Video Transcript
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.
