### Video Transcript

In this video, we will learn how to
relate the mechanical, kinetic, and gravitational potential energies for an object
that work is done to. Along the way, we’ll come to
understand what mechanical energy is.

But before we talk about categories
of energy, let’s consider the term work. This is an everyday familiar word,
but in physics, work has a very specific meaning. It is defined as the energy
transferred when a force moves an object some distance. So work in physics is energy, and
it’s the energy transferred by a force moving an object over a distance. For example, say a box is at rest
on the ground. We come along and lift the box. Raising it to some height requires
that we exert a force on the box. So there’s a force involved, and
this force moves the box through a distance equal to its height above ground. We say then that we’ve done work on
the box. And notice that by doing this work,
we have indeed transferred energy to the box. Its gravitational potential energy
has increased.

Gravitational potential energy — we
can call it potential energy here for short — is one energy category. Another one is kinetic energy,
which is energy due to motion. Imagine that the box is back at
ground level. Now, instead of lifting it, we push
it along the ground. Our push makes the box move, so we
are once again doing work on the box. Now, though, the energy transferred
is kinetic energy; the box has started moving after being stationary. So doing work on an object,
exerting a force on it to make it move some distance, can transfer both potential
and kinetic energy to the object. If we combine these two categories
of energy, we get a third category called mechanical energy. The mechanical energy of an object
equals the object’s kinetic and potential energies added together.

Let’s think about the mechanical
energy of the box when it’s in different positions. At rest on the ground, the kinetic
and potential energy of the box are both zero, so its mechanical energy is zero. We’ve seen we can lift the box
straight up. When we bring it to rest again, the
box’s kinetic energy is zero, but its potential energy has increased to a positive
value. So the box’s mechanical energy in
this position is positive. We’ve also started with the box on
the ground and given it horizontal motion. By pushing on the box, we made it
slide along the ground. Though its potential energy remains
zero all during this time, its kinetic energy increases from zero to a positive
value. So overall, its mechanical energy
is positive.

Lastly, imagine we pick up the box
and moved it diagonally. This way, the box has positive
potential and kinetic energy. Therefore, the box’s mechanical
energy is positive.

We can use a formula with symbols
to express the fact that mechanical energy is the sum of kinetic and potential
energy for an object. Let’s say that 𝐾 stands for
kinetic energy. Then we’ll say 𝑃 stands for
potential energy. Lastly, if we say that 𝑇 stands
for mechanical energy, then we can write 𝑇 equals 𝐾 plus 𝑃. Let’s look now at a few examples to
help us better understand mechanical energy.

In which of the following cases is
the mechanical energy of an object zero? (A) The object is at rest in any
position. (B) The object is at rest on the
surface of Earth. (C) The object is on the surface of
Earth with any motion.

Let’s remember that the mechanical
energy of an object is defined as the sum of that object’s kinetic and potential
energies. So in order for mechanical energy
to be zero, the object’s potential energy and its kinetic energy must add up to
zero. One way for this to happen is if
both of those energies are zero.

If we picture this object in
relation to Earth’s surface, we can see that the object’s gravitational potential
energy is zero when it is on the surface. An object that has zero
gravitational potential energy must also have no kinetic energy for its mechanical
energy to be zero. This means the object on Earth’s
surface must be at rest. We see this agrees with answer
choice (B). If an object is at rest on the
surface of Earth, its mechanical energy is zero.

Let’s look now at another
example.

Which of the following is correct
about the difference between the initial and final mechanical energies of a ball
that is thrown upward and then caught and held at the same height that it was thrown
from? (A) The difference is zero. (B) The difference is nonzero.

Let’s picture this situation. Here we’re holding a ball, and we
imagine throwing it straight up from a certain height. The released ball rises, slows
down, and comes to a stop. Then it falls back down to our
waiting hand. We’re told that we catch the ball
at the same height from which it was released. The question is, in this process,
does the mechanical energy of the ball change or not? Let’s recall that an object’s
mechanical energy is the sum of the object’s kinetic and potential energies. So to find our answer, we’ll want
to look at the kinetic and potential energy of the ball at the start and end of its
journey.

At the instant the ball is released
upward, it has some amount of gravitational potential energy. This is because the ball is some
height above Earth’s surface. Let’s label that energy GPE
one. Also at that time, the ball has
kinetic energy because it’s in motion. We don’t know exactly what the
kinetic energy of the ball is, but we do know that it is some specific amount, and
we can call that quantity KE one. If we add GPE one and KE one
together, we have the mechanical energy of the ball when it is thrown upward. Let’s call this energy ME one.

We want to compare ME one with the
ball’s mechanical energy at the instant we catch it when it’s falling downward. That energy, which we can call ME
two, equals the ball’s gravitational potential energy plus its kinetic energy at
this instant. Notice that we catch the ball at
the same height it was released from. So if GPE two is the ball’s
gravitational potential energy when it lands in our hand, then GPE two equals GPE
one.

Now how about the ball’s kinetic
energy? In general, an object’s kinetic
energy is one-half its mass times its speed squared. If we think about the ball’s motion
as it rises and falls, we know that as it rises, it slows down, and as it falls, it
speeds up. If we ignore the effect of the air
slowing the ball down, then the amount the ball slows down while ascending and the
amount it speeds up while descending are the same. All this means that the ball’s
speed upward when we first release it will be the same as its speed downward when we
catch it. The speeds are the same. And since the mass of the ball
doesn’t change, the ball’s kinetic energy at both of these two instants must be
equal as well.

We’re finding then that ME two, the
mechanical energy of the ball when we catch it, is equal to ME one, the mechanical
energy of the ball at the instant it’s released. For our answer, we’ll choose option
(A). The difference between the
mechanical energy of the ball at these two instants is zero.

Let’s now work an example where we
calculate a specific energy quantity.

The mechanical energy of an object
that is at Earth’s surface is 36 joules. What is the kinetic energy of the
object?

These two energy categories,
kinetic energy and mechanical energy, are related. An object’s mechanical energy
equals its kinetic energy plus its potential energy. We can write this as an equation
using symbols: 𝑇, mechanical energy, equals 𝐾, kinetic energy, plus 𝑃, potential
energy. In this example, we’re given 𝑇,
the object’s mechanical energy, as 36 joules. We also know that the object is at
Earth’s surface. This means its gravitational
potential energy is zero. That’s true for every object at the
surface of the Earth.

So in our equation, 𝑇 is 36 joules
and 𝑃 is zero joules. 𝑃 plus 𝐾 is 36 joules. Therefore, it must be the case that
𝐾 is 36 joules. Our answer is that the kinetic
energy of the object is 36 joules.

Let’s look now at one last
example.

A car drives up a hill, slowing
down as it climbs the hill. The kinetic energy of the car
decreases by 12 joules and the gravitational potential energy of the car increases
by 12 joules. What is the change in the
mechanical energy of the car?

As this car drives uphill, it slows
down. This change in speed impacts its
kinetic energy. The change in the car’s elevation
changes its gravitational potential energy. We want to find the change in the
car’s mechanical energy. This is equal to the change in the
sum of its potential and kinetic energies.

We are told that as the car
ascends, its kinetic energy decreases by 12 joules. This means that the change in the
car’s kinetic energy is negative 12 joules. The reason this change is negative
is because the car is slowing down, so its kinetic energy is decreasing. We also know that the change in the
car’s potential energy equals positive 12 joules. This change is positive because the
car gets higher up as it climbs, which increases its potential energy. Adding these changes in energies
together gives zero joules.

Therefore, the change in the car’s
mechanical energy is zero joules.

Let’s now summarize what we’ve
learned in this video. Doing work on an object will cause
its kinetic energy or potential energy or both to change. An object’s mechanical energy is
the sum of its kinetic energy and potential energy. Written as an equation with
symbols, mechanical energy, 𝑇, equals kinetic energy, 𝐾, plus potential energy,
𝑃.