17. Periodic Motion

Energy in Pendulums

# Example

Patrick Ford

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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 m pendulum and then as it's making a five 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. Mass is equal to 0.4. The length is equal to two. Now, what does the second this third? Yeah, seconds. I mean, so it makes a five degree angle with the vertical, and it has a speed of 1.5 m per second. So if I just go ahead and draw on my little pendulum diagram, so what happens is so this is gonna be my pendulum. And at this specific point here, where it makes five degrees, we're told that this thing already has a velocity in this direction or speed of 1.5 m per second. So what this is this 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 That's the rial amplitude and this is state API. So where they're telling us is that Data P is equal to five degrees, and at that specific point, the velocity as is 1.5 m 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 pendulum. And really the H Max Onley pops up in one place this MGH max. So we know we're using energy conservation, but which equations that we're gonna use. 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 em. But I don't have what V Max squared is I'm not told what the's speed is at the bottom of the swing not told 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 em n g. So let's go ahead and check that off M and G, I've got em and the velocity at a point squared. But what about this hp? 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 M G H. Max is equal to M G. Whoops. M g h at a point 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 one minus co sign of theta P. We just have to remember that all of this stuff we're using coastline has to mean radiance. Okay, so if you take a look here, I've got the length of the pendulum. And I know what they Tapie is in terms of degrees at least. So that means I could actually figure out what this HP is. I can figure out that height at that specific point is, so let's go ahead and figure that out. The first thing I'm gonna do is I'm gonna convert this five degrees over the radiance, so it means that a P is five times pi over 80. That's what I get is an angle of zero point 087 and that's gonna be in rads. So now I'm just gonna put that in there and then make sure that my calculator is in degrees mode. So I've got that the height at a point. So HP is the length of the pendulum, which is two. And then I've got one minus the co sign of 0. 87. What you're gonna get is H P is equal to something very, very small. So 0.76 it's important to keep those extra decimal places because I don't want to round off too early. Okay, so now that we've got this number, I could just plug it into this formula and now I'm good to go. 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. 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 0.0.76 Then I got plus one half m on Sorry, em is equal. 2.4 again. And then I've got V P squared. So that is the 1.5 m 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 gonna get is you're gonna get 0.4 times 9. h max is equal to 0.48. So if you just divide this stuff over to the other side, you're just gonna get that H Max is equal to 0. m. 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

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