Conservation is the third and last primary category of problem
you'll encounter in this course. The other two categories were dvats and net
force problems. Of these three types, conservation is the most important.
Here are some fundamental concepts needed for solving conservation of energy
problems.

Forces come in two general types as far as energy
is concerned: conservative and nonconservative. The textbook
gives two definitions of conservative force. Review these as needed
in section 81.
For our purposes, the conservative forces that we'll deal with the most
this term are gravity and spring forces. In regard to spring
forces, it's important to realize that the Hooke's Law force (F_{spring} = kΔx)
is the conservative one. (It's possible for a spring to obey a
different force law that isn't conservative.) The most common
nonconservative force that we'll deal with is friction.

Potential energy is just a different way to
deal with work. Potential energy is defined in terms of i) a change and ii) the work done by a conservative force. (Nonconservative forces don't have potential energy associated with
them.) If a conservative force does work, W_{c }, on an
object, the object's potential energy changes. The work is the
negative of the change in the potential energy: W_{c} = ΔU.
Note that potential energy doesn't have meaning except in terms of how it
changes.

Energy is always associated with a system. Consider, for example, an apple
falling to the Earth. The gravitational potential energy is a
property of the appleEarth system. Without both the apple and the
Earth, there would be no associated gravitational potential energy.

Mechanical energy is a term that means the
sum of the kinetic and potential energies of a system. All forms of
potential energy are included. For example, if an object is dropped
onto a spring, there are 3 forms of energy that change: kinetic energy
of the object, gravitational potential energy, and elastic (or spring)
potential energy.
Energy can change
forms, but the total mechanical energy doesn't change with time when
conservative forces only are present. (We'll see later how to modify
this statement when nonconservative forces do work on the system.)
In the
last example of an object dropped onto a spring, the system initially has only gravitational potential energy, U_{g}. As the object falls, some of the U_{g} is
converted to kinetic, K. As the object strikes and begins compressing the
spring, both U_{g} and K are being converted to elastic potential
energy, U_{e}. When the spring is compressed as far as
possible and the object is momentarily at rest, all of the original U_{g} has been converted to U_{e} and there is no K. 
The goal in what follows is to present and apply a general conservation of
energy equation to the solution of problems. We're going to use a more
general statement than is used in
the text. You need only one equation for all of your conservation of
energy problems. Here's the general statement.
The first thing to recognize in interpreting this relationship is that
it applies to a system. ΔE_{sys} represents all the energy changes within the system, while W_{ext} represents the work done on the system by all forces external to the system.
One must specify the system before applying the relationship in order to differentiate between what goes on the
left and right sides of the equation.
While this is not required, it's the practice that the textbook author uses;
therefore, we will usually follow this procedure. 
Before looking at applications, let's see how the general conservation of
energy equation above is consistent with that given in the
text. One finds in many conservation of energy problems that no work is done
by external forces on the system. In that case 0 = ΔE_{sys}. The term on the left is what the textbook author calls ΔE without the sys subscript. We choose to use the subscript to
emphasize that the righthand side of the equation applies to the system.
When all the objects in the system interact through conservative forces only, then ΔE_{sys} is simply the mechanical energy of the system; that is, ΔE_{sys} is the sum of the kinetic and potential energies associated with the system.
Thus, the following statements in the textbook are all the same as saying
"Mechanical energy is conserved when external forces do no work
on the system."
Derived equations (assume W_{ex}_{t} = 0): 
E = constant 
ΔE = 0 
E_{i} = E_{f} 
Do not start with the less general derived equations. 
 Identify the system.
 Identify the forces acting
external to the system.
 For internal conservative forces, include a term for the potential energy change
corresponding to that force on the righthand side of the equation. Also
include a ΔK term.
 For external forces, include a term for the work done by that force
on the lefthand side of the equation.

What you should learn is the method of applying conservation of energy to any situation. You'll learn the method by doing problems that use
it. 
Example 1. Ball falling vertically in a vacuum
Consider a ball that falls from rest in the absence of air friction. Suppose
the goal is to find the velocity of the ball after it has fallen a certain
distance. What are the interacting objects? The answer is the ball and the
Earth. They interact through gravity, which is a conservative force. The
Earth pulls on the ball, and the ball pulls back on the Earth. Therefore, the system
here includes the ball and the Earth. Mechanical energy, which includes U_{g} and K, remains constant for this system, since
the only force is a conservative one. Since gravity is internal to this system, and there are no external forces, W_{ext} = 0. ΔE_{sys} is composed of two terms, the change in kinetic energy, ΔK, and the change in gravitational potential energy,
ΔU_{g}. Thus, the conservation of energy
equation becomes
0 = ΔK +
ΔU_{g}. 
Click here to see how the energies change with time.
The energy of the system is conserved (total
system energy remains constant). 

Example 2. Ball falling vertically in air
Suppose now that
the ball is falling in air. This means there will be a force of air
friction. In conformance with the note in the yellow box above, we
choose not to
include the air in the system, because air friction is not a conservative
force. Therefore, the system is the same as that chosen for Example 1. You may
wonder how we know that air friction is a nonconservative force. The test
to apply is whether the total work done over a closed path is 0. This means
that there must be as much negative work done as positive work so that the
total work is 0. This can never be the case for air friction. That's because
the force of air friction always opposes the motion of the object;
therefore, the work done by air friction is always negative. Of course, the
presence of air friction means that the ball achieves a lower velocity than
it would in falling the same distance in a vacuum. This means that
mechanical energy isn't conserved. W_{ext} will be the
work done by friction, W_{f}, and will not be 0. The righthand side of the
equation will remain the same as in Example 1. Therefore, the result is
W_{f} = ΔK +
ΔU_{g}. 
Click here to see how the energies change with time.
The energy of the system decreases with time
as a result of the work done by friction on the system. 

Example 3. Block sliding down a frictionless inclined plane
A box slides down a frictionless, inclined plane. Again, the goal is to find
the speed of the box after it has slid a particular distance. Interestingly,
we need only include the box and the Earth in the system. They interact
through the conservative force of gravity. Mechanical energy is conserved
for this system. Let's see why the plane isn't included in the system.
The plane exerts a normal force on the box, but that force doesn't change
the energy of the box. Work must be done on a system to change its energy.
The normal force can't do work, because it's always perpendicular to the
displacement of the object.
We select as the system the block and the Earth. Gravity is a force internal
to the system as in the previous two examples. However, the normal force on
the block is external. Thus, the work done by the external force is W_{n}, and
we have
W_{n} = ΔK +
ΔU_{g}.
Since the normal force is always perpendicular to the displacement of the
block, W_{n} = 0 and
0 = ΔK + ΔU_{g}.
The result is the same as for Example 1. Note that you would
also get the same result if the block were sliding on a curved surface such
as the interior of a bowl. The surface still does no work on the block.
Perhaps you can begin to see how seemingly different problems are
actually the same problem in terms of energy conservation. When you see such
similarities, you're really understanding physics. This makes your problem
solving more efficient. 
Click here to see how the energies change with time.
The energy of the system is conserved. 

Example 4. Block oscillating on a frictionless surface
A block oscillates horizontally across a frictionless surface under the
action of a spring. Knowing the block's maximum displacement from the
equilibrium position, the goal is to find the speed of the block as it
passes through the equilibrium position. What would you pick for the
system? The spring force is conservative, so it's convenient to include the
block and the spring in the system. We don't need to pick the Earth this
time, because gravity doesn't change the energy of the block. We don't
include the plane, because the normal force can't change the energy of the
block either. For the system of block and spring, mechanical energy is
conserved. Consider what
is included in ΔE_{sys}. There
will of course be a ΔK, since the block is
accelerating and experiences a change in kinetic energy. There will
also be a change in the elastic potential energy, ΔU_{e},
since the compression of the spring is changing.
There will be no ΔU_{g} term,
since the block isn't changing height. Therefore, the conservation of energy
equation applied to this situation is
0 = ΔK + ΔU_{e}. 
Click here to see how the energies change with time.
The energy of the system is conserved. 

An example for you to do. The simple pendulum
Consider a bob swinging in a pendulum motion at
the end of a string and in the absence of friction. The goal is to find the
speed of the bob after it has swung through a particular arc. What would you
pick for the system and why? Would you include the string which exerts
tension force on the bob? Would mechanical energy be conserved? Apply the general conservation of energy equation W_{ext} = ΔE_{sys} to the solution.
Use the above examples as a guide. Try drawing the energy bars at different
times. 

