Torque and kinematics conceptual example

We are told a student hangs blocks with different masses from a pulley of mass m and radius r and releases them from rest. The student measures the time of the fall t and the magnitude of the angular velocity omega sub f when the block reaches a distance d below the axle. There is considerable friction between the axle and pulley.

Which of the following plots could show the student's measurements as a function of the applied torque on the pulley? So pause this video and see if you can figure it out on your own.

Alright, so let's first understand what these axes are and then we'll talk a little bit about what's going on with this mass-pulley system. On our horizontal axis here, we have our applied torque. Now, where is the applied torque in this system? Well, you're going to have some mass here and we assume we're on a planet, or probably on Earth, so you're going to have gravity pulling down on this.

You're going to have the force of the weight of this object pulling down, and we know that this has radius r. So that force times that radius is going to be the applied torque. Of course, the more mass you have, the more weight you're going to have, and you're going to have more torque. So this is really showing that as the student applies larger and larger masses, right over here, we're going to have more and more torque.

Now, what is in our vertical axis right over here? Omega sub f divided by t. So this is our final angular velocity once this thing has gone down a distance d, that's what they tell us, divided by the time it took to get there. Well, if we assume that it started from a standstill, you could view this as your angular acceleration.

So this is the angular acceleration as a function of applied torque. I could write angular acceleration right over there, or you could say your average angular acceleration over that time period from when you start to time t.

So what do we think would happen? Well, if you have small masses here, it would create a small torque, but if you have considerable friction, which they talk about right over here, then there's considerable friction between the axle and pulley. For small masses right over here, the torque will likely not be strong enough to overcome the friction.

In that situation, you would not have any angular acceleration; everything would just stay still. But then at some point, once your mass gets large enough, right over here, then your torque will be large enough to overcome the friction between the axle and the pulley.

So which of these graphs describe that, or which of these plots? Well, this first one describes a situation when you have a low applied torque. You have a high angular acceleration; that does not make sense, so we should rule that out.

This one kind of shows angular acceleration is indifferent to applied torque, and that doesn't make sense. The more applied torque you have, the larger masses here should indicate more angular acceleration, so that doesn't make sense.

Now, both of these show an upward trend, where as you have more applied torque, you have more angular acceleration. So which of these would you pick? Well, this one here on the right kind of paints a picture that as soon as you have any applied torque, you start having angular acceleration.

And if this was a frictionless system, if there was very little friction here, maybe this would make sense. But as we talked about, for low masses which generate low weight, they would generate low applied torque and wouldn't be able to overcome friction. So you should not have some angular acceleration for low torques, but then as the torque gets larger, you will overcome the friction and then start to accelerate the system.

So I would pick B.