Chapter 7

Work and Kinetic Energy

http://people.virginia.edu/~kdp2c/downloads/WorkEnergySelections.html

2

Conservative and Nonconservative Forces

Conservative force:

- the work it does is stored in the form of energy that can be released at a later time

-the work done by a conservative force moving an object around a closed path is zero

-Force depends upon position only

Example of a conservative force: gravity

Example of a nonconservative force: friction

3

Work done by gravity on a closed path is zero

4

Work done by friction on a closed path is not zero

5

The work done by a conservative force is zero on any closed path

So the work must be reversible (opposite when taking the same path) AND path independent (same amount of work

for any two different paths connecting two points)

Go A-B on path 1, the back B-A.Wt = W1 + -W1

Go A-B on path 1, the B-A on path 2.

Wt = W1 + -W2

6

Potential Energy

If we pick up a ball and put it on the shelf, we have done work on the ball. We can get that energy back if the ball falls back off the shelf (gravity does positive work on the ball, “releasing” the work that we put in before).

Until that happens, we say the energy is stored as potential energy.

7

Potential Energy

Consider the process in which the book goes from h=0 to h=0.50 m

Work done by gravity: W = - (mg)h = -13.5 J

For the book to go up against gravity, another force must be applied to overcome the weight. This other force did a (minimum) work of 13.5 J

If I lft the book steadily, the “external force” is provided by my hand with F~mg, work done by me: W=(mg)h = 13.5 J

The book’s potential energy changed by: 13.5 J

8

Potential EnergyThe work done against a conservative force is stored in the form of (potential) energy that can be released at a later time.

Note the minus sign:

•positive Wc (work by the conservative force) is negative potential energy (energy is released)

•negative Wc is positive potential energy (another force as done work against the conservative force)

9

Gravitational Potential Energy

Q: What does “UG = 0” mean?

10

Work Done by a Variable Force

The force needed to stretch a spring an amount x is F = kx.

Therefore, the work done in stretching (or compressing) the spring is

on t

he s

prin

g

with positive work applied leading to a positive change in potential: W = Uf - Ui

11

Potential energy in a spring

The corresponding conservative force is the force of the spring acting on the hand: positive work by the spring releases potential energy Wc = - ΔU

So, taking U=0 at x=0:

12

Up the Hill

a) the same

b) twice as much

c) four times as much

d) half as much

e) you gain no PE in either case

Two paths lead to the top of a

big hill. One is steep and direct, while the other is twice as long but less steep. How much more potential energy would you gain if you take the longer path?

13

Because your vertical position (height) changes by the

same amount in each case, the gain in potential energy

is the same.

Up the Hill

a) the same

b) twice as much

c) four times as much

d) half as much

e) you gain no PE in either case

Two paths lead to the top of a

big hill. One is steep and direct, while the other is twice as long but less steep. How much more potential energy would you gain if you take the longer path?

Follow-up: How much more work do you do in taking the steeper path?

Follow-up: Which path would you rather take? Why?

14

Is it possible for the

gravitational potential

energy of an object to

be negative?

a) yes

b) no

Sign of the Energy

15

Is it possible for the

gravitational potential

energy of an object to

be negative?

a) yes

b) no

Gravitational PE is mgh, where height h is measured relative to some

arbitrary reference level where PE = 0. For example, a book on a table

has positive PE if the zero reference level is chosen to be the floor.

However, if the ceiling is the zero level, then the book has negative PE on

the table. Only differences (or changes) in PE have any physical

meaning.

Sign of the Energy

16

You and your friend both solve a problem involving a skier going down a slope, starting from rest. The two of you have chosen different levels for y = 0 in this problem. Which of the following quantities will you and your friend agree on?

a) only B

b) only C

c) A, B, and C

d) only A and C

e) only B and C

KE and PE

A) skier’s PE B) skier’s change in PE C) skier’s final KE

17

You and your friend both solve a problem involving a skier going down a slope, starting from rest. The two of you have chosen different levels for y = 0 in this problem. Which of the following quantities will you and your friend agree on?

a) only B

b) only C

c) A, B, and C

d) only A and C

e) only B and C

The gravitational PE depends upon the reference level, but the

difference PE does not! The work done by gravity must be

the same in the two solutions, so PE and KE should be the

same.

A) skier’s PE B) skier’s change in PE C) skier’s final KE

KE and PE

18

Mechanical Energy

It is useful to define the mechanical energy:

Consider the total amount of work done on a body by the conservative and the non-conservative forces. This is the change in kinetic energy (work-energy theorem)

Then:

The work done by all non-conservative forces is the change in the mechanical energy of a body

19

Conservation of Mechanical Energy

The work done by all non-conservative forces is the change in the mechanical energy of a body

If there are only conservative forces doing work during a process, we find:

20

Work-Energy Theorem vs. Conservation of Energy?

Work-Energy Theorem

total work done (by both conservative and non-conservative forces) = change in kinetic energy

Conservation of mechanical energytotal work done by non-conservative forces = change in mechanical energy

These two are completely equivalent. The difference is only how to treat conservative forces. Do NOT use both potential energy AND work by the conservative force... that’s double-counting!

In general, energy conservation makes kinematics problems much easier to solve...

21

Runaway Truck

A truck, initially at rest, rolls

down a frictionless hill and attains a speed of 20 m/s at the bottom. To achieve a speed of 40 m/s at the bottom, how many times higher must the hill be?

a) half the height

b) the same height

c) 2 times the height

d) twice the height

e) four times the height

22

Runaway Truck

A truck, initially at rest, rolls down a frictionless hill and attains a speed of 20 m/s at the bottom. To achieve a speed of 40 m/s at the bottom, how many times higher must the hill be?

a) half the height

b) the same height

c) 2 times the height

d) twice the height

e) four times the height

Use energy conservation:

initial energy: Ei = PEg = mgH

final energy: Ef = KE = mv2

Conservation of Energy:

Ei = mgH = Ef = mv2

therefore: gH = v2

So if v doubles, H quadruples!

23

Cart on a Hill

A cart starting from rest rolls down a hill

and at the bottom has a speed of 4 m/s. If the cart were given an initial push, so its initial speed at the top of the hill was 3 m/s, what would be its speed at the bottom?

a) 4 m/s

b) 5 m/s

c) 6 m/s

d) 7 m/s

e) 25 m/s

24

Cart on a Hill

When starting from rest, thecart’s PE is changed into KE:

PE = KE = m(4)2

A cart starting from rest rolls down a hill

and at the bottom has a speed of 4 m/s. If the cart were given an initial push, so its initial speed at the top of the hill was 3 m/s, what would be its speed at the bottom?

a) 4 m/s

b) 5 m/s

c) 6 m/s

d) 7 m/s

e) 25 m/s

When starting from 3 m/s, thefinal KE is:

KEf = KEi + KE= m(3)2 + m(4)2

= m(25) = m(5)2

Potential Energy CurvesThe curve of a hill or a roller coaster is itself essentially a plot of the gravitational potential energy:

Q: at what point is speed maximized?

Q: where might apparent weight be minimized?

26

Potential Energy for a Spring

27

Potential Energy Curves and Equipotentials

Contour maps are also a form of potential energy curve:

Each contour is an equal height, and so an “equipotential” for gravitational potential energy

28

A mass attached to a vertical spring causes the spring to stretch and the mass to move downwards. What can you say about the spring’s

potential energy (PEs) and the

gravitational potential energy

(PEg) of the mass?

a) both PEs and PEg decrease

b) PEs increases and PEg decreases

c) both PEs and PEg increase

d) PEs decreases and PEg increases

e) PEs increases and PEg is constant

Question 8.5 Springs and Gravity

29

A mass attached to a vertical spring causes the spring to stretch and the mass to move downwards. What can you say about the spring’s

potential energy (PEs) and the

gravitational potential energy

(PEg) of the mass?

a) both PEs and PEg decrease

b) PEs increases and PEg decreases

c) both PEs and PEg increase

d) PEs decreases and PEg increases

e) PEs increases and PEg is constant

The spring is stretched, so its elastic PE increases,

because PEs = kx2. The mass moves down to a

lower position, so its gravitational PE decreases,

because PEg = mgh.

Question 8.5 Springs and Gravity

30

8-4 Work Done by Nonconservative Forces

In this example, the nonconservative force is water resistance:

30

31

Chapter 9

Linear Momentum

Linear Momentum

Momentum is a vector; its direction is the same as the direction of the velocity.

Momentum is a vector

Change in momentum:

(a) mv

(b) 2mv

Going Bowling I

p

p

a) the bowling ball

b) same time for both

c) the Ping-Pong ball

d) impossible to say

A bowling ball and a Ping-Pong ball are rolling toward you with the same momentum. Which one of the two has the greater kinetic energy?

Going Bowling I

p

p

a) the bowling ball

b) same time for both

c) the Ping-Pong ball

d) impossible to say

A bowling ball and a Ping-Pong ball are rolling toward you with the same momentum. Which one of the two has the greater kinetic energy?

Momentum is p = mv

so the ping-pong ball must have a much greater velocity

Kinetic Energy is KE = 1/2 mv2

so (for a single object): KE = p2 / 2m

Momentum and Newton’s Second Law

Newton’s second law, as we wrote it before:

is only valid for objects that have constant mass. Here is a more general form (also useful when the mass is changing):

A net force of 200 N acts on a 100-kg

boulder, and a force of the same

magnitude acts on a 130-g pebble.

How does the rate of change of the

boulder’s momentum compare to

the rate of change of the pebble’s

momentum?

a) greater than

b) less than

c) equal to

Momentum and Force

A net force of 200 N acts on a 100-kg

boulder, and a force of the same

magnitude acts on a 130-g pebble.

How does the rate of change of the

boulder’s momentum compare to

the rate of change of the pebble’s

momentum?

a) greater than

b) less than

c) equal to

The rate of change of momentum is, in fact, the force.

Remember that F = p/t. Because the force exerted

on the boulder and the pebble is the same, then the

rate of change of momentum is the same.

Momentum and Force

Impulse

Impulse is a vector, in the same direction as the average force.

The same change in momentum may be produced by a large force acting for a short time, or by a smaller force acting for a longer time.

Impulse quantifies the overall change in momentum

Impulse

We can rewrite

as

So we see that

The impulse is equal to the change in momentum.

Why we don’t dive into concreteThe same change in momentum may be

produced by a large force acting for a short time, or by a smaller force acting for a longer time.

Going Bowling II

p

p

a) the bowling ball

b) same time for both

c) the Ping-Pong ball

d) impossible to say

A bowling ball and a Ping-Pong ball are rolling toward you with the same momentum. If you exert the same force to stop each one, which takes a longer time to bring to rest?

Going Bowling II

We know:

Here, F and p are the same for both balls!

It will take the same amount of time to stop them. p

p so p = Fav t

a) the bowling ball

b) same time for both

c) the Ping-Pong ball

d) impossible to say

A bowling ball and a Ping-Pong ball are rolling toward you with the same momentum. If you exert the same force to stop each one, which takes a longer time to bring to rest?

av tp

F

Going Bowling III

p

p

A bowling ball and a Ping-Pong

ball are rolling toward you with

the same momentum. If you

exert the same force to stop each

one, for which is the stopping

distance greater?

a) the bowling ball

b) same distance for both

c) the Ping-Pong ball

d) impossible to say

Going Bowling III

p

p

Use the work-energy theorem: W = KE. The ball with less mass has the greater speed, and thus the greater KE. In order to remove that KE, work must be done, where W = Fd. Because the force is the same in both cases, the distance needed to stop the less massive ball must be bigger.

A bowling ball and a Ping-Pong

ball are rolling toward you with

the same momentum. If you

exert the same force to stop each

one, for which is the stopping

distance greater?

a) the bowling ball

b) same distance for both

c) the Ping-Pong ball

d) impossible to say

Conservation of Linear Momentum

The net force acting on an object is the rate of change of its momentum:

If the net force is zero, the momentum does not change!

•A vector equation•Works for each coordinate separately

With no net force:

Internal Versus External Forces

Internal forces act between objects within the system.

As with all forces, they occur in action-reaction pairs. As all pairs act between objects in the system, the internal forces always sum to zero:

Therefore, the net force acting on a system is the sum of the external forces acting on it.

Momentum of components of a systemInternal forces cannot change the momentum of a system.

However, the momenta of pieces of the system may change.

An example of internal forces moving components of a system:

With no net external force:

Kinetic Energy of a SystemAnother example of internal forces moving components of a system:

The initial momentum equals the final (total) momentum.

But the final Kinetic Energy is very large

50

Birth of the neutrino

Beta decay fails momentum conservation?

Pauli “fixes” it with a new ghost-like, undetectable particle

Bohr scoffs

First detection 1956