A 70.0-kg person stands on a tiny rotating platform with arms ...

Question

A 70.0-kg person stands on a tiny rotating platform with arms outstretched.

(a) Estimate the moment of inertia of the person using the following approximations: the body (including head and legs) is a 62.0-kg cylinder, 16 cm in radius and 1.70 m high; and each arm is a 4.0-kg thin rod, 60.0 cm long, attached to the cylinder.

Accepted Answer

Hello, fellow physicists today, we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. A 75.0 kg skater spins on the ice with arms outstretched. The skater's body is considered a 50.0 kg cylinder with a radius of 0.300 m and a height of 1.60 m. Each arm is modeled as a 3.00 kg thin rod 50.0 centimeter long attached to the cylinder, calculate what the moment of inertia of the skater will be. So that's our end goal. Ultimately, we're trying to figure out what the moment of inertia for the skater will be. Awesome. We're also given some multiple choice answers. Let's read them off to see what our final answer might be. And let's note that they're all in units of kilograms multiplied by meters squared A is 0.970 B is 1.94 C is 2.25 and D is 4.19. Awesome. First off let us draw a diagram of this problem to help us better visualize what's going on. So I've gone ahead and drawn a figure of our cylinder because we're gonna be modeling the skater as a cylinder. And as we know the cylinder is 50.0 kg. And I put that, I labeled that in green in the center of our cylinder. And then we have two rods sticking out on either end of our cylinder, think of it like a person. So like think of this as like a stick figure. So the cylinder is the torso and the rods are the arms and the rods are each 3.00 kg. But I've simply just put it as 3.0 to keep things simple. And then we know the distance between the arm to the cylinder is 0.50 m as labeled in red. Awesome. And then we also have the radius of our cylinder which is 0.3 m which is from the center of our cylinder out to the edge because it's modeling the radius. Awesome. Now that we have a visualization of this problem, let us write down all of our given variables now. So we know that the mass of the arm and let's call this ma. So this is the mass of the arm, which is the rod, which is equal to 3.0 kilograms. We also know the mass of the body which is the mass of the cylinder. So MB is equal to 50 0.0 kg. And we also know the radius of the body. So capital RB is how we're gonna denote the radius of the body is 0.30 meters or we could just simply put 0.3 m. Awesome moving right along here. Now, we must note that for this particular problem, the inertia of the skater is equal to the inertia of the body plus the inertia of the arms which you can write like. So, so I'm gonna denote S to be the skater. So the inertia of the skater is equal to the inertia of the body plus the inertia of the arms. Awesome. Let us also and recall and note that the inertia of the body, which we're going to denote as I subscript B is equal to one half multiplied by the mass of the body multiplied by the radius of the body squared. So now at this point, we can plug in all of our known variables and solve for the numerical value of IB. So let's do that. So we know one half multiplied by, we know that the mass of the body is 50 0.0 kg. And we also know that RB, which is the radius of the body is equal to 0.3 meter square. So when we plug that into our calculator, we will find that it's equal to 2.2 five and its units are kilograms multiplied by meters squared. Awesome. So now at this point, we need to recall and use the parallel axis theorem to help us calculate the value of I A which is the inertia of the arms. So the inertia of the arms is equal to two multiplied by the mass of the arms multiplied by uh length of the arms squared divided by 12 plus the mass of the arms multiplied by the radius of the body plus the length of the arms divided by two. And this value is squared Awesome. So now at this point, we can plug in all of our known variables to solve for I A. So let's do that. So I A is equal to two multiplied by. So we know that M is equal to 3.0 kg. And we know that the length of the arms is 0.50 meters as stated in our diagram. And don't forget that that value is squared divided by 12 plus the mass of our arms again, which is 3.0 kg. Awesome. And this is multiplied by the value of our radius of the body which is 0.3 m multiplied or actually plus this is plus our value for the length of the arms which we know is 0.50 m divided by two. Awesome. And this value is all squared Awesome. So when you plug that into our calculator, we will find that I A is equal to 1.94 when we round the two decimal places and its units will be kilograms multiplied by meters squared. Awesome. So now we can take this and plug this back into our inertia of the skater equation. So we need to take that our inertia of the skater is equal to the inertia of the body, which we just found to be 2.25 kilograms multiplied by meters squared. And we need to add this value to the inertia of the arms which we found to be 1.94 kilograms multiplied by meters squared. Awesome. And when you add those two values together, we will get 4.19 and of course, its units will be kilograms multiplied by meters squared. Awesome. We did it we solved for this problem. Hooray. So looking at our multiple choice answers, the correct answer has to be the letter D 4.19 kg multiplied by meters squared. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

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