Angular velocity graphs due to multiple torques

A disc is initially rotating clockwise around a fixed axis with angular speed omega naught. At time t equals 0, the two forces, F₁ is equal to 20 newtons and F₂ is equal to 10 newtons, are exerted on the disk as shown in the figure below. So these are the magnitudes, but here they show the magnitude and the direction, and they say which graph could show the angular velocity of the disk over time.

So before I even show you the options, let's see if we can just sketch it ourselves. So pause this video and see if you can sketch the angular velocity of the disk over time.

All right, so in our vertical axis here, let's put our angular velocity, and then in our horizontal axis here, we want to see how that changes over time. So what is our initial angular velocity? Well, they tell us that our angular speed is omega naught, and the direction is in the clockwise direction. They sit right over there, and the typical convention is that clockwise is negative, and counterclockwise would be positive.

So we're starting at a negative angular velocity. Let's see, right over there, and it actually would be negative omega naught. I'll just write negative omega naught. Remember, omega naught is the speed, the angular speed, so it's the magnitude of the angular velocity, but since we're going clockwise, it would be negative. So we're going to start right there.

Now, let's see what the net torque is. F₁, we're assuming, goes straight down, and it's perpendicular to the radius. Let's call that r, that length r, and F₂ is straight down; it's perpendicular to the radius. But F₁ is providing a torque in the counterclockwise direction, while F₂ is providing a torque in the clockwise direction. And the way it's depicted, F₁ is larger, and they're both away from the center of rotation.

So your net torque is going to be counterclockwise. It's going to be against the direction of the starting direction of rotation. If your net torque is counterclockwise, we could say our net torque is going to be positive. If our net torque is positive, that means that our angular acceleration is going to be positive.

Our angular acceleration is just the rate of change of our angular velocity, and this is going to be constant, assuming that these forces are constant. So you're going to see this: we're going to be rotating clockwise at a slower and slower rate until all of a sudden, the wheel switches directions, or the disc switches directions, and then all of a sudden, we are rotating counterclockwise.

So it would look something like that. So let's see which of these choices fit. If we look at all of them, this looks almost exactly like we drew. But let's just review what these other ones are saying.

This one is implying here that you're starting off going clockwise, but then you're accelerating in the clockwise direction, and that wasn't the direction that the net torque was being applied in, so we ruled that out. This assumes that you're starting in the counterclockwise direction, which we know is not the case.

This assumes that you're starting in the counterclockwise direction, and then you are accelerating in the clockwise direction, and you would have picked this if you got your convention mixed up between counterclockwise and clockwise, and which one is positive or negative, but we can rule that one out, and the choice is A.