### Video Transcript

In this video, our topic is the
mechanical energy of a pendulum. We’re going to see how the energies
that make up mechanical energy change over time as a pendulum swings back and
forth. We’ll learn how to calculate those
energies, and we’ll also see how to interpret graphs of the energy of a pendulum
over time.

As we get started, let’s first
recall a bit about the motion of a pendulum. We know that in general a pendulum
consists of a mass, sometimes called a bob, attached to the end of an arm of fixed
length, where this arm might be a solid rod or it might be a length of cord. We know that if we move the
pendulum bob away from what we could call its equilibrium position, once we release
the pendulum, it will swing back to this position and then past it until it’s gone
as far away from equilibrium on this side as it did over here at which point it
swings back down through its equilibrium position and so forth and so on. If we assume that there are no
losses to friction anywhere in this process, then the bob will continue to move back
and forth indefinitely.

If we think about the speed of the
pendulum bob at these different snapshots in time, we know that at its extreme
positions on the left and on the right, at these moments, the speed of the pendulum
is zero. But then, in either case, as the
pendulum moves back down to its equilibrium position, by the time it reaches that
position, its speed is some maximum value, the highest it will ever get. Right away then, we can see that
when it comes to the kinetic energy of this pendulum over time, that energy
changes. When the pendulum is as high as it
ever gets, then that energy is zero. And when it’s as low as it ever
gets at what we’ve called its equilibrium position, that energy is at a maximum. So, that’s the story of the kinetic
energy of the pendulum, which we can recall is equal to one-half its mass times its
speed squared.

But we can also consider the
pendulum’s gravitational potential energy. Looking at this relationship, we
know that the mass 𝑚 of the pendulum is fixed. And we can assume that the
gravitational field the pendulum moves through is uniform. So that means lowercase 𝑔 is
constant. And therefore, for a pendulum, the
only factor that changes with time is the height ℎ. We can specify height values in
this scenario by drawing a dotted line through the center of mass of our pendulum
when it’s at its equilibrium position. And what we’ll say is that we
define this height to be zero. Knowing this, if we draw a parallel
line through the center of the pendulum when it’s at its maximum height values, then
we know that the perpendicular distance between these two lines is the maximum
height the pendulum will achieve. As the pendulum swings back and
forth then, we can see that not only does kinetic energy change but so does
gravitational potential energy.

We can see this more clearly if we
make a graph of energy on the vertical axis and time on the horizontal. Let’s say that our pendulum is held
at this position and then released at time 𝑡 equals zero. So, the very first data points on
our graph will show what’s going on at that moment. If we think about the kinetic
energy of the pendulum at this moment, we’ve said that the speed of the pendulum is
zero and, therefore, its kinetic energy must be zero too. On our graph, we can start to
create a kinetic-energy-versus-time curve. And the first point on that curve
will be at the origin.

Similarly, we can plot the
gravitational potential energy of the pendulum over time. Unlike kinetic energy, at this
initial moment, the gravitational potential energy is at a maximum value. If we make a GPE-versus-time curve
in orange, then the first data point of that curve, we can say, is up here. So far then, we see how these two
energies of the pendulum start out. We know, though, that as soon as we
release the bob, it will start to descend until it reaches this position. Over that interval, two things take
place. First, the bob’s speed gets faster
and faster until it reaches some maximum value. And second, the height of the bob
descends until it reaches its minimum value, a height we’ve called zero.

Now, let’s say that our pendulum
reaches this equilibrium position at this tick mark on our time axis. Because its height at this moment
in time is zero, as we’ve defined it, we can say that it no longer has any
gravitational potential energy. Starting from the release of the
bob, the GPE has gone like this. And the fact that it’s now zero at
this moment in time tells us something about the kinetic energy of the pendulum. This is because the energy of our
pendulum system must be conserved over time. That is, at any moment, if we add
together the pendulum’s kinetic energy and its gravitational potential energy, that
sum, which is called the mechanical energy of the pendulum, must be constant. That is, at any instant in time, if
we add together these two values, we get the same result. That’s what it means for the energy
of the system to be conserved.

Getting back to our graph then, if
at this instant in time the gravitational potential energy is zero, then that must
mean the kinetic energy of our pendulum, which is now at a maximum value, is
here. That way, the total energy of our
system at this moment in time is the same as it was at the outset. That is, the mechanical energy of
our system is the same. Now, once our pendulum reaches this
position where its speed is maximum, we know that it won’t stop there but will
continue ascending until its height equals its original height. We can say that that instant
corresponds to this tick mark on our time axis. And once the bob is here, once
again its speed is zero and its height is back to a maximum value. And therefore, its kinetic energy
is back to zero, and its gravitational potential energy is back to its maximum.

As the pendulum continues to swing
back and forth, this cycle continues. And what we’re seeing is that if we
were to plot the mechanical energy of the pendulum on this graph, then that curve
would be a flat line that looks like this. It’s always equal to the sum of KE
and GPE. And since, by energy conservation,
that sum is constant. The gradient of the mechanical
energy curve is zero. Considering this graph, there’s
actually more than it tells us about the pendulum’s motion beyond how its energies
change over time. For example, we’ve seen that by
considering our kinetic energy curve by, looking at where this curve has a minimum
and then a maximum and then a minimum, we can infer from that the corresponding
position of the pendulum.

We know that at these minimum
values of kinetic energy, the pendulum is at its maximum height and then at the
maximum kinetic energy value, it’s back at what we’ve called its equilibrium
position. Therefore, as we moved on our time
axis from a time equals zero up to this value here, our pendulum has gone through
one-half of a complete cycle. That is, it’s gone from its maximum
height on one side of the equilibrium position to its maximum height on the
other. To complete one cycle, it would
then have to do this in reverse. And we know that if it did complete
this cycle, it would do that in a period of time called the period and represented
capital 𝑇. On our graph then, we’re seeing
that this interval of time is equal to 𝑇 divided by two, one-half the pendulum’s
period. And then, as we consider its
period, we can recall that this is equal to the inverse of the frequency. So, one-half the period of
oscillation is equal to one divided by two times the pendulum’s frequency.

And note that we could discover the
same relationship by looking at GPE instead of KE. For this energy, instead of looking
at the time interval between minimum values, we look at that interval between
maximum values. The sinusoidal changes in GPE and
KE then can give us information about the pendulum’s period and its frequency.

Now, there’s one more thing that we
can see about the gravitational potential energy of our pendulum. We recall that in general, the GPE
of an object in a uniform gravitational field is equal to the object’s mass
multiplied by the acceleration due to gravity times the object’s height above some
standard level. On our diagram, we’ve called that
maximum height ℎ sub max. And it turns out, we’re able to
solve for this value using other parameters in this scenario. Specifically, when we work with a
pendulum swinging back and forth, we’re often told the length of the pendulum arm,
we’ll call that 𝑙. In addition, the maximum angular
deviation from equilibrium may be known. We can call that angle 𝜃. We can see that this maximum height
achieved is equal to the length of the pendulum minus this distance right here. That difference is equal to ℎ sub
max. And so we’d like to solve for this
length that we’ve marked out.

To do that. Let’s consider this right triangle,
the hypotenuse of which we know to be 𝑙. Knowing that, along with the angle
𝜃, tells us that the length of this leg of the triangle is equal to 𝑙 times the
cos of 𝜃. And so, to solve for ℎ sub max,
we’ll subtract 𝑙 times the cos of 𝜃 from the overall length of our pendulum arm;
that’s 𝑙. Returning to our equation for GPE,
this means we can replace the factor ℎ with 𝑙 minus 𝑙 times the cos of 𝜃. And then, if we factor out one
multiple of 𝑙 from this expression, we see that for a pendulum of length 𝑙 with
maximum angle of deviation 𝜃, gravitational potential energy is equal to 𝑚 times
𝑔 times 𝑙 times the quantity one minus the cos of 𝜃.

Knowing all this about pendulum
energy, let’s get some practice now with these ideas through an example.

Which of the lines on the graph
correctly shows how the gravitational potential energy of a pendulum compared to
that at its equilibrium position varies with time?

On this graph, we see gravitational
potential energy in joules plotted against time in seconds. There are a number of different
lines on the graph. There’s a black one here. That’s a flat line. Then, there’s a red one, a yellow
one, a blue one, and here a purple one. We want to know which line
correctly shows how the gravitational potential energy of a pendulum compared to
that at its equilibrium position varies with time. So, say that this is our pendulum,
and we see it here at three different snapshots in time. The center position of the pendulum
here is what we call its equilibrium position. This is where the pendulum
naturally moves if it’s not perturbed. But then, if we do move it to a
side, say over here, and then release the pendulum, we know that it will start this
back-and-forth swinging motion. And if we imagine no friction in
our system, then this motion goes on indefinitely.

If we imagine that all of the mass
of the pendulum is at its end, called the pendulum bob, and none of it is in the arm
that supports the bob as that swings back and forth, then in that case we can
understand the gravitational potential energy of this pendulum by tracking the
motion of the bob as it moves. In general, the GPE of an object in
a uniform gravitational field is equal to the mass of that object multiplied by the
acceleration due to gravity times the object’s height above some reference. For our pendulum, that reference
level, we can say, goes right through the middle of the bob when it’s at its
equilibrium position. We’ll say that this height
corresponds to a height of zero. This definition is important
because it means that at instants in time when our pendulum is at its equilibrium
position, its gravitational potential energy is zero.

And therefore, whatever line on our
graph correctly shows GPE must reach the horizontal axis. We see that enforcing that
condition eliminates two of the possible lines, the black line and the purple
one. Neither of these lines crosses the
horizontal axis. And we see that they fail for
another reason. Note that they show us a constant
gravitational potential energy over time, whereas really we know that the height of
our pendulum bob, as it moves up and down, is changing the gravitational potential
energy of this system. For a few reasons then, we won’t
choose the black line or the purple line as our answer.

Considering once more our bob in
this equilibrium position, we said that at that point the gravitational potential
energy of the pendulum is zero. The question is, how does that
amount of GPE relate to the GPE of the system at other times? In other words, is it a minimum
value, a maximum value, or somewhere in between? Our sketch shows us that at every
instant other than times where the bob is at this equilibrium position, the height
value of the pendulum, as we’ve defined it, will be positive. And therefore, since 𝑔 and 𝑚 are
both positive as well, at all those instants the gravitational potential energy of
the pendulum will be positive too. This shows us that the zero points
we expect on our line should be minima, that is, the low points of the gravitational
potential energy of this pendulum.

Now, if we go and look at the blue
line, we see that this has maximum values at zero, while the yellow line shows us
zero values between the max and mid values of the line. It’s only the red line which does
have zero values where those values correspond to the low points on the curve that
satisfies this condition. And so, this is our answer. It’s the red line on this graph
that correctly shows how the gravitational potential energy of a pendulum compared
to that at its equilibrium position varies with time.

Let’s summarize now what we’ve
learned about the mechanical energy of a pendulum. In this lesson, we saw that when a
swinging pendulum is at its equilibrium position, then its gravitational potential
energy is zero and its kinetic energy reaches its maximum value. And then, when a swinging pendulum
is at its maximum height, it’s kinetic energy equals zero and its GPE reaches a
maximum value.

We saw further that the mechanical
energy of a pendulum, that is, the sum of its kinetic and gravitational potential
energy, is constant and that when we look at a pendulum-energy-versus-time graph,
from this graph, we can understand the pendulum’s period 𝑇 and its frequency
𝑓. And lastly, we saw that for a
pendulum of mass 𝑚 and arm length 𝑙 moved an angle 𝜃 away from its equilibrium
position, the gravitational potential energy of that pendulum is equal to 𝑚 times
𝑔 times 𝑙 multiplied by the quantity one minus the cos of 𝜃. This is a summary of the mechanical
energy of a pendulum.