A woman with mass 50 kg is standing on the rim of a large disk that is rotating at 0.80 rev/s about an axis through its center. The disk has mass 110 kg and radius 4.0 m. Calculate the magnitude of the total angular momentum of the woman–disk system. (Assume that you can treat the woman as a point.)

Video transcript

Hey everyone, welcome back in this problem. We have a revolving gate. Okay, It has four arms that are at right angles and it rotates around a central axis. Okay, so we have some sort of central axis here, like this and then we have four arms that are all the same, so pretend that these are all the same despite my poor drawing skills. Okay, Alright. They have a massive 8.2 kg. Okay, so each arm has a mass of 8.2 kg And they are 2.6 m long. Okay, so this distance here is 2.6 m. Alright, now we know that a vandal, we're also told a vandal that is 59 kg is gonna sit on the outer end of one of the arms. Okay, so we have the vandal sitting here on the outer end and they are 59 kg. Okay, they're gonna push against the wall and then this entire thing is going to rotate and let's have it rotating the positive direction, It's going to be rotating 3. revolutions per second. 3.6 revolutions per second. What we want to know is find the angular momentum magnitude of this vandal gate system. Okay. And we're told that we can treat the arms as uniform rods in the vandal as a point mass. Okay, so that's important when we're talking about angular momentum, when we're doing the moment of inertia calculations will need that information. Alright, so we have our little diagram here, let's just write out the information we're given. Okay, so we have the mass of one of the arms Of the gate is 8.2 kg. Okay. The radius of one of the arms. Okay, so the length of the arm is 2.6 m. The radius when we're talking about rotation from the central axis is going to be 2. m. Okay. And then the omega, It's going to be 3.6 revolutions per second. And we talk about a revolution. We talk about going all the way around one entire turn. Okay, well if we go one entire turn, that's two pi radiance. Okay, so we essentially have 3.6 times two pi um radiance per second. So that's our omega. That's our angular velocity or angular speed. We also have the mass of the vandal. So we'll call it M. V. That's 59 kg In the radius of the vandal RV while he's sitting at the very edge. Okay, so his radius or the distance from him to the central access to the axis of rotation is 2.6 m. Alright, perfect. So that's all the information we're given in a little diagram. And what we want to know is what is the angular momentum. L Now let's recall the angular momentum. L is given by I the moment of inertia times omega. Alright, well, moment of inertia I has to be made up of two components. It's made up of the inertia. Of the actual arms that are spinning around plus the moment of inertia of the vandal. Okay, so we have the eye of the arms plus i of the vandal all times omega. And we have a single omega. We can kind of factor these. We have one omega because the vandal is sitting on the gate and they're moving together so they have the same speed and that's why we're treating it as just one omega. All right. Now, when we think about the mass or the sorry, moment of inertia of the arms? I Well, this is really we have four arms that are all the same. So this is just gonna be four times the moment of inertia of a single arm plus the moment of inertia of the vandal times. Omega. Alright, let's just scroll down, we're just going to give ourselves a little bit more room to write. All right now, the moment of inertia of the arm, we're told to treat the arm as a uniform rods. Okay? So, if we look at our moment of inertia table for uniform rod rotating about its end. Right? So the rods are here and the it's rotating about the end of the rod. Well, that's one third M L squared. Okay, so we're gonna have four for the four arms Then we're gonna multiply by 1/3. Um L squared. Ok, And this is going to be m Of the arm a and L squared. Now the moment of inertia for the vandal V. That's just going to be M R squared. We're told to treat it as a point mass. Okay, So again, looking at our moment of inertia table whether that be in your textbook or provided by your professor, Okay, We're going to have m the mass of the vandal, r the radius of the vandal squared. Okay. All of this times omega. Now let's substitute the values we know. So we're gonna end up with four thirds em we're talking massive the arm and in this case we're doing one arm. Okay, 8.2 kg L Is the length of the Rod, m all squared. The mass of the vandal is 59 kg. The radius. The distance between the vandal and The central axis is 2. m because he's sitting on the very edge of the Rod. And then we're gonna take all of that times are 3.6 times two pi radiance per second. Okay? And so here we're going to get 7.2 pi per second. All right. Now, it's just a matter of working out all this multiplication edition. Okay. On our calculator and what we'll see is that we end up with 73.9093 kg meters squared Plus 0.84 kilogram meters squared. Okay. And then that's going to be all times 7.25 radiance per second. Alright. So Here we go. 10693. kilogram meter squared per second. Okay, just moving up just a little bit Writing This in scientific notation. Okay. We're gonna have 1.07 times 10 to the four kilogram meters squared per second, which is the unit we want for our momentum. L Okay, so our angular momentum of the vandal gate system is going to be 1.7 times 10 to the four kg meters per second squared. That is going to be answer. E. Thanks everyone for watching. See you in the next video.