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Physics Chapter 11 Energy
11.1 Energy in Its Many Forms
We have defined energy as the ability to change an object or its environment. Change can occur in several
ways. A speeding automobile can change itself, people, and objects in its path. The energy that causes the car’s
motion is normally stored in the chemical bonds of gasoline. Energy is stored in many other forms. A heavy
object placed high on a shelf has a potential to change almost anything on which it might fall. A stretched spring
or rubber band has the ability to give an object a high velocity.
Forms of Energy
Anything moving, from a roller coaster car to a falling leaf, is capable of causing some kind of change in an
object it touches. Moving objects have a form of energy called kinetic energy. Eureka! Episode 9 - Kinetic
Energy
Where did the roller coaster car get its kinetic energy? The work done by the electric motor was saved or stored
by raising the cars to the top of the hill. The cars were said to have potential energy. In the roller coaster
example, potential energy is stored in the cars as a result of their height; because work was done on the cars
against gravity. When they go downhill, the potential energy is changed into kinetic energy.
Another example of energy transfer is the kicking of a football. As the football is being kicked the kicker’s foot
does work by compressing the ball. Energy is transferred from the kicker into the ball in the form of potential
energy. In an instant, that potential energy will be converted into kinetic energy, and the ball will move with great
velocity off the foot and into the air.
Work is the transfer of energy by mechanical means. In the example above, work was done, transferring energy
into potential energy of an object. From where did this energy come? In the case of the place kicker, the body
obtains energy from food and stores it in certain chemical compounds until needed. The compounds can
produce motion in muscles, transferring their stored energy to kinetic energy. Energy in both food and the body
is stored in chemical bonds, and so is called chemical energy. In the case of the roller coaster, the energy came
from electricity, which was probably generated by burning coal. Coal and gasoline also store chemical energy,
which is released when they burn.
There are many forms of potential energy. No matter what the form, the amount of potential energy depends
on the position, shape, or form of an object. Some examples of potential energy being converted into kinetic
energy are walking, cooking, sliding down a sliding board, a pitcher throwing a fastball and avalanches.
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Doing Work to Change Kinetic Energy
A pitcher winds up and throws the baseball. During the throw, a force is exerted on the ball. As a result, the ball
leaves the pitcher’s hand with a high velocity and considerable kinetic energy. How can you find the amount of
energy? How is the kinetic energy related to the work done on the ball?
The amount of kinetic energy of an object has is determined by the objects mass and velocity which is
expressed below mathematically.
KE = 1/2mv2
where m is the mass of the object and v is its velocity. The kinetic energy is proportional to the mass of the
object. Thus a 7.26-kg shot has much more kinetic energy than a 148-g baseball with the same velocity. Kinetic
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energy is also proportional to the square of the velocity. A car speeding at 30.0 m/s has four times the kinetic
energy of the same car traveling at 15.0 m/s. Kinetic energy, like any form of energy, is measured in joules
Imagine you are the pitcher. How can you give the ball more kinetic energy? You can either pitch the ball with
greater force or exert that force over a larger distance. In other words, you can do more work on the ball.
Newton’s second law shows the connection between work and kinetic energy. According to Newton’s second
law, F = ma, an object is accelerated with a constant acceleration if a constant net force is exerted on it. The
work done on an object is given by W = Fd. Thus the work done is W = (ma)d. Assume the object was originally
at rest, v1 = 0. As it accelerates, V2 = v12 + 2ad
V2 = 2ad, or d = v2/2a, since a is constant.
Therefore, W = Fd and F =ma so W = mad but d= v2/2a so
W = ma (v2/2a) = mv2 /2 or 1/2 mv2 which = KE.
That is, W = KE; the work done equals the kinetic energy gained by the ball.
Not all objects start at rest. They may already have kinetic energy when additional work is done on them. If
we define initial kinetic energy, KEi , and final kinetic energy, KEf , as the energies the object has before and
after the work is done, we can write
W net = KEf – KEi = Δ KE.
That is, the change in the kinetic energy of an object is equal to the network done on it. This equation
represents the work- energy theorem. It can be stated: The network done on an object is equal to its change in
kinetic energy. Note that the work in the work-energy theorem is the work done on an object by a net force. It is
the algebraic sum of work done by all forces.
The work-energy theorem indicates that if the network is positive, the kinetic energy increases. Network is
positive when the net force acts in the same direction as the motion. For example, consider a ball pitched in a
baseball game. The net force on the pitched ball is in the same direction as the motion of the ball. The network
is positive and the kinetic energy of the ball increases.
If the network is negative, the kinetic energy decreases. When the catcher catches the ball, the net force
acting on it is in the direction opposite its motion. The kinetic energy decreases to zero as the ball stops in the
catcher’s mitt. The catcher does negative work on the ball.
The kinetic energy of several common moving objects is shown in table below:
Item
Aircraft carrier
Orbiting satellite
Trailer truck
Compact car
Football linebacker
Pitched baseball
Remarks
91 400 tons at 30 knots (34.5mph)
100 kg at 7.8 km/s
5700 kg at 100 km/h
750 kg at 100 km/h
110 kg at 9.0 m/s
148 g at 45 m/s
KE(J)
9.90e9
3.00e9
2.20e6
2.90e5
4.50e3
1.50e2
2
Falling nickel
Bumblebee
Snail
5 g from 50-m height
2g at 2m/s
5 g at 0.05 km/h
2.50
4.00e-3
4.50 e-7
Try example problem – then do practice problems 11-1a.
Kinetic Energy and Work
A shot putter heaves a 7.26-kg shot with a final velocity of 7.50 m/s. a. What is the kinetic energy of the shot? b.
The shot was initially at rest. How much work was done on it to give it this kinetic energy?
Potential Energy Eureka! Episode 10 - Potential Energy
If you have juggled or watched someone juggle you have witnessed kinetic and potential energy constantly
being interchanged. When you throw a ball up into the air, you do work on it. As it leaves your hand, it has
kinetic energy. As the ball rises, its speed is reduced because of the downward force of Earth’s gravity. The ball
moves up, but the force is down, so the work done on the ball is negative, and the kinetic energy of the ball
becomes smaller. At the top of its flight, its speed is instantaneously zero and it has no kinetic energy. By the
time it returns to your hand, however, gravity has done an equal amount of positive work, and the ball has
regained its original speed. Thus its kinetic energy is the same as it was when it left your hand.
The kinetic energy you give the ball is transferred to potential energy and then back into kinetic energy. It
makes sense, then, to describe the total energy, E, as the sum of the kinetic energy and potential energy.
E = KE + PE
During the flight of the ball, the sum of kinetic and potential energy is constant. At the start and end of the
flight, the energy is totally kinetic. Potential energy is zero. At the top, the energy is fully potential, and the kinetic
energy is zero. In between, the energy is partially kinetic and partially potential.
How does potential energy depend on height? As long as the ball is close to Earth, the gravitational
acceleration, g, is constant, where g = 9.80 m/s2. Using the equation for motion with constant acceleration, we
find the velocity of the ball at any height, vf, is given by
vf2 = vi2 + 2gh
In this equation, h is the vertical distance measured from the launching height of the ball. Multiplying each term
in this equation by ½m gives the kinetic energy, 1/2mvf2, at any height, h,
1/2mvf2 = 1/2mvi2 + mgh.
At the start of the flight, h = 0. The energy is all kinetic, E = 1/2mvi 2. Because the total energy of the ball does
not change,
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E = 1/2mvi2
which is = ½mvf2 + mgh.
Since total E = KE + PE, and KE = ½ mv2 the gravitational potential energy is given by
PE = mgh.
Only changes in potential energy can be measured; thus the total amount of energy cannot be determined.
This means that gravitational potential energy can be set equal to zero at any height you choose. Often potential
energy is conveniently measured from the surface of Earth or from the floor of a room. That is, h is set equal to
zero at the ground or floor level. In fact, any height, called the reference level, can be used. It is important,
however, that the reference level not be changed in the middle of a problem.
The formula for gravitational potential energy, PE = mgh, is valid only if the gravitational force, and thus the
acceleration, is constant. When the distance is far above Earth, the gravitational force, mg, is reduced, and the
potential energy no longer increases linearly with the height.
Energy also can be stored in the bending or stretching of an object. The stretching, squeezing, or bending of
objects such as metal springs, slingshots, and trampolines stores potential energy. The fiberglass pole used in
the pole vault has led to greatly increased records in competition. The pole vaulter runs with the pole, then
plants the end of it into a socket in the ground. The kinetic energy of the runner is first stored in the bending of
the pole. Then, as the pole straightens, the stored energy is converted into gravitational potential energy and
kinetic energy as the vaulter is lifted up to 6 m above the ground.
Try example problem on Gravitational Potential Energy – then do practice problems 11-1b.
A 2.00-kg textbook is lifted from the floor to a shelf 2.10 m above the floor, a. What is its gravitational potential
energy relative to the floor? b. What is its gravitational potential energy relative to the head of a 1.65 m tall
person?
11.2 Conservation of Energy
We found the equation for gravitational potential energy by assuming that the total energy of a ball in flight
was constant. Although the energy changed from kinetic to potential and back again, the total amount of energy
stayed the same. Unlike the conservation of momentum, the constancy of energy is not predicted by Newton’s
laws. This separate fact of nature is defined in the Law of conservation of Energy. LCE states that energy can
change from one form to another but the total amount of energy in a closed isolated system does not change.
Systems
In your experience, you have probably seen few examples of energy remaining constant. The kinetic energy
of a ball rolling on the grass is soon lost. Even on smooth ice, a hockey puck eventually stops. The key to
understanding and using the constancy of energy is in selecting the system, the collection of objects we want to
study. Just as in the case of conservation of momentum, we need a special kind of a system, a closed, isolated
one.
Objects do not enter or leave a closed, isolated system. It is isolated from external forces, and so no work can
be done on it. The law of conservation of energy states that within a closed, isolated system, energy can change
form, but the total amount of energy is constant. That is, energy can be neither created nor destroyed. A ball
alone, acted on by gravity, is not an isolated system. A ball and Earth, however, is an example of a closed,
isolated system. The kinetic energy of the ball can change, but the sum of gravitational potential and kinetic
energy is constant. The sum of potential and kinetic energy is often called the mechanical energy. Suppose the
ball has a weight of 10.0 N. If it is at rest on a shelf 2.00 m above Earth’s surface, it has no kinetic energy, but a
potential energy related to Earth’s surface given by
PE = mgh
= (10.0 N)(2.00 m) = 20.0 J.
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If the ball rolls from the shelf, there are no forces on the ball other than the gravitational force of Earth, so the
ball falls. When it is 1.00 m above Earth’s surface, its potential energy is
PE = mgh
= (10.0 N)(1.00 m) = 10.0 J.
The ball has lost half its potential energy falling 1.00 m. The ball is moving, however, and has gained kinetic
energy. The change in kinetic energy can be found from the work-energy theorem:
W = Δ KE = KEf - KEi
The work done on the falling ball by the gravitational force, mg, is given by
W = Fd = (10.0 N)(1 .00 m) = 10.0 J
The work is positive because the force and motion are in the same direction. Thus the kinetic energy of the ball
is
KEf= 10.0J.
The decrease in the potential energy of the ball is equal to the increase in its kinetic energy. Thus the sum of
potential and kinetic energies is not changed. The mechanical energy is constant. When the ball reaches Earth’s
surface, its potential energy will be zero. All its energy will be kinetic. The final kinetic energy will equal the initial
potential energy. The equation describing the conservation of energy is
KEi + PEi = KEf + PEf.
Changing the path an object follows as it falls does not change its potential energy. As an example, if a 10.0N ball is at the top of a 2.0 m high frictionless inclined plane. As it slides down the plane, it moves horizontally as
well as vertically, but its change in height above Earth is still 2.00 m. Its potential energy depends only on its
height above Earth’s surface. The kinetic energy of the ball at the bottom of the plane is the same whether the
ball falls vertically or slides down without friction.
Thus, in the case of a roller coaster that is almost at rest at the top of the first hill, if any hill were higher than
the first, the potential energy needed to reach the top of that hill would be larger than the mechanical energy
stored in the car at the top of the first hill.
The simple harmonic motion of a pendulum also demonstrates the conservation of energy. Usually the
gravitational potential energy is chosen to be zero at the equilibrium position of the bob. The initial gravitational
potential energy of a raised pendulum bob is transferred to kinetic energy as the bob moves along its path. At
the equilibrium point, the gravitational potential energy is at a minimum (zero) and kinetic energy is at a
maximum. The figure below is a graph of the changing potential and kinetic energy of a pendulum bob during
one-half period of its oscillation. The sum of potential and kinetic energies is constant.
For the SHM of a pendulum the bob(a), the sum of the PE and KE is a constant
As you know, the oscillations of a pendulum bob eventually die away, and a bouncing ball finally comes to
rest. Where did the mechanical energy go? Work was done against friction, and in the case of the ball, to
change its shape when it bounces. The system was not isolated so its mechanical energy was not conserved.
If total energy is conserved, the potential and kinetic energies must have changed into other forms. When a
heavy crate is dropped on the floor, you feel the floor tremble. Some energy has changed into the motion of the
floor. If you measure the temperature of the dropped crate or of the stopped pendulum bob very accurately, you
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will find that they are slightly warmer. The energy has changed into a different form, the increased motion of the
particles that make up the object. This form of energy is thermal energy.
Albert Einstein recognized another form of potential energy, mass itself. This equivalence is expressed in his
famous equation E0 = mc2. Mass, by its very nature, has energy, E0, called its rest energy. Further, energy has
mass. Stretching a spring or bending a bow causes them to gain mass. In these cases, the mass change is too
small to be easily detected. When the strong nuclear forces are involved, however, the energy released by
changes in mass can be very large indeed.
When solving conservation of energy problems:
Carefully identify the system. Make sure it is closed; no objects can leave or enter it. It must also be isolated; no
external forces can act on any object in the system. Thus, no work can be done on or by objects outside the
system.
1. Is friction present? If it is, then the sum of kinetic and potential energies will not be constant. But, the
sum of the kinetic energy, potential energy, and the work done against friction will be constant.
2. Finally, if there is no friction, find the initial and final total energies and set them equal.
Try example problem – then do practice problems 11-2a.
Conservation of Energy
A large chunk of ice with mass 15.0 kg falls from a roof 8.00 m above the ground. a.Find the kinetic energy of
the ice when it reaches the ground. b. What is the speed of the ice when it reaches the ground? c. Is the answer
the same as you would determine by solving as a constant acceleration problem?
vf2 = vi2 +2gd
vf = 2gd
2 x 9.8 x 8 = 12.5m/s
Thermal energy can be increased either by adding heat or by doing work on a system. Thus, the total increase in the thermal
energy of a system is the sum of the work done on it and the heat added to it. This fact is called the first law of
thermodynamics. Thermodynamics is the study of the changes in thermal properties of matter. The first law is merely a
restatement of the law of conservation of energy.
The second law of thermodynamics states that natural processes go in a direction that increases the total entropy of the
universe. Entropy and the second law can be thought of as statements of the probability of events happening. When food
coloring is placed in water an increase in entropy is shown as food color molecules, originally separate from clear water,
are thoroughly mixed with the water molecules after a time.
Analyzing Collisions
During a collision between two objects, forces act that slightly change the shape of the colliding bodies.
The kinetic energy of motion is changed into potential energy. If the potential energy is completely converted
back into kinetic energy after the collision, the collision is called an elastic collision. For this special case, the
kinetic energy of the bodies before the collision is equal to their kinetic energy after the collision. The collision
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between two billiard balls or glass marbles is very nearly elastic.
What conservation laws can be used to analyze collisions? Momentum is conserved whenever bodies
collide with no external forces present. Suppose a red billiard ball of mass m = 95 g moves with a velocity of
v = + 0.75 m/s. It collides head-on with a white billiard ball of equal mass that is at rest. The total momentum
before the collision is mv = (0.095 kg)(+0.75 m/s) = + 0.071 kg m/s. What are the speeds of the two balls after
the collision? Using only the momentum equation, the two speeds cannot be determined—with two unknowns,
two equations are needed. If kinetic energy is also conserved, we have a second equation and can solve the
two equations simultaneously to find the final speeds. The results show that the moving red ball must come to
rest and the white ball must move with the final speed vf = + 0.75 m/s, the initial speed of the red ball. If, during a
collision, some kinetic energy is changed into other forms, the collision is called an inelastic collision. In the case
of real billiard balls, some energy is usually needed to do work against the surface of the table. The red ball will
not stop, and the white ball will have a smaller final speed. If, instead of billiard balls, two balls of soft clay, each
with a mass of 0.095 kg, collide and stick together, the collision is said to be completely inelastic. The two clay
balls will have the same final speed. As was shown in Chapter 9, because of conservation of momentum, the
final speed is half the initial speed. The momentum of the balls is not changed.
2m(v/2) = (0.190 kg)((0.75 m/s)/2) = 0.07125 kg m/s
The energy after the collision is
1/2(2m)(v/2)2 = 1/2(0.190 kg)(0.375 m/s)2 = 0.01336 J.
This result indicates that half the initial kinetic energy has changed into other forms of energy.
Many collisions are neither completely elastic nor completely inelastic. The velocities can then be
calculated only if the amount of energy loss is known.
Elastic Collisions - Example problem
Block A with mass 12.2 kg moving at +2.45 m/s makes a perfectly elastic head-on collision with a block B, mass
36.3 kg, at rest. Find the velocities of the two blocks after the collision. Assume all motion is in one dimension.
Energy loss in a Collision – Example problem
In an accident due to slippery roads, a compact car, mass 575 kg, moving at + 15.7 m/s, smashes head-on into
the rear end of a car with a mass of 1575 kg, originally moving at +5.01 m/s. They lock together and slide
forward.
a. What is the final velocity of the wrecked cars? b.How much kinetic energy was lost in the collision? c.What
fraction of the original energy was lost? d. Into what forms did this energy most likely go?
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Do problems 11-2b and Concept Review 11-2
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