


Forms of Energy
Though energy is always measured in joules, and though
it can always be defined as a capacity to do work, energy manifests
itself in a variety of different forms. These various forms pop
up all over SAT II Physics, and we will look at some additional
forms of energy when we discuss electromagnetism, relativity, and
a number of other specialized topics. For now, we will focus on
the kinds of energy you’ll find in mechanics problems.
Kinetic Energy
Kinetic energy is the energy a body in motion has by virtue
of its motion. We define energy as the capacity to do work, and
a body in motion is able to use its motion to do work. For instance,
a cue ball on a pool table can use its motion to do work on the
eight ball. When the cue ball strikes the eight ball, the cue ball
comes to a stop and the eight ball starts moving. This occurs because
the cue ball’s kinetic energy has been transferred to the eight
ball.
There are many types of kinetic energy, including vibrational,
translational, and rotational. Translational kinetic energy,
the main type, is the energy of a particle moving in space and is
defined in terms of the particle’s mass, m,
and velocity, v:
For instance, a cue ball of mass 0.5 kg moving
at a velocity of 2 m/s has a kinetic energy of
^{1}/_{2 } (0.5 kg)(2 m/s)^{2} =
1 J.
The WorkEnergy Theorem
If you recall, work is a measure of the transfer of energy.
An object that has a certain amount of work done on it has that
amount of energy transferred to it. This energy moves the object
over a certain distance with a certain force; in other words, it
is kinetic energy. This handy little fact is expressed in the workenergy
theorem, which states that the net work done on an object
is equal to the object’s change in kinetic energy:
For example, say you apply a force to a particle, causing
it to accelerate. This force does positive work on the particle
and increases its kinetic energy. Conversely, say you apply a force
to decelerate a particle. This force does negative work on the particle
and decreases its kinetic energy. If you know the forces acting
on an object, the workenergy theorem provides a convenient way
to calculate the velocity of a particle.
Example

If we know the puck’s kinetic energy after it has glided 32 m,
we can calculate its velocity. To determine its kinetic energy at
that point, we need to know its initial kinetic energy, and how
much that kinetic energy changes as the puck glides across the ice.
First, let’s determine the initial kinetic energy of the
puck. We know the puck’s initial mass and initial velocity, so we
just need to plug these numbers into the equation for kinetic energy:
The friction between the puck and the ice decelerates
the puck. The amount of work the ice does on the puck, which is
the product of the force of friction and the puck’s displacement, is
negative.
The work done on the puck decreases its kinetic energy,
so after it has glided 32 m, the kinetic energy of
the puck is 50 – 32 = 18 J. Now that we know the final
kinetic energy of the puck, we can calculate its final velocity
by once more plugging numbers into the formula for kinetic energy:
We could also have solved this problem using Newton’s
Second Law and some kinematics, but the workenergy theorem gives
us a quicker route to the same answer.
Potential Energy
As we said before, work is the process of energy transfer.
In the example above, the kinetic energy of the puck was transferred
into the heat and sound caused by friction. There are a great number
of objects, though, that spend most of their time neither doing
work nor having work done on them. This book in your hand, for instance,
is not doing any work right now, but the second you drop it—whoops!—the
force of gravity does some work on it, generating kinetic energy.
Now pick up the book and let’s continue.
Potential energy, U, is
a measure of an object’s unrealized potential to have work done on
it, and is associated with that object’s position in space, or its
configuration in relation to other objects. Any work done on an
object converts its potential energy into kinetic energy, so the
net work done on a given object is equal to the negative change
in its potential energy:
Be very respectful of the minus sign in this equation.
It may be tempting to think that the work done on an object increases
its potential energy, but the opposite is true. Work converts potential
energy into other forms of energy, usually kinetic energy. Remove
the minus sign from the equation above, and you are in direct violation
of the law of conservation of energy!
There are many forms of potential energy, each of which
is associated with a different type of force. SAT II Physics usually
confines itself to gravitational potential energy and the potential
energy of a compressed spring. We will review gravitational potential
energy in this section, and the potential energy of a spring in
the next chapter.
Gravitational Potential Energy
Gravitational potential energy registers the
potential for work done on an object by the force of gravity. For
example, say that you lift a water balloon to height h above
the ground. The work done by the force of gravity as you lift the
water balloon is the force of gravity, –mg,
times the water balloon’s displacement, h.
So the work done by the force of gravity is W
= –mgh. Note that there is a negative amount
of work done, since the water balloon is being lifted upward, in the
opposite direction of the force of gravity.
By doing –mgh joules of
work on the water balloon, you have increased its gravitational
potential energy by mgh joules (recall
the equation ). In other words, you have increased its
potential to accelerate downward and cause a huge splash. Because the
force of gravity has the potential to do mgh joules
of work on the water balloon at height h,
we say that the water balloon has mgh joules
of gravitational potential energy.
For instance, a 50 kg mass held at a height
of 4 m from the ground has a gravitational potential
energy of:
The most important thing to remember is that the
higher an object is off the ground, the greater its gravitational
potential energy.
Mechanical Energy
We now have equations relating work to both kinetic and
potential energy:
Combining these two equations gives us this important
result:
Or, alternatively,
As the kinetic energy of a system increases, its potential
energy decreases by the same amount, and vice versa. As a result,
the sum of the kinetic energy and the potential energy in a system
is constant. We define this constant as E,
the mechanical energy of the system:
This law, the conservation of mechanical energy, is one
form of the more general law of conservation of energy, and it’s
a handy tool for solving problems regarding projectiles, pulleys,
springs, and inclined planes. However, mechanical energy is not conserved
in problems involving frictional forces. When friction is involved,
a good deal of the energy in the system is dissipated as heat and
sound. The conservation of mechanical energy only applies to closed
systems.
Example 1

Before the object is released, it has a certain amount
of gravitational potential energy, but no kinetic energy. When it
hits the ground, it has no gravitational potential energy, since h =
0, but it has a certain amount of kinetic energy. The mechanical
energy, E, of the object remains constant,
however. That means that the potential energy of the object before
it is released is equal to the kinetic energy of the object when
it hits the ground.
When the object is dropped, it has a gravitational potential
energy of:
By the time it hits the ground, all this potential energy
will have been converted to kinetic energy. Now we just need to
solve for v:
Example 2

The answer to question 1 is point B. At the top of the
tomato’s trajectory, the tomato is the greatest distance above the
ground and hence has the greatest potential energy.
The answer to question 2 is point B. At the top of the
tomato’s trajectory, the tomato has the smallest velocity, since
the ycomponent of the velocity is zero, and hence
the least kinetic energy. Additionally, since mechanical energy
is conserved in projectile motion, we know that the point where
the potential energy is the greatest corresponds to the point where
the kinetic energy is smallest.
The answer to question 3 is point E. At the bottom of
its trajectory, the tomato has the greatest velocity and thus the
greatest kinetic energy.
The answer to question 4 is point A. At this point, the
velocity is decreasing in magnitude and the tomato is getting higher
in the air. Thus, the kinetic energy is decreasing and the potential
energy is increasing.
The answer to question 5 is point C. From our study of
kinematics, we know that the speed of a projectile is equal at the
same height in the projectile’s ascent and descent. Therefore, the
tomato has the same kinetic energy at points A and C. Additionally,
since the tomato has the same height at these points, its potential
energy is the same at points A and C.
Keep this example in mind when you take SAT II Physics,
because it is likely that a similar question will appear on the
test.
Thermal Energy
There are many cases where the energy in a system seems
simply to have disappeared. Usually, this is because that energy
has been turned into sound and heat. For instance, a coin sliding
across a table slows down and comes to a halt, but in doing so,
it produces the sound energy of the coin scraping along the table
and the heat energy of friction. Rub your hands together briskly
and you will feel that friction causes heat.
We will discuss thermal energy, or heat, in greater detail
in Chapter 9, but it’s worth noting here that it is the most common
form of energy produced in energy transformations. It’s hard to
think of an energy transformation where no heat is produced. Take
these examples:
 Friction acts everywhere, and friction produces heat.
 Electric energy produces heat: a light bulb produces far more heat than it does light.
 When people talk about burning calories, they mean it quite literally: exercise is a way of converting food energy into heat.
 Sounds fade to silence because the sound energy is gradually converted into the heat of the vibrating air molecules. In other words, if you shout very loudly, you make the air around you warmer!
