energy and its conservation learn that energy is a property of an object that can change the...

Energy and Its Conservation

Learn that energy is a property of an object that can change the object’s position, motion, or its environment.

Learn that energy changes from one form to another, and that the total amount of energy in a closed system remains constant.

Chapter

11

In this chapter you will:

Table of Contents

Chapter 11: Energy and Its Conservation

Section 11.1: The Many Forms of Energy

Section 11.2: Conservation of Energy

Chapter

11

Assignments

Read Chapter 11.

HW 11.A: p.307: 54,55,57,59,62,63,65 – 68.

HW 11.B: p.308: 73-77, 79, 82.

Energy Study Guide is due before the test.

The Many Forms of Energy

Identify different forms of energy

Define kinetic energy and gravitational potential energy.

Explain the relationship between work and kinetic energy.

Solve problems with potential and kinetic energy.

In this section you will:

Section

11.1


The word energy is used in many different ways in everyday speech.

Some fruit-and-cereal bars are advertised as energy sources. Athletes use energy in sports.

Companies that supply your home with electricity, natural gas, or heating fuel are called energy companies.

Section

11.1

Scientists and engineers use the term energy much more precisely.

energy – the ability to produce a change in itself or its surroundings.

The SI unit of energy is the joule (J).


Work causes a change in the energy of a system.

That is, work transfers energy between a system and the external world.

Work-Energy Theorem – the work done on an object is equal to its change in kinetic energy.

W = KE = KEf – KEi

Section

11.1


Kinetic Energy

Section

11.1

The kinetic energy is proportional to the object’s mass.

A 7.26-kg shot put thrown through the air has much more kinetic energy than a 0.148-kg baseball with the same velocity, because the shot put has a greater mass.

where m is the mass of the object and v is the

magnitude of its velocity.

kinetic energy (KE) – the energy of motion

Anything that is moving (v 0) has KE.

KE = ½ mv2


The kinetic energy of an object is also proportional to the square of the object’s velocity.

A car speeding at 20 m/s has four times the kinetic energy of the same car moving at 10 m/s.

Kinetic energy also can be due to rotational motion.

If you spin a toy top in one spot, does it have kinetic energy?

Kinetic Energy

Section

11.1


You might say that it does not because the top is not moving anywhere.

However, to make the top rotate, someone had to do work on it.

Therefore, the top has rotational kinetic energy. This is one of the several varieties of energy.

The kinetic energy of a spinning object is called rotational kinetic energy.

Rotational Kinetic Energy

Section

11.1

Rotational kinetic energy depends on an object’s moment of

inertia and the object’s angular velocity.


A diver does work as she is diving off the diving board. This work produces both linear and rotational kinetic energies.

When the diver’s center of mass moves as she leaps, linear kinetic energy is produced. When she rotates about her center of mass, rotational kinetic energy is produced.

Because she is moving toward the water and rotating at the same time while in the tuck position, she has both linear and rotational kinetic energy.

When she slices into the water, she has linear kinetic energy.

Kinetic Energy

Section

11.1


Kinetic Energy

Throw a ball through the air. Your hand does work on the ball because you apply a force through a distance. The work done on the ball results in the ball’s kinetic energy.

Section

11.1

What happens when you catch a ball?

Before hitting your hands or glove, the ball is moving, so it has kinetic energy.

In catching it, you exert a force on the ball in the direction opposite to its motion.

Therefore, you do negative work on it, causing it to stop.

Now that the ball is not moving, it has no kinetic energy.


Imagine a group of boulders high on a hill.

These boulders have been lifted up by geological processes against the force of gravity; thus, they have stored energy.

In a rock slide, the boulders are shaken loose.

They fall and pick up speed as their stored energy is converted to kinetic energy.

In the same way, a small, spring-loaded toy, such as a jack-in-the-box, has stored energy, but the energy is stored in a compressed spring.

While both of these examples represent energy stored by mechanical means, there are many other means of storing energy.

Stored Energy

Section

11.1


Automobiles, for example, carry their energy stored in the form of chemical energy in the gasoline tank.

Energy is made useful or causes motion when it changes from one form to another.

There are different ways energy can be stored as potential energy.

Stored Energy

Section

11.1

potential energy (PE) – stored energy; energy of position


A pole-vaulter runs with a flexible pole and plants its end into the socket in the ground.

When the pole-vaulter bends the pole, as shown in the figure, some of the pole-vaulter’s kinetic energy is converted to elastic potential energy.

Section

11.1

Elastic Potential Energy


When a string on a bow is pulled, work is done on the bow, storing energy in it.

Thus, the energy of the system increases.

Identify the system as the bow, the arrow, and Earth.

When the string and arrow are released, energy is changed into kinetic energy.

The stored energy in the pulled string is called elastic potential energy, which is often stored in rubber balls, rubber bands, slingshots, and trampolines.

Elastic Potential Energy

Section

11.1


Albert Einstein recognized yet another form of potential energy: mass itself.

He said that mass, by its very nature, is energy.

This energy, E0, is called rest energy and is represented by the following famous formula.

Mass

Section

11.1

Rest Energy E0 = mc2

The rest energy of an object is equal to the object’s mass times the speed of light squared.


gravitational potential energy – energy stored as a result of the gravitational attraction between an object and the earth.

Gravitational potential energy is usually referred to simply as potential energy, or PE.

reference level - the place in a system where the PE is defined to be zero.

examples: the bottom of a mountain; a pendulum’s lowest point

It is convenient to pick a reference level so the PE is always positive; however it is possible for it to be negative.

Section

11.1


The formula for gravitational potential energy is:

PE = mgh or mgh

where m is mass in kilograms g is the acceleration due to gravity (9.8 m/s2)

h is the height above the reference level in meters

example: In an experiment, Galileo dropped a rock off the top of the Leaning Tower of Pisa. The rock had a mass of 0.5 kg and the tower is 20 m high. How much potential energy did the rock have before it was dropped?

PE = mgh = (0.5 kg) ( 9.8 m/s2) ( 20 m)

PE = 98 J

Section

11.1

The Many Forms of EnergySection

11.1

Potential Energy at Varying Locations


Gravitational Potential Energy

You lift a 7.30-kg bowling ball from the storage rack and hold it up to your shoulder. The storage rack is 0.610 m above the floor and your shoulder is 1.12 m above the floor.

a. When the bowling ball is at your shoulder, what is the bowling ball’s gravitational potential energy relative to the floor?

b. What is the bowling ball’s gravitational potential energy relative to the storage rack?

c. How much work was done by gravity as you lifted the ball from the rack to shoulder level?

Section

11.1


Sketch the situation.


11.1

Choose a reference level.



Section

11.1

Known:

m = 7.30 kg

hs rel r = 0.610 m (relative to the floor)

hs rel f = 1.12 m (relative to the floor)

g = 9.80 m/s2

Unknown:

PEs rel f = ?

PEs rel r = ?

Identify the known and unknown variables.


Set the reference level to be at the floor. Solve for the potential energy of the ball at shoulder level.


11.1

Substitute m = 7.30 kg, g = 9.80 m/s2, hs rel f = 1.12 m

PEs rel f = (7.30 kg)(9.80 m/s2)(1.12 m)

PEs rel f = mghs rel f

PEs rel f = 80.1 J

Set the reference level to be at the rack height. Solve for the height of your shoulder relative to the rack.

h = hs – hr


Solve for the potential energy of the ball.


11.1

PEs rel r = mgh

Substitute h = hs – hr

PE = mg (hs – hr)

Substitute m = 7.3 kg, g = 9.80 m/s2, hs = 1.12 m, hr = 0.610 m

PE = (7.30 kg)(9.80 m/s2)(1.12 m – 0.610 m)

PE = 36.5 J

The work done by gravity is the weight of the ball times the distance the ball was lifted.


11.1

Because the weight opposes the motion of lifting, the work is negative.

W = Fd

W = –(mg)h

W = – (mg) (hs – hr)


Substitute m = 7.30 kg, g = 9.80 m/s2, hs = 1.12 m, hr = 0.610 m

W = -(7.30 kg)(9.80 m/s2)(1.12 m – 0.610 m)

W = -36.5 J


Are the units correct?

The potential energy and work are both measured in joules.

Is the magnitude realistic?

The ball should have a greater potential energy relative to the floor than relative to the rack, because the ball’s distance above the reference level is greater.

Section

11.1




Section

11.1

Ch11_1_movanim


Consider the energy of a system consisting of an orange used by the juggler plus Earth.

The energy in the system exists in two forms: kinetic energy and gravitational potential energy.

Kinetic Energy and Potential Energy of a System

Section

11.1


At the beginning of the orange’s flight, all the energy is in the form of kinetic energy, as shown in the figure.

On the way up, as the orange slows down, energy changes from kinetic energy to potential energy.

At the highest point of the orange’s flight, the velocity is zero.

Thus, all the energy is in the form of gravitational potential energy.


Section

11.1


On the way back down, potential energy changes back into kinetic energy.

The sum of kinetic energy and potential energy is constant at all times because no work is done on the system by any external forces.

Section

11.1



In the figure, the reference level is the juggler’s hand.

That is, the height of the orange is measured from the juggler’s hand.

Thus, at the juggler’s hand, h = 0 m and PE = 0 J.

You can set the reference level at any height that is convenient for solving a given problem. In general, pick a reference level so your numbers stay positive.

Section

11.1

Reference Levels


Practice Problems p. 287: 1-3; p.291: 5-8

Section

11.1

Section Check

A boy running on a track doubles his velocity. Which of the following statements about his kinetic energy is true?

Question 1

Section

11.1

A. Kinetic energy will be doubled.

B. Kinetic energy will reduce to half.

C. Kinetic energy will increase by four times.

D. Kinetic energy will decrease by four times.

Answer: C

Answer 1

Reason: The kinetic energy of an object is equal to half times the mass of the object multiplied by the speed of the object squared.

Section

11.1 Section Check

21 =

2KE mv

Kinetic energy is directly proportional to the square of velocity. Since the mass remains the same, if the velocity is doubled, kinetic energy will increase by four times.

Section Check

If an object moves away from the Earth, energy is stored in the system as the result of the force between the object and the Earth. What is this stored energy called?

Question 2

Section

11.1

A. Rotational kinetic energy

B. Gravitational potential energy

C. Elastic potential energy

D. Linear kinetic energy

Answer: B

Answer 2

Reason: The energy stored due to the gravitational force between an object and the Earth is called gravitational potential energy.

Section

11.1 Section Check

Gravitational potential energy is represented by the equation, PE = mgh. That is, the gravitational potential energy of an object is equal to the product of its mass, the acceleration due to gravity, and the distance from the reference level. Gravitational potential energy is measured in joules (J).

Section Check

Two girls, Sarah and Susan, having same masses are jumping on a floor. If Sarah jumps to a greater height, what can you say about the gain in their gravitational potential energy?

Question 3

Section

11.1

A. Since both have equal masses, they gain equal gravitational potential energy.

B. Gravitational potential energy of Sarah is greater than that of Susan.

C. Gravitational potential energy of Susan is greater than that of Sarah.

D. Neither Sarah nor Susan possesses gravitational potential energy.

Answer: B

Answer 3

Reason: The gravitational potential energy of an object is equal to the product of its mass, the acceleration due to gravity, and the distance from the reference level.

Section

11.1 Section Check

Gravitational potential energy is directly proportional to height. As the masses of Sarah and Susan are the same, the one jumping to a greater height (Sarah) will gain greater potential energy.

PE = mgh

Section

11.1

Type of Energy Definition Example

energy of motion

energy that depends on an object’s moment of inertia and its angular velocity

vibration of air particles

due to breaking bonds between atoms in a molecule

release of photons

fusion – combining of atomsfission – splitting of atoms

energy of position; stored energy

energy stored in an Earth-object system as a result of gravitational attraction between the object and Earth

movement of electrons

energy stored in an object as a result of a change in its shape

makes temperature rise; usually due to friction

Einstein’s famous equation; potential energy of mass itself

Section

11.1

Type of Energy Definition Example

kinetic (KE) energy of motion throw a ball

rotational kinetic energy that depends on an object’s moment of inertia and its angular velocity

top

sound vibration of air particles speakers

chemical due to breaking bonds between atoms in a molecule

batteries, photosynthesis

light release of photons light bulb

nuclear fusion – combining of atomsfission – splitting of atoms

nuclear power plant, the Sun, nuclear weapons

potential energy of position; stored energy yo-yo

gravitational potential

energy stored in an Earth-object system as a result of gravitational attraction between the object and Earth

playground slide

electrical movement of electrons wiring in a building

elastic potential energy stored in an object as a result of a change in its shape

bow & arrow, slingshot

thermal makes temperature rise; usually due to friction rub hands together

rest energyE0 = mc2

Einstein’s famous equation; potential energy of mass itself

nuclear explosion

Conservation of Energy

Solve problems using the law of conservation of energy.

Analyze collisions to find the change in kinetic energy.

Define the Law of Conservation of Energy.

In this section you will:

Section

11.2


Consider a ball near the surface of Earth. The sum of gravitational potential energy and kinetic energy in that system is constant.

As the height of the ball changes, energy is converted from kinetic energy to potential energy, but the total amount of energy stays the same.


Section

11.2

The law of conservation of energy states that in a closed, isolated system, energy can neither be created nor destroyed; rather, energy is conserved.

closed system – no mass in or out

isolated system – not forces in our out

Under these conditions, energy changes from one form to another while the total energy of the system remains constant.


The sum of the kinetic energy and gravitational potential energy of a system is called mechanical energy.

Section

11.2

The mechanical energy of a system is equal to the sum of the kinetic energy and potential energy if no other forms of energy are present.

In any given system, if no other forms of energy are present, mechanical energy is represented by the following equation.

Mechanical Energy of a System E = KE+PE

Law of Conservation of Energy


Conservation of Mechanical Energy

Section

11.2

ch11_3_movanim


In the case of a roller coaster that is nearly at rest at the top of the first hill, the total mechanical energy in the system is the coaster’s gravitational potential energy at that point.

Suppose some other hill along the track were higher than the first one.

The roller coaster would not be able to climb the higher hill because the energy required to do so would be greater than the total mechanical energy of the system.

Roller Coasters

Section

11.2


Suppose you ski down a steep slope.

When you begin from rest at the top of the slope, your total mechanical energy is simply your gravitational potential energy.

Once you start skiing downhill, your gravitational potential energy is converted to kinetic energy.

Skiing

Section

11.2


As you ski down the slope, your speed increases as more of your potential energy is converted to kinetic energy.

In ski jumping, the height of the ramp determines the amount of energy that the jumper has to convert into kinetic energy at the beginning of his or her flight.

Skiing

Section

11.2


The simple oscillation of a pendulum also demonstrates conservation of energy.

The system is the pendulum bob and Earth. Usually, the reference level is chosen to be the height of the bob at the lowest point, when it is at rest. If an external force pulls the bob to one side, the force does work that gives the system mechanical energy.

Pendulums

Section

11.2


When the bob is at the lowest point, its gravitational potential energy is zero, and its kinetic energy is equal to the total mechanical energy in the system.

Note that the total mechanical energy of the system is constant if we assume that there is no friction.

Pendulums

Section

11.2


The oscillations of a pendulum eventually come to a stop, a bouncing ball comes to rest, and the heights of roller coaster hills get lower and lower.

Where does the mechanical energy in such systems go?

Any object moving through the air experiences the forces of air resistance.

In a roller coaster, there are frictional forces between the wheels and the tracks.

Loss of Mechanical Energy

Section

11.2


When a ball bounces off of a surface, all of the elastic potential energy that is stored in the deformed ball is not converted back into kinetic energy after the bounce.

Some of the energy is converted into thermal energy and sound energy.

As in the cases of the pendulum and the roller coaster, some of the original mechanical energy in the system is converted into another form of energy within members of the system or transmitted to energy outside the system, as in air resistance.

Loss of Mechanical Energy

Section

11.2

The total energy of the molecules is called thermal energy. Thermal energy causes the temperature of objects to rise slightly.


During a hurricane, a large tree limb, with a mass of 22.0 kg and at a height of 13.3 m above the ground, falls on a roof that is 6.0 m above the ground.

Section

11.2

a. Ignoring air resistance, find the kinetic energy of the limb when it reaches the roof.

b. What is the speed of the limb when it reaches the roof?


Sketch the initial and final conditions. Choose a reference level.

Section

11.2 Conservation of Energy



Identify the known and unknown variables.

Section

11.2

Unknown:

PEi = ? KEf = ?

PEf = ? vf = ?

Known:

m = 22.0 kg g = 9.80 m/s2

hlimb = 13.3 m Δvi = 0.0 m/s2

hroof = 6.0 m KEi = 0.0 J


Conservation of EnergySection

11.2

a. Set the reference level as the height of the roof. Solve for the initial height of the limb relative to the roof.

h = hlimb – hroof


Substitute hlimb = 13.3 m, hroof = 6.0 m

= 13.3 m – 6.0 m

= 7.3 m


11.2

Identify the initial potential energy of the limb.

PEi = mgh

Substitute m = 22.0 kg, g = 9.80 m/s2, h = 7.3 m

PEi = (22.0 kg) (9.80 m/s2) (7.3 m)

= 1.6×103 J


The tree limb is initially at rest.


11.2

Solve for the initial kinetic energy of the limb.

KEi = 0.0 J


PEf = 0.0 J because h = 0.0 m at the reference level.

The kinetic energy of the limb when it reaches the roof is equal to its initial potential energy because energy is conserved.

KEf = PEi

KEf = 1.6×103 J


11.2

b. Solve for the speed of the limb.

2

ff 2

1KEKE

m

KEv f

f

2


Substitute KEf = 1.6×103 J, m = 22.0 kg

32(1.6 10 J)

22.0 kg

m/s12


Are the units correct?

Velocity is measured in m/s and energy is measured in kg·m2/s2 = J.

Do the signs make sense?

KE and the magnitude of velocity are always positive.

Section

11.2



A collision between two objects, whether the objects are automobiles, hockey players, or subatomic particles, is one of the most common situations analyzed in physics.

Because the details of a collision can be very complex during the collision itself, the strategy is to find the motion of the objects just before and just after the collision.

Analyzing Collisions

Section

11.2


What conservation laws can be used to analyze such a system?

If the system is closed and isolated, then momentum and energy are conserved.

However, the potential energy or thermal energy in the system may decrease, remain the same, or increase.

Therefore, you cannot predict whether or not kinetic energy is conserved.


Section

11.2


Consider the collision shown in the figure. “Before” is the same in all three cases.

Section

11.2



In case 1, the momentum of the system before and after the collision is represented by the following:


Section

11.2

pi = pCi+pDi = (1.00 kg)(1.00 m/s)+(1.00 kg)(0.00 m/s)

= 1.00 kg·m/s

Pf = pCf+pDf = (1.00 kg)(–0.20 m/s)+(1.00 kg)(1.20 m/s)

= 1.00 kg·m/s

Thus, in case 1, the momentum is conserved.



Section

11.2

Is momentum conserved in case 2 and in case 3? (Yes)


Consider the kinetic energy of the system in each of these cases.

The kinetic energy of the system before and after the collision is represented by the following equations:


Section

11.2

J50.0

J74.0

KE KE2 2

Ci Di1 1

1.00kg 1.00m/s 1.00kg 0.00m/s2 2

KE KE2 2

Cf Df1 1

1.00kg 0.20m/s 1.00kg 1.20m/s2 2


In case 1, the kinetic energy of the system increased.

If energy in the system is conserved, then one or more of the other forms of energy must have decreased.

Perhaps when the two carts collided, a compressed spring was released, adding kinetic energy to the system.

This kind of collision is sometimes called a superelastic or explosive collision.


Section

11.2


After the collision in case 2, the kinetic energy is equal to:


Section

11.2

J50.0m/s00.0kg00.121

m/s00.1kg00.121 22

DfCf KEKE

Kinetic energy remained the same after the collision.

This type of collision, in which the kinetic energy does not change, is called an elastic collision.


Collisions between hard, elastic objects, such as those made of steel, glass, or hard plastic, often are called nearly elastic collisions.

After the collision in case 3, the kinetic energy is equal to:


Section

11.2

KE KE2 2

Cf Df1 1

1.00kg 0.50m/s 1.00kg 0.50m/s 0.25J2 2

Kinetic energy decreased and some of it was converted to thermal energy. This kind of collision, in which kinetic energy decreases, is called an inelastic collision.

Objects made of soft, sticky material, such as clay, act in this way.


In collisions, you can see how momentum and energy are really very different.

Momentum is almost always conserved in a collision.

Energy is conserved only in elastic collisions.

Momentum is what makes objects stop.


Section

11.2


It is also possible to have a collision in which nothing collides.

If two lab carts sit motionless on a table, connected by a compressed spring, their total momentum is zero.

If the spring is released, the carts will be forced to move away from each other.

The potential energy of the spring will be transformed into the kinetic energy of the carts.

The carts will still move away from each other so that their total momentum is zero.


Section

11.2


The understanding of the forms of energy and how energy flows from one form to another is one of the most useful concepts in science.

The term energy conservation appears in everything from scientific papers to electric appliance commercials.


Section

11.2

Practice Problems p. 297: 15,17; p.301: 27

Section Check

Write the law of conservation of energy and state the formula for mechanical energy of the system.

Question 1

Section

11.2

Section Check

The law of conservation of energy states that in a closed, isolated system, energy can neither be created nor destroyed; rather, energy is conserved. Under these conditions, energy changes from one form to another, while the total energy of the system remains constant.

Since energy is conserved, if no other forms of energy are present, kinetic energy and potential energy are inter-convertible.

Mechanical energy of a system is the sum of the kinetic energy and potential energy if no other forms of energy are present.

That is, mechanical energy of a system E = KE + PE.

Answer 1

Section

11.2

Section Check

Two brothers, Jason and Jeff, of equal masses jump from a house 3-m high. If Jason jumps on the ground and Jeff jumps on a platform 2-m high, what can you say about their kinetic energy?

Question 2

Section

11.2

A. The kinetic energy of Jason when he reaches the ground is greater than the kinetic energy of Jeff when he lands on the platform.

B. The kinetic energy of Jason when he reaches the ground is less than the kinetic energy of Jeff when he lands on the platform.

C. The kinetic energy of Jason when he reaches the ground is equal to the kinetic energy of Jeff when he lands on the platform.

D. Neither Jason nor Jeff possesses kinetic energy.

Answer: A

Answer 2

Reason: The kinetic energy at the ground level is equal to the potential energy at the top. Since potential energy is proportional to height, and since the brothers have equal masses, Jason will have greater kinetic energy since he falls from a greater height.

Section

11.2 Section Check

Section Check

A car of mass m1 moving with velocity v1 collides with another car of

mass m2 moving with velocity v2. After collision, the cars lock

together and move with velocity v3. Which of the following equations

about their momentums is true?

Question 3

Section

11.2

A. m1v1 + m2v2 = (m1 + m2)v3

B. m1v2 + m2v1 = (m1 + m2)v3

C. m1v1 + m2v2 = m1v2 + m2v1

D. m1v12 + m2v2

2 = (m1 + m2)v3

2

Answer: A

Answer 3

Reason: By the law of conservation of momentum, we know that total momentum before collision is equal to total momentum after collision.

Section

11.2 Section Check

Before collision, the car of mass m1 was moving with velocity

v1 and the car of mass m2 was moving with velocity v2.

Hence, the total momentum before collision was ‘m1v1 +

m2v2’. After collision, the two cars lock together. Hence, the

mass of the two cars is (m1 + m2) and the velocity with which

the two cars are moving is v3. Hence, the total momentum

after collision is ‘(m1 + m2)v3.’ Therefore, we can write, m1v1 +

m2v2 = (m1 + m2)v3.

Chapter 10 & 11 Test Information

The test is worth 50 points total.

Matching: 10 questions, 1 point each.Multiple Choice: 14 questions, 1 point each.Problems: 7 questions for a total of 28 points.

Know how to use the formulas on the green formula sheet.

When Mechanical Energy is Conserved: Ei = Ef

KEbefore +PE before = KEafter + PEafter

When Mechanical Energy is Lost to Heat due to Friction: Ei > Ef

KEbefore +PE before = KEafter + PEafter + heat

energy and its conservation learn that energy is a property of an object that can change the...

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