A composite disc is made out of two concentric cylinders, as s...

Question

A composite disc is made out of two concentric cylinders, as shown. The inner cylinder has radius 30 cm. The outer cylinder has radius 50 cm. If you pull on a light rope attached to the edge of the outer cylinder (shown left) with 100 N, how hard must you pull on a light rope attached to the edge of the inner cylinder (shown right) so the disc does not spin?

Accepted Answer

Hey, guys, hope you try this practice problem here. We have a composite disk, and we're trying to figure out how much force we need to apply to balance this thing out. So let's check it out. So it says we have a composite disk made out of two concentric cylinders. Concentric. This means that you have an inner circle in an outer circle. Um, it's shown over here. The inner circle has radius of 30 centimeters, so this guy right here has a radius inner of points three and the outer. This one has a radius outer of m. It's as you pull in the light rope attached to the edge of the outer cylinder. That's this one. Let's call this F, um, outs with 100 Newtons, and it's asking, how hard must you pull on a light rope attached to the edge of the inner circle inner cylinder? This one F n so that the disk doesn't spin. So if things are the radio and you're pushing with this on the left, how much must you push on the right? So the whole thing balance, um, discs are in this case, the discs are, um they could spin around themselves as a result of your pulling. You don't want them to spin. So what you want is you want rotational equilibrium. You want the torques to cancel. So you want the sum of all torques to be zero. Now, if you look here 11 force this force here is trying to get you to spend this way. Um, this is torque out, which would be counter clockwise so positive and this is trying to get you a torque in. That would be negative. We want the two of them. Two equals zero. There's two ways you can write this u can write torque out, has a positive plus negative torque in equals zero on. Then you can move this over here. Torque out equals torque in or so you can start here and work your way through here. Or you can actually already start here by looking at this picture and say, Hey, the talks of canceling. So I'm gonna say all the torques going one direction. Will you go all the torques going in the other direction? Okay, Now all we gotta do is expand both sides, so it's gonna be torque is f R sign of data. So what we're gonna do is we're gonna write f out are out Sign of data out equals f n are in sign of data and okay, so let's draw the are vectors real quick. Are vectors from the axis of rotation to the point where the force happens. So this is your are out, and this is your are in notice that both of them make an angle of our of 90 degrees with their respective forces. So are out. F out are in f n. Okay, so both of these guys will be simply one now f out is 100 are out is the happens to be the radius because it's at the edge. So it's 1000.5 f n is what we're looking for and f out. I'm sorry. Are in is this distance here which is the radius of the smaller in or one which is points three. So F n is 50 divided by 500.3 50 divided by 500.3, which is 167 Newton's. Okay, notice that the the inner one turned out to be a greater force that should make sense. Torques are larger if you're pulling farther from the axis of rotation on the second case here, on the in, um, you're pulling closer to the axis of rotation. So to compensate for that, your force has to be stronger. Okay, so that's it. Very straightforward. Hopeful. This made sense. Let me do you have any questions and let's keep going.

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