Energy in Pendulums - Online Tutor, Practice Problems & Exam Prep

In pendulum motion, energy oscillates between kinetic energy and gravitational potential energy. At maximum height, gravitational potential energy is highest, while kinetic energy is zero. Conversely, at the lowest point, kinetic energy peaks and gravitational potential energy is zero. The height can be calculated using the equation: h_max = l(1−cosθ). The conservation of energy principle applies, allowing the derivation of maximum speed as v_max = √2g(l(1−cosθ)).

Energy in Pendulums

Hey, guys. Remember when we talked about the energy of mass spring systems? We're going to do the same thing for pendulums in this one. So remember that just a little refresher about mass spring systems, you have this object that's moving back and forth and it's oscillating between due to different kinds of energies. We've got the elastic potential energy that's the maximum at the endpoints, and then you've got the kinetic energy that's maximum at the equilibrium position. But for pendulums, it's the same thing. So as you pull this thing back and it's swinging back and forth, it's got 2 types of energy. It's got kinetic energy depending on how fast it's moving, and because it's going up and then coming back down again, it's changing this vertical height, it's also got gravitational potential energy. And those are the 2 things we're going to be looking at in this video.

So as this thing is swinging back and forth, right, you're going to take this pendulum that's length l, and you're going to pull on it so that it's on some θ angle, and you're going to release it. But when you release it, that θ is equal to its maximum value because it's just going to swing back and forth between those angles. So what does the gravitational potential look like? Well, the gravitational potential we always have to establish like a zero point. And for the zero point in pendulums, it's going to be at the bottom of the swing. So that's h0 = 0. So now what happens is that when you pull this thing all the way out to θ maximum, now you have the maximum height right here, where the height is measured relative to that zero point. So that means at the very, very top, the gravitational potential energy is maximum and the kinetic energy is equal to 0. It's not moving.

So let's take a look at the equilibrium position. The equilibrium is the exact opposite. So now you're at the bottom of the swing right here. You're at the bottom of the swing. So your gravitational potential energy when θ=0 is just 0, and then your kinetic energy is at its maximum value.

So that means the total mechanical energy for these things, in the first case, is just mghmax. And then for this guy, it's just 12 mvmax^2. And then at any other point in between, I've got this, like, point right here, θp. So this is θ p, and the energy is just going to be a combination of both of them. So I'm just going to have whatever the height is at that specific point, plus whatever the kinetic energy is at that specific point, and that's it. But a lot of the times, what happens in problems is we don't have what the height is specifically. We're actually going to see that in the example that we're going to get to. So how do we figure out what that height is if we're only just given l and the θ angle itself? Here between the zero point and wherever high we are. So let's figure that out.

If you've got the total entire length of the pendulum, and that's equal to l, and I'm trying to figure out what this highlighted distance is, then all I need to do is just figure out what this little distance is in the triangle. So using SOHCAHTOA, I'm going to have l × cosθ = that little vertical piece right there. So that means that this little highlighted distance is actually going to be the difference between the length and the lcosθ. So that means that for any θ, the height is just equal to l−lcosθ. This is sometimes called the pendulum equation. And the other way you might see it is l1−cosθ, and that's it.

So let's rewrite our mechanical energy formula. So if all of these things are conserved between all these cases, then that means that mghmax = 12 mvmax^2, and that's just equal to mgh@acertainpoint, and then plus 12 mvv@acertainpoint^2. And that's basically the energy conservation formula for pendulums. So that's it. Let's take a look at an example. So like I said before, a lot of times in problems, you won't see what the actual height of this thing is because then you could just use normal energy conservation. So notice how in our energy conservation for pendulums, it's exactly the same thing as what we did for energy conservation for, like, gravity and kinetic energy and stuff.

Okay. So we've got this mass and it's attached to this pendulum. It's some length l. So let me go ahead and start filling stuff in. So I've got this as l and it's pulled up an angle θ and then it's released. So that means that that angle is already at its maximum. It's just going to swing back and forth. That's the amplitude. And we're supposed to figure out what the maximum speed is. We're supposed to derive an expression for the max speed. So here's what I'm going to do. I'm going to go up here and I'm just going to copy down my energy conservation formula. I'm going to put that guy, like, right over here. So that's my energy conservation formula. And what am I looking for? So this is equals, this is equal, and this is a plus sign. So I'm actually looking for what the V_max is. And what am I given? Am I given the energy at a specific point, or am I given the amplitude's energy or the energy at the amplitude? Well, let's see. When θ = its maximum, now we just have to establish what our 0, potential energies are. So let me just move this over a second. And so I've got the zero energy is going to be at the bottom of the swing, and the maximum energy is going to be at hmax. So if I've got hmax because I've got θmax, I'm going to use these two equations right here. The problem is that I'm not asked for hmax, and I'm not given what hmax is. I'm supposed to get this in terms of l and θ.

So here's what I'm going to do. I'm going to take my pendulum equation that I just derived when h=l one - cosθ. And when θ = maximum, so hmax = when 1 - cosθ is at its maximum value. So now I'm just going to take this expression here and plug it in for H_Max. So let's go ahead and do that. So I've got mg, and this is going to be l one - cosθmax = 12 m vmax^2. And that's what I'm looking for, v_max. So if we're looking for this vmaximum right here, take a look. I've got these m's that actually cancel, and then I can move the one half to the other side. So that means I'm going to get 2 g l one -cosθ max is equal to vmax^2. Now all I have to do is just take the square root. So if I move that over, I get that v_max = √2 g l one -cosθ maximum. So this is how problems will usually go. They'll add they'll tell you the length of the pendulum, and they'll tell you the angle that they pull it back, and you can find out what the energy is based on you just using the energy conservation formula. So let me know if you guys have any questions. Let's keep moving on for now.

Example

Hey, guys. Let's get some more practice with these energy and pendulum questions. So we're told the mass of this some object that's hanging from a 2 meter pendulum. And then as it's making a 5-degree angle with the vertical, it has a speed of something. And we're supposed to figure out what the maximum height is. So let's just start writing some stuff out. The mass is equal to 0.4. The length is equal to 2. Now what does this second set mean? So it makes a 5-degree angle with the vertical, and it has a speed of 1.5 meters per second. So if I just go ahead and draw my little pendulum diagram, so what happens is, this is going to be my pendulum. And at this specific point here where it makes a 5 degrees, we're told that this thing already has a velocity in this direction or a speed of 1.5 meters per second. So what this is telling us is this is not the amplitude. This is actually a very, very specific point where the amplitude is actually somewhere out here probably. So that's the real amplitude and this is theta p. So what they're telling us is that theta p is equal to 5 degrees. And at that specific point, the velocity is 1.5 meters per second. And now we're supposed to find out what the maximum height is. So what does that mean? That represents h max. So let's go through our energy conservation equations for a pendulum. And really, the h max only pops up in one place, this mgh_max. So we know we're using energy conservation, but which equations are we gonna use? Well, let's take a look. So I've got mgh_max, and I've got the m, and I've got the g. So I've got m, but I don't have what v_max squared is. I'm not told what the speed is at the bottom of the swing. I'm not told at v_max. So I'm not sure I'd be able to figure that out. And then let's look at the second part. So I've got m and g. So let me go ahead and check that off. I got m and g. I've got m and the velocity at a point squared. But what about this h_p? What about this height at a specific point? Well, if I've got the angle and I've got the length of the pendulum, I can probably figure out what the height is. So let's go ahead and use that relationship. So let me set up that equation. I've got mgh_max is equal to mg whoops. Mgh_@apoint plus one-half m v at a point squared. So what I'm really looking for here is what is this maximum height. So again, I've got all of these variables. All I have to do is just figure out what the height at a specific point is. So how do I do that? Let's go ahead and bring that over here. So how do I figure out what the height is given some masses, lengths, and then theta angles? Well, I can use the pendulum equation. Remember that the height at any point is given as the length of the pendulum, 1 minus the cosine of theta_p. We just have to remember that all of this stuff when we're using cosine has to be in radians. Okay. So if you take a look here, I've got the length of the pendulum and I know what theta_p is in terms of degrees at least. So that means I can actually figure out what this h_p is. I can figure out what that height at that specific point is. So let's go ahead and figure it out. The first thing I'm going to do is I'm going to convert this 5 degrees over to radians. So it means theta_p is 5 times pi over 180. And so what I get is an angle of 0.087, and that's gonna be in radians. So now I'm just gonna put that in there. So I've got that the height at a point, so h_p, is the length of the pendulum, which is 2. And then I've got 1 minus the cosine of 0.087. What you're gonna get is h_p is equal to something very, very small. So 0.0076. It's important to keep those extra decimal places because you don't want to round off too early. Okay. So now that we've got this number, I can just plug it into this formula and now I'm good to go. So the rest of it is just plugging in numbers and then dividing stuff over. Okay. So if I write all this stuff out, I get 0.4 and then I get 9.8 and then I get h_max is equal to 0.4, then I get 9.8. The height at that specific point is 0.0076. Then I've got plus 1 half m, and sorry, m is equal to 0.4 again. And then I've got v_p squared. So that is 1.5 meters per second. Right? That's the velocity at a specific point. So if you go ahead and just plug all of this stuff into your calculator, what you're going to get is you're going to get 0.4 times 9.8. h_max is equal to 0.48. So now if you just divide this stuff over to the other side, you're just gonna get that h_max is equal to 0.12 meters. So that's the maximum height that this pendulum will reach. Alright, guys. Let me know if you have any questions. That's it for this one.

A mass swinging at the end of a pendulum has a speed of 1.32m/s at the bottom of its swing. At the top of its swing, it makes a 9° with the vertical. What is the length of the pendulum?

• A. 6.83 m
• B. 7.22 m
• C. 1.77 m
• D. 5.18 m

Here’s what students ask on this topic:

What is the energy conservation formula for a pendulum?

The energy conservation formula for a pendulum states that the total mechanical energy (sum of kinetic and potential energy) remains constant throughout its motion. At any point in the pendulum's swing, the total energy can be expressed as:

$E_{\mathrm{total}}=U_g+K_e$

Where $U_g$ is the gravitational potential energy and $K_e$ is the kinetic energy. At the highest point, the energy is all potential:

$E_{\mathrm{total}}=U_g=mgh$

At the lowest point, the energy is all kinetic:

$E_{\mathrm{total}}=K_e=\frac{1}{2}m{v}^2$

How do you calculate the maximum height of a pendulum?

The maximum height $h_{\max }$ of a pendulum can be calculated using the equation:

$h_{\max }=l(1-cos\theta )$

Here, $l$ is the length of the pendulum and $\theta$ is the maximum angle from the vertical. This equation derives from the geometry of the pendulum's swing, where the height is the difference between the pendulum's length and the vertical component of the length when displaced by angle $\theta$.

What is the maximum speed of a pendulum?

The maximum speed $v_{\max }$ of a pendulum can be derived using energy conservation principles. The formula is:

$v_{\max }=\sqrt{2gl(1-cos\theta )}$

Here, $g$ is the acceleration due to gravity, $l$ is the length of the pendulum, and $\theta$ is the maximum angle from the vertical. This formula shows that the maximum speed occurs at the lowest point of the swing, where all the potential energy has been converted to kinetic energy.

How does the length of a pendulum affect its energy?

The length $l$ of a pendulum significantly affects its energy. The gravitational potential energy at the maximum height is given by:

$U_g=mg{h}_{\max }$

Since $h_{\max }=l(1-cos\theta )$, the potential energy depends directly on the length of the pendulum. A longer pendulum will have a higher maximum height for the same angle $\theta$, resulting in greater potential energy. Consequently, the kinetic energy at the lowest point, which is equal to the initial potential energy, will also be higher for a longer pendulum.

What is the relationship between kinetic and potential energy in a pendulum?

In a pendulum, kinetic energy (KE) and potential energy (PE) oscillate between each other. At the highest point of the swing, the pendulum has maximum potential energy and zero kinetic energy:

$U_g=mg{h}_{\max }$

At the lowest point, the pendulum has maximum kinetic energy and zero potential energy:

$K_e=\frac{1}{2}m{v}^2$

Throughout the swing, the total mechanical energy (sum of KE and PE) remains constant, illustrating the conservation of energy principle. As the pendulum moves, energy continuously converts between kinetic and potential forms.