq1q2 /r we get back the 1/r2 forces

Conservation of Energy Energy is conserved

This means that energy cannot be created nor destroyed

If the total amount of energy in a system changes, it can only be due to the fact that energy has crossed the boundary of the system by some method of energy transfer!

Isolated System For an isolated system, Emech = 0

Remember Emech = K + U This is conservation of energy for an isolated system with

no nonconservative forces acting

If nonconservative forces are acting, some energy is transformed into internal energy

Conservation of Energy becomes Esystem = 0 Esystem is all kinetic, potential, and internal energies This is the most general statement of the isolated system

model

Isolated System, cont (example book falling) The changes in energy Esystem = 0 Or K +U=0 K=-U Ie. Kf - Ki = -(Uf –Ui) can be written out and rearranged Kf + Uf = Ki + Ui

Remember, this applies only to a system in which conservative forces act

Or 1/2mvf2 +mghf =1/2mgvi

2+mghi

Example – Free Fallexample 8-1

Determine the speed of the ball at y above the ground

Conceptualize Use energy instead of

motion

Categorize System is isolated Only force is gravitational

which is conservative

Example – Free Fall, cont Analyze

Apply Conservation of Energy Kf + Ugf = Ki + Ugi

Ki = 0, the ball is dropped

Solving for vf

Finalize The equation for vf is consistent with the results

obtained from kinematics

2 2f iv v g h y

For the electric force

Total energy Is K+U=1/2mv2 +kq1q2 /r Specifically in a hydrogen atom using charge

units e (CALLED ESU we get rid of K) and the proton and electron both have the same charge =e

Or total energy for electron in orbit =1/2mv2 +e2 /r we will use this in chapter 3

Instantaneous Power Power is the time rate of energy transfer The instantaneous power is defined as

Using work as the energy transfer method, this can also be written as

dE

dt

avg

W

t

Power The time rate of energy transfer is called

power The average power is given by

when the method of energy transfer is work Units of power: what is a Joule/sec called ? Answer WATT! 1 watt=1joule/sec

WP

t

Instantaneous Power and Average Power

The instantaneous power is the limiting value of the average power as t approaches zero

The power is valid for any means of energy transfer

NOTE: only part of F adds to power ?

lim0t

W dW d

t dt dt

rF F v

Units of Power The SI unit of power is called the watt

1 watt = 1 joule / second = 1 kg . m2 / s2

A unit of power in the US Customary system is horsepower 1 hp = 746 W

Units of power can also be used to express units of work or energy 1 kWh = (1000 W)(3600 s) = 3.6 x106 J

Example 8.10 melev =1600kg passengers =200kgA constant retarding force =4000 N

How much power to lift at constant rate of 3m/sHow much power to lift at speed v with a=1.00 m/ss

T

W

f

lim0t

W dW d

t dt dt

rF F v

USE F =0 in first part and =ma in second then useNext equation