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Chapter 7 Work and Kinetic Energy

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Chapter 7

Work and Kinetic Energy

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

Work Done by a Constant Force

The definition of work, when the force is parallel to the displacement:

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

2

Convenient notation: the dot product

The work can also be written as the dot product of the force and the displacement:

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

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

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, thenW < 0. Note that the work done on the ball is negative, and its speed decreases.

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

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?

T

v

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?

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, thenW = 0.

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

N

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?

Fgx = Fg sinθ

W = Fd = Fgx L = (Fg sinθ) (h / sinθ)

W = Fd = Fgx h

a

W = mgh

h = L sinθ

Fg

a

W = Fg h = mgh

Fg

h

h

θ

Work by gravity

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

Path doesn’t matter when asking “how much work did gravity do?” Only the change in height!

Motion and energy

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

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:

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)

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

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.

Work Done by a Variable Force

We can interpret the work done graphically:

Work Done by a Variable Force

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

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

Chapter 8

Potential Energy and Conservation of Energy

Recall

Work is the force directed along a displacement:

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

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

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?

T

N

f

displacement

mg

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?

Any force not perpendicularto the motion will do work:

N doesno work

T doespositivework

f does negative work

mg does negative work

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?

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?

a) quarter as much

b) half as much

c) the same

d) twice as much

e) four times as much

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 thegreater masshas thegreater

KE, which istwiceas big for the heavy stone.

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”

Power

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

a)energy

b) power

c) current

d) voltage

e) none of the above

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

When you pay the electric company by the kilowatt-hour, what are you actually paying for?

We have defined: Power = energy/ time

So we see that:Energy = power× time

This means that the unit ofpower× time (watt-hour) is a unit ofenergy !!

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?

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

(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 N

F = -250 Ny

(b) massless pully!

Upper:

Tch - 2Trope = 0

Tch = 500 N

Lower:

Tch -2Trope =0 Tch = 500 N

Mechanical Advantage!

What about work?

(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?

Trope = 250 N

Tchain = 500 N

F = -250 Ny

(a) P = Fv = 500 N * vbox

(b) P = Fv = 250 N * vhand

vhand = 2 vbox!

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!

N

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?

Fgx = Fg sinθ

W = Fd = Fgx L = (Fg sinθ) (h / sinθ)

W = Fd = Fgx h

a

W = mgh

h = L sinθ

Fg

a

W = Fg h = mgh

Fg

h

h

θ

Work by gravity

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

Path doesn’t matter when asking “how much work did gravity do?” Only the change in height!

Condition:Fcp > mg at top of loop

Fcp = mv2/r = mg

v2 = gr

KE = mv2 / 2 = mgr/2

Gravity must provide this energy

Wg = mgh = KE

h = r/2

above the top of the loop!

Application: ball on a track

how high must I place the ball so that it can complete a 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

Work done by gravity on a closed path is zero

Work done by friction on a closed path is not zero

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

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

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)

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.

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

Potential Energy

The 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)

Gravitational Potential Energy

Q: What does “UG = 0” mean?

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

Work Done by a Variable Force

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

on the spring

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

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:

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?

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?

Because your vertical position (height) changes by the same amount in each case, the gain in potential energy is the same.

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

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

Is it possible for the gravitational potential energy of an object to be negative?

a) yes

b) no

Sign of the Energy

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.

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

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

KE and PE

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

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

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.

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)

It is useful to define the mechanical energy:

Then:

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

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:

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 energy

total 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...

a)half the height

b) the same height

c) < 2 times the height

d) twice the height

e) four times the height

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

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?

- 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!

a) 4 m/s

b) 5 m/s

c) 6 m/s

d) 7 m/s

e) 25 m/s

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

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?

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

PE = KE = m(4)2

When starting from 3 m/s, the

final KE is:

KEf= KEi + KE

= m(3)2 + m(4)2

= m(25)

= m(5)2

Potential Energy Curves

The 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

Potential Energy for a Spring

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

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

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.

8-4 Work Done by Nonconservative Forces

In this example, the nonconservative force is water resistance:

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