A large wooden turntable in the shape of a flat uniform disk has ...

Question

A large wooden turntable in the shape of a flat uniform disk has a radius of 2.00 m and a total mass of 120 kg. The turntable is initially rotating at 3.00 rad/s about a vertical axis through its center. Suddenly, a 70.0-kg parachutist makes a soft landing on the turntable at a point near the outer edge. (a) Find the angular speed of the turntable after the parachutist lands. (Assume that you can treat the parachutist as a particle.)

Accepted Answer

Hey everyone welcome back in this problem. We have a wooden dining table. Okay. And it has two concentric circular surfaces. Kid has a big one. And outer surface that's going to be stationary. That's where people eat. Okay. And then on the inner surface replacing all the food and drinks. Okay. And the inner surface is going to be rotating about the vertical axis. Okay? We're gonna put a dish on the edge right on its outer edge. Okay. And what we wanna do is we want to calculate the angular speed of the inner surface once that dish has been placed on it. Okay. We're told that we can treat the dish as a pointless. Alright, so let's just draw what's going on here. So pretend that we're looking top down at this table, we have this big stationary outer circle and then we have the inner surface. Okay? The inner surface rotates and it rotates within omega initial of 2.5 Iranians per second. Hey, that's given here. We're told that it has a radius of 1.2 m. Okay, so the distance from the middle to the outside here, the radius 1.2 m. And that at 6.75 kg. So this inner disc 6.75 kg. Okay, now we have the table again. And the inner desk. And what we're gonna do is we're gonna put a dish right on the edge this and we want to know what's going on. What is the angular speed when we do that. Okay, so what is omega? All right, well, we know that we have no external torques acting on this system. So we're gonna have conservation of angular momentum. That tells us that we have L. I. K. The initial angular momentum equal to L. F. The final angular momentum. Let's recall what is angular momentum. Well, when we're talking about linear momentum, we have M V. Mass velocity. We're talking about angular momentum. We have I the moment of inertia times. Omega the angular velocity. Okay, so in this case we're going to have I initial omega initial is equal to I final. Oh my God. Final and again, omega final. That's what we're looking for. So let's fill in the details and see what we can find. Alright, let's start with I initial. So initially again we have this surface it's a disk. So we can look at our moment of inertia table for a disc or cylinder And we'll see that the moment of inertia for that shape is 1/2 mm r squared. And then we have omega I now for the final moment of inertia or it's going to be made up of two components. Okay, We're gonna have the moment of inertia from the surface itself. It's gonna be the same as before. And then we're gonna have the moment of inertia from the dish that we added to the edge. Okay. And both of those things are going to be moving together with the same. Oh my God. Okay. So we have one speed where both the disk is traveling at the speed and the dish on top of the surface is going to be traveling at the same speed. Alright, so let's fill in what we know. So we have one half we know that the mass of the disk is 6.75 kg. The radius of the disk we have is 1.2 m Squared. OK? Oh my God, I were given his 2.5 radiance per second. Okay, so that's everything on the left hand side now, on the right hand side. Well, I and the moment of inertia of the surface again and that's the same surface as we had before. So the moment of inertia for that portion is gonna be the same. It's gonna be one half 6. kg Times 1.2 m two. Now, the moment of inertia of the disc, we're told of the dish, sorry, not the disc of the dish that we place on the outside, we're told to treat it as a point mass. Okay, so when we have a point mass, the moment of inertia is given by M R. Squared. Okay, Where this is the mass of the dish? Alright. And we have oh my God, Alright working some of this out on the left hand side, we're gonna get 12.5 12.15 sorry, 12.15 in our units we have kilogram meters squared times, radiant per second. So kilogram meter squared. Okay. And it looks messy now but it will simplify. It's a good idea to include your unit. So you make sure that you haven't made any mistakes. Okay, on the right hand side working this out, we're gonna have four 0. And then plugging in the information. We know the mass of the dish that we add to the edges five kg. And we forgot our units here, kilogram meters squared. Alright, so then five kg times the radius 1. m squared. Alright, time's up Omega F. Okay, so on the left hand side we still have 12.15 kg meter squared radiance per second. And let us just move down. So we have a little bit more room to work. Alright. On the right hand side we are going to have 12.6. And again the unit kilogram meter squared kilogram meters squared. So we have 12.6 kg meters squared times on my Gaff. Okay, we're gonna divide And when we do that, the unit of kilogram meter squared is going to cancel? We're gonna be left with omega F is equal to 1.01 in our unit. That's left is radiant for a second. Okay. Which is the unit that we want? We're talking about angular speed. So that's great. Omega f. radiance per second. And that's going to correspond with answer. D Thanks everyone for watching. I hope this video helped see you in the next one.

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