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.