Lesson Video: The Mechanical Energy of a Pendulum Physics

In this video, we will learn how to calculate the gravitational potential energy and kinetic energy of a pendulum at different points in its motion.

14:16

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.

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