Chapter 7

Work and Kinetic Energy

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

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Work Done by a Constant ForceThe definition of work, when the force is parallel to the displacement:

SI unit: newton-meter (N·m) = joule, J

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3

Convenient notation: the dot productThe work can also be written as the dot product of the force and the displacement:

vector “dot” operation: project one vector onto the other

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In a baseball game, the

catcher stops a 90-mph

pitch. What can you say

about the work done by the

catcher on the ball?

a) catcher has done positive work

b) catcher has done negative work

c) catcher has done zero work

Play Ball!

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In a baseball game, the

catcher stops a 90-mph

pitch. What can you say

about the work done by the

catcher on the ball?

a) catcher has done positive work

b) catcher has done negative work

c) catcher has done zero work

The force exerted by the catcher is opposite in direction to the displacement of the ball, so the work is negative. Or using the definition of work (W = F (Δr)cos ), because = 180º, then W < 0. Note that the work done on the ball is negative, and its speed decreases.

Play Ball!

Follow-up: What about the work done by the ball on the catcher?

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Tension and Work

a) tension does no work at all

b) tension does negative work

c) tension does positive work

A ball tied to a string is

being whirled around in

a circle with constant

speed. What can you

say about the work

done by tension?

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Tension and Work

a) tension does no work at all

b) tension does negative work

c) tension does positive work

A ball tied to a string is

being whirled around in

a circle with constant

speed. What can you

say about the work

done by tension?

v

T

No work is done because the force

acts in a perpendicular direction to

the displacement. Or using the

definition of work (W = F (Δr)cos ),

= 90º, then W = 0.

Follow-up: Is there a force in the direction of the velocity?

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Work by gravity

Fg a

h

A ball of mass m drops a distance h. What is the total work done on the ball by gravity?

N

Fg

A ball of mass m rolls down a ramp of height h at an angle of 45o. What is the total work done on the ball by gravity?

h

a

θ

Fgx = Fg sinθ

h = L sinθ

W = Fd = Fgx L = (Fg sinθ) (h / sinθ)

W = Fg h = mgh

W = Fd = Fgx h

W = mgh

Path doesn’t matter when asking “how much work did gravity do?”

Only the change in height!

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Motion and energyWhen positive work is done on an object, its speed increases; when negative work is done, its speed decreases.

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Kinetic Energy

As a useful word for the quantity of work we have done on an object, thereby giving it motion, we define the kinetic energy:

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Work-Energy Theorem

Work-Energy Theorem: The total work done on an object is equal to its change in kinetic energy.

(True for rigid bodies that remain intact)

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By what factor does the

kinetic energy of a car

change when its speed is

tripled?

a) no change at all

b) factor of 3

c) factor of 6

d) factor of 9

e) factor of 12

Kinetic Energy I

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By what factor does the

kinetic energy of a car

change when its speed is

tripled?

a) no change at all

b) factor of 3

c) factor of 6

d) factor of 9

e) factor of 12

Because the kinetic energy is mv2, if the speed increases by a factor

of 3, then the KE will increase by a factor of 9.

Kinetic Energy I

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Work Done by a Variable Force

We can interpret the work done graphically:

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Work Done by a Variable Force

We can then approximate a continuously varying force by a succession of constant values.

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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 the spring is

Potential Energy and

Conservation of Energy

Chapter 8

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Recall

The total work done on an object is equal to its change in kinetic energy:

Work is the force directed along a displacement:

Lets wrap up our discussion of work and kinetic energy...

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Force and Work

a) one force

b) two forces

c) three forces

d) four forces

e) no forces are doing work

A box is being pulled up a

rough incline by a rope

connected to a pulley. How

many forces are doing work on

the box?

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Force and Work

N

f

T

mg

displacementAny force not perpendicular

to the motion will do work:

N does no work

T does positive work

f does negative work

mg does negative work

a) one force

b) two forces

c) three forces

d) four forces

e) no forces are doing work

A box is being pulled up a

rough incline by a rope

connected to a pulley. How

many forces are doing work on

the box?

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Free Fall I

a) quarter as much

b) half as much

c) the same

d) twice as much

e) four times as much

Two stones, one twice the mass of

the other, are dropped from a cliff.

Just before hitting the ground,

what is the kinetic energy of the

heavy stone compared to the light

one?

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Consider the work done by gravity to make the stone fall

distance d:

KE = Wnet = F d cos KE = mg d

Thus, the stone with the greater mass has the greater

KE, which is twice as big for the heavy stone.

Free Fall I

a) quarter as much

b) half as much

c) the same

d) twice as much

e) four times as much

Two stones, one twice the mass of

the other, are dropped from a cliff.

Just before hitting the ground,

what is the kinetic energy of the

heavy stone compared to the light

one?

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Power

Power is a measure of the rate at which work is done:

SI unit: J/s = watt, W

1 horsepower = 1 hp = 746 W

if work is energy, then you would think of “energy flow”

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Power

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Power

If an object is moving at a constant speed in the face of friction, gravity, air resistance, and so forth, the power exerted by the driving force can be written:

Question: what is the total work per unit time done on this object (by all

forces)?

This expression, P = Fv, gives the instantaneous power applied, even if the object is not moving at constant

speed

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a) energy

b) power

c) current

d) voltage

e) none of the above

Electric Bill

When you pay the electric

company by the kilowatt-hour,

what are you actually paying

for?

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We have defined: Power = energy / time

So we see that: Energy = power × time

This means that the unit of power × time

(watt-hour) is a unit of energy !!

Electric Bill

When you pay the electric

company by the kilowatt-hour,

what are you actually paying

for?

a) energy

b) power

c) current

d) voltage

e) none of the above

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A block rests on a horizontal frictionless surface. A string is attached to the block, and is pulled with a force of 45.0 N at an angle above the horizontal, as shown in the figure. After the block is pulled through a distance of 1.50 m, its speed is 2.60 m/s, and 50.0 J of work has been done on it. (a) What is the angle (b) What is the mass of the block?

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The pulley system shown is used to lift a 52 kg crate. Note that one chain connects the upper pulley to the ceiling and a second chain connects the lower pulley to the crate. Assuming the masses of the chains, pulleys, and ropes are negligible, determine (a) the force F required to lift the crate with constant speed, and(b) the tension in two chains

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(a) the force F required to lift the crate with constant speed, and(b) the tension in two chains

(a) constant velocity, a=0, so net force =0.

2T - (52kg)(9.8m/s2) = 0

T = 250 NF = -250 Ny

(b) massless pully!Upper:Tch - 2Trope = 0 Tch = 500 N

Lower:Tch -2Trope =0 Tch = 500 N

Mechanical Advantage!

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(a) how much power is applied to the box by the chain?(b) how much power is applied on the rope by the applied force?

What about work?

Trope = 250 NTchain = 500 NF = -250 Ny

(a) P = Fv = 500 N * vbox

(b) P = Fv = 250 N * vhand

vhand = 2 vbox!

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You can explain what is about to happen, in terms of force (elastic band is about to pull on the rock, accelerating it toward our camera lens)...

...or using work/energy: work has been done on the elastic band, and it now “contains” energy.

Energy wants to be “free” - in fact, physics can be described in terms of the rules that govern stored energy

Also: energy really is a “thing”. E = mc2 relates mass to energy, and stored energy counts... it’s not just an accounting rule!

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Work by gravity

Fg a

h

A ball of mass m drops a distance h. What is the total work done on the ball by gravity?

N

Fg

A ball of mass m rolls down a ramp of height h at an angle of 45o. What is the total work done on the ball by gravity?

h

a

θ

Fgx = Fg sinθ

h = L sinθ

W = Fd = Fgx L = (Fg sinθ) (h / sinθ)

W = Fg h = mgh

W = Fd = Fgx h

W = mgh

Path doesn’t matter when asking “how much work did gravity do?”

Only the change in height!

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Application: ball on a trackhow high must I place the ball so

that it can complete a loop?

Condition: Fcp > mg at top of loop

Fcp = mv2/r = mg v2 = gr

KE = mv2 / 2 = mgr/2Gravity must provide this energy Wg = mgh = KE

h = r/2 above the top of the loop!

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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

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Work done by gravity on a closed path is zero

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Work done by friction on a closed path is not zero

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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

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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.

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Potential EnergyConsider 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

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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)

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Gravitational Potential Energy

Q: What does “UG = 0” mean?

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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

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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:

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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?

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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?

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Is it possible for the

gravitational potential

energy of an object to

be negative?

a) yes

b) no

Sign of the Energy

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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

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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

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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

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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

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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:

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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...

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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

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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!

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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

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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

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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?59

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Potential Energy for a Spring

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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

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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

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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

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8-4 Work Done by Nonconservative Forces

In this example, the nonconservative force is water resistance:

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