chapter 3 energy and conservation laws. 2 conservation laws the most fundamental ideas we have in...

Chapter 3

Energy and Conservation Laws

2

Conservation laws

The most fundamental ideas we have in physics are conservation laws. Statements telling us that some quantity

does not change. Conservation of mass states:

The total mass of an isolated system is constant.

To apply these, we must define a “system.”

3

Conservation laws, cont’d

A system is just a collection of objects we decide to treat at one time. The tanker and fighter can represent a

system. The fuel leaving

the tanker goes into the fighter:mass is conserved.

4

Linear momentum

Linear momentum is defined as the product of an object’s mass and its velocity.

We typically just say momentum.

linear momentum mass velocity

p mv

5

Linear momentum, cont’d

Momentum is a measure of an object’s state of motion. Consider an object whose momentum is

1 kg·m/s This could be a 0.005 kg bullet traveling at 200

m/s. This could be a 0.06 kg tennis ball traveling at

16.7 m/s.

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Newton’s 2nd law is closely related to momentum. The net external force acting on an object

equals the rate of change of linear momentum:

change in momentumforce

change in time

pF

t

7


How is this related to F = ma?

So, F = ma holds only if the object’s mass remains constant. Not a rocket. Not really a car but its close enough.

mvp vF m ma

t t t

8

ExampleExample 3.1Let’s estimate the average force

on a tennis ball as it is served. The ball’s mass is 0.06 kg and it leaves the racquet with a speed of 40 m/s. High-speed photography indicates that the contact time is about 5 milliseconds.

9

ANSWER:

The problem gives us:

The force is:

ExampleExample 3.1

0.06 kg

0 m/s

40 m/s

0.005 s

i

f

m

v

v

t

0.06 kg 40 m/s

0.005 s480 N 108 lb

mvF

t

10


This tells why we must exert a force to stop an object or get it to move. To stop a moving object, we have to bring its

momentum to zero. To start moving an object, we have to impart

some momentum to it.

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It also tells us that we can change the momentum using various forces and time intervals:

Use a large force for a short time, or Use a small force for a long time.

p F t

12

Conservation of linear momentum The Law of Conservation of Linear

Momentum states:

The total linear momentum of an isolated system is constant.

Isolated implies no external force:

0 0p

F pt

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Conservation of linear momentum, cont’d

This law helps us deal with collisions. If the system’s momentum can not change,

the momentum before the collision must equal that after the collision.

14


We can write this as:

To study a collision: Add the momenta of the objects before the

collision. Add the momenta after the collision. the two sums must be equal.

before afterp p

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ExampleExample 3.2

A 1,000 kg car (car 1) runs into the rear of a stopped car (car 2) that has a mass of 1,500 kg. Immediately after the collision, the cars are hooked together and have a speed of 4 m/s. What was the speed of car 1 just before the collision?

16

ANSWER:


The momentum before:

The momentum after:

ExampleExample 3.2

1

2

1,000 kg

1,500 kg

4 m/sf

m

m

v

before 1 1 11,000 kgp m v v

after 1 2 2 2,500 kg 4 m/sp m m v

17

ANSWER:

Conserving momentum

ExampleExample 3.2

1

1

1,000 kg 2,500 kg 4 m/s

2,500 kg4 m/s

1,000 kg

10 m/s

v

v

18

DISCUSSION:

Both cars together have more mass than just car 1.

Since both move away at 4 m/s, the lighter car 1 must have a greater speed before the collision.

ExampleExample 3.2

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How do rockets work? The exhaust exits the rocket at high speed.

We need high speed because the gas has little mass.

The rocket moves in the opposite direction. Not as fast as the

gas b/c more mass.

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Work

Imagine using a lever to lift a heavy object. The lever allows us to exert less force than

the object actually weighs.

This sounds like “free money.”

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Work, cont’d

There’s a catch: We have to apply our force through a greater

distance than the rock moves. So there must be

some connection between force and distance.

22

Work, cont’d

The force multiplied by the distance moved is the same for both:

input output

3 3F d F d

23

Work, cont’d

We have the same situation for placing a barrels on a loading dock:

2 2

2 2

F d Fd

F d Fd

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Work, cont’d

Work is defined as the product of force and the distance through which the force moves an object in the direction of the force.

work w Fd

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Work, cont’d

The units of work: Metric

SI: joule (J = N·m), erg (= 10-7 J), calorie (cal = 4.186 J), kilowatt-hour (kWh = 0.278 J).

English: foot-pound (ft·lb), British thermal unit (Btu).

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Work, cont’d

From the definition of work:

1 joule (1 newton) (1 meter)

1 J 1 N m

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ExampleExample 3.3

Because of friction, a constant force of 100 newtons is needed to slide a box across a room. If the box moves 3 meters, how much must be done?

28

ANSWER:


The required work is:

ExampleExample 3.3

100 N

3 m

F

d

100 N 3 m

300 J

W Fd

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Work, cont’d

Recall that force is a vector. Involves magnitude and direction.

Work is just that part of the force in the direction of the displacement. Work is not a vector — it’s a scalar.

But the sign of the work does depend on the relative directions.

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Work, cont’d

If the force and distance are in the same direction, the force does positive work.

If the force and distance are in the opposite direction, the force does negative work.

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Work, cont’d

If the force is not in the direction of the direction, the force does no work. The string’s tension is toward the center of

the circle. The ball moves along

the circle’s circumference.

So, the tension does no work.

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Work, cont’d

You do positive work (in the physics-sense) when you lift the create.

You do NO work (in the

physics-sense) when you carry the crate.

You do negative work when you set the crate down.

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Work, cont’d

When you throw or catch a ball, you do work on the ball. Your hand exerts a force on the ball. You exert that force through the throwing or

catching distance. If you’re strong, you

don’t need the same distance because of the larger force.

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ExampleExample 3.4

Let’s say that the barrel has a mass of 30 kg and that the height of the dock is 1.2 meters. How much work would you do when lifting the barrel?

35

ANSWER:


The required work is:

ExampleExample 3.4

30 kg

1.2 m

m

d

230 kg 9.8 m/s 1.2 m

353 J

W Fd mgd

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DISCUSSION:

You would do the same amount of work rolling the barrel up the ramp.

You would only have to exert a force of 150 N instead of the entire 300 N.

But you have to exert that smaller force over a distance of 2.4 m.

ExampleExample 3.4

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ExampleExample 3.5

In Example 2.2 we used Newton’s 2nd law to compute the force needed to accelerate a 1,000-kg car from 0 to 27 m/s in 10 seconds. The answer was 2,700 N. How much work is done?

38

ANSWER:


To find work we use:

But we need the distance the car moved.

ExampleExample 3.5

27 m/s

1,000 kg

10 s

2,700 N

v

m

t

F

W Fd

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ANSWER:

Recall that

The work is

ExampleExample 3.5

2,700 N 135 m

364,500 J

W Fd

2 21 12 2

212

27 m/s10 s 135 m.

10 s

vd at t

t

40

DISCUSSION:

In reality, this is smaller than the energy the engine must generate.

The engine must overcome its internal friction — a loss of energy.

Most cars are about 30% efficient.

So you need (364 kJ)/(0.3) = 1.2 MJ to actually accelerate this car.

ExampleExample 3.5

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Energy

Energy is defined as the measure of a system’s ability to do work. We use the symbol E to represent energy. Energy has the same units as work:

Joule for SI, ft·lb for English.

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Energy, cont’d

There are various types of energy. Kinetic energy is the energy associated with

an object’s motion. We use the symbol KE.

Potential energy is energy associated with the system’s position or orientation.

We use the symbol PE.

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Kinetic energy

The formula for kinetic energy is:

m is the object’s mass. v is the object’s speed.

212KE mv

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ExampleExample 3.6

In Example 3.5 we computed the work that is done on a 1,000-kg car as it accelerates from 0 to 27 m/s. Find the car’s kinetic energy when it is traveling at 27 m/s.

45

ANSWER:


The kinetic energy is:

ExampleExample 3.6

27 m/s

1,000 kg

v

m

212

212 1000 kg 27 m/s

364,500 J

KE mv

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DISCUSSION:

This equals the (ideal) work required to get the car up to speed.

We could determine how much work is required by finding the kinetic energy of the car.

This is the idea of energy conservation.

ExampleExample 3.5

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Gravitational potential energy

Gravitational potential energy equals the work done by the gravity. If you lift an object, you must apply a force at

least equal to the object’s weight:

Lifting it through a distance d, the work is

F mgPE W Fd mgd

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Gravitational potential energy, cont’d

Note that we only deal with the distance through which the object moves. The brick has 14.7 J

of PE relative to the table top.

It has 44.1 J of PE relative to the floor.

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Gravitational potential energy, cont’d

Where you say an object has zero PE is arbitrary. We only care about the change.

Let’s say the ground is at zero PE.

In the hole, the ball has negative PE.

It is below the reference level.

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ExampleExample 3.7

A 3-kg brick is lifted to a height of 0.5 meters above a table that is 1.0-m tall. Find the gravitational potential energy relative to the table and the floor.

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ANSWER:


The PE relative to the table is:

ExampleExample 3.7

above table

above floor

3 kg

0.5 m

1.5 m

m

h

h

rel to table above table

23 kg 9.8 m/s 0.5 m

14.7 J

PE mgd mgh

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ANSWER:

The PE relative to the table is:

ExampleExample 3.7

rel to floor above floor

23 kg 9.8 m/s 1.5 m

44.1 J

PE mgd mgh

53

DISCUSSION:

The PE is meaningless without specifying the reference level.

If we drop the brick:It has a lower speed upon impact with the table than it

does with the floor.

The brick has more energy to convert to KE relative to the floor.

ExampleExample 3.7

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Internal energy

Internal energy is that energy associated with the internal structure of the object. A hot rock has more internal energy than a

cold rock. Lifting either increases the external PE but not

the internal energy.

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Internal energy, cont’d

Internal energy can be used to do work. If you turn on an electric stove, the internal

energy of the filament increases. This energy can be used to boil water.

You increase the internal energy of the water. The produced steam can be used to power a

turbine/generator.

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Conservation of energy

The Law of Conservation of Energy: energy cannot be created or destroyed. The total energy of an isolated system is

constant. The energy of the Universe is constant.

Energy can only be transformed from one form to another.

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Conservation of energy, cont’d

Here are examples of transforming energy:

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If the energy of an isolated system is constant, the energy before an event must the same as the energy after an event.

total energy before total energy after

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To deal with energy conservation, we need the total energy:

total energy:

constantE KE PE

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We can use this to solve problems. The initial energy of the ball is all PE:

The final energy is KE:

initial 0E KE PE PE

mgd

final

212

E KE PE

mv

61


Conserving energy:

Solving for the speed:

initial final

212

E E

mgd mv

2122 2

2

v gd

v gd

62

ExampleExample 3.8

In 2003, a man went over Horseshoe Falls, part of Niagara Falls, and survived. The height of the falls is about 50 meters. Estimate the speed of the man when he hit the water at the bottom of the falls.

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ANSWER:


The initial energy of the man is:

The final energy of the man is:

ExampleExample 3.8

50 mh

i i iE KE PE mgh

212f f fE KE PE mv

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ANSWER:

Conserve energy:

ExampleExample 3.8

212

212

i fE E

mgh mv

gh v

65

ANSWER:

Solving for the speed gives:

ExampleExample 3.8

2 2 2

2

2 9.8 m/s 50 m 980 m /s

31.3 m/s.

v gh

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DISCUSSION:

The speed does not depend on the man’s mass.

If you tried this, you’d hit the bottom with the same speed.

This is obviously ideal:We do not consider air resistance. That would convert some

of his KE into heat and sound. The real speed would be slower.

ExampleExample 3.8

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Conserving energy:

Or solve for the distance:

initial final

212

E E

mgd mv

212

2

2

vgd

g g

vd

g

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Example

I toss a 0.06-kg tennis ball straight up. When it leaves my hand, it has a speed of 20 m/s. Find how high the ball rises.

69

ANSWER:


The initial energy of the ball is:

The final energy of the ball is:

Example

0.06 kg

v 20 m/s

m

212i i iE KE PE mv

f f fE KE PE mgh

70

ANSWER:

Conserve energy:

Example

212

212

i fE E

mgh mv

gh v

71

ANSWER:

Solving for the height gives:

Example

22

2

20 m/s

2 2 9.8 m/s

20.4 m.

vh

g

72

DISCUSSION:

Again, since we neglect air resistance the height would be the same if I tossed a tennis ball, bowling ball or brick.

Example

73


We can understand a roller-coaster by as an example of energy conservation.

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Collisions

A collision is when two objects interact to: exchange energy, and/or exchange momentum.

75

Collisions, cont’d

An elastic collision is one in which the total kinetic energy remains constant.

An inelastic collision is one in which the total kinetic energy does not remain constant.

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The first case shows an elastic collision. The second

case shows a perfectly inelastic collision.

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In general, the collision does not have to involve physical contact.

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ExampleExample 3.9

Recall the automobile collision analyzed in Example 3.2. Compare the amounts of kinetic energy in the system before and after the collision.

79

ANSWER:


The initial KE is:

The final KE is:

ExampleExample 3.9

1

1

2

2

1,000 kg

10 m/s

2,500 kg

4 m/s

m

v

m

v

212 1000 kg 10 m/s 50 kJiKE

212 2500 kg 4 m/s 20 kJfKE

80

DISCUSSION:

30 kJ of energy was lost during the collision.Converted to heat and sound, used to deform the cars,

etc.

Since KE was lost, the collision was inelastic.It is perfectly inelastic because they stuck together.

ExampleExample 3.8

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Power

Energy tells you about the position and speed.

But it contains no information about time. Power is the rate at which energy is

transferred or transformed. The rate of doing work.

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Power, cont’d

Mathematically,

In SI units, we use the watt: 1 W = 1 J/s.

In English units, we use horsepower: 1 hp = 746 W.

Power: work E

Ptime t

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ExampleExample 3.10

In Examples 2.2 and 3.5, we computed the acceleration, force and work for a 1,000-kg car that goes from 0 to 27 m/s in 10 s. We can now determine the required power output of the engine. What is its value?

84

ANSWER:


The power is:

ExampleExample 3.10

1,000 kg

0 m/s

27 m/s

10 s

i

f

m

v

v

t

364,500 J36,450 W

10 s

EP

t

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DISCUSSION:

This is the ideal value.

For a 30% efficient car to reach 27 m/s, to overcome the losses due to friction the engine must generate a power of

ExampleExample 3.10

36,450 W122,000 W

0.30

86

Rotation and angular momentum We defined linear momentum as

Consider an object moving in a circle.

p mv

87

Rotation and angular momentum, cont’d

If we shorten the string, the object’s speed increases. There must

be some relation between the speed and radius.

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Angular momentum is the momentum of an object moving in a circle.

For a point particle, e.g., a ball, the angular momentum is

mvr

89


Conservation of Angular Momentum states that the total angular momentum of an isolated system is constant. For our ball on a string:

initial: i imv r

final: f fmv r

90


By conservation of momentum:

The right hand side must equal the left hand side.

So as rf gets smaller, vf must get larger.

i i f f

i i f f

mv r mv r

v r v r

91


This also holds for orbits. As the satellite

gets to B, it must move faster than at A.

92

Example

You spin a ball attached to the end of a 1.0-meter string with a speed of 10 m/s. Find the ball’s speed as you shorten the string to 10 centimeters.

93

ANSWER:


From angular momentum conservation:

Example

10 m/s

1 m

0.10 m

i

i

f

v

r

r

ii i f f f i

f

rv r v r v v

r

94

ANSWER:

Inserting the numbers:

Example

1.0 m1 m/s

0.10 m

10 m/s.

if i

f

rv v

r

95

DISCUSSION:

Just a figure skater pulls in her arms, the ball’s speed increases as the string’s length shortens.

Example

chapter 3 energy and conservation laws. 2 conservation laws the most fundamental ideas we have in...

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