# Constant angular velocity in vertical circular motion?

I want to do the following vertical circular motion experiment, where I "hand spin" a rotating mass (red) that is attached to the same string with the slotted mass at the other end. I spin the mass by grabbing the hollow tube (blue).

My question is what would happen to the velocity of the rotating mass? Would rotating mass adopt constant angular velocity? Why or why not? Based on observation, it seems to me that if a lighter rotating mass was used, it does somewhat adopts constant angular velocity. Why is that?

If the rotating mass does not adopt constant angular velocity, would that mean the velocity changes at every point? And the velocity of the rotating mass would be the highest at the bottom of the circle( $$u \geq \sqrt{5gr}$$) and lowest on top of the circle ( $$u\geq \sqrt{gr}$$)?

And for calculations, can I just use minimum velocity (ie use $$u = \sqrt{gr}$$ as the velocity on top)?

If the ball is rotating at the end of a "string," which generally exerts a tension force but no shear force, then consider the points where the string is horizontal and the velocity of the ball is vertical. Gravity is, at those points, exerting a force parallel to the ball's velocity, and that vertical force can have no cancelling component from the horizontal string. Therefore the magnitude of the velocity vector isn't a constant, and the circular motion is not uniform.

If the path is circular, the speed is uniform, and the string is attached at the center of the motion, then a similar argument shows that the tension in the string must be nonuniform. If the string is connected by some low-friction method to a constant-mass hanging weight, as in your diagram, the change in tension will cause the weight to move up and down. But if the length of the string is changing (as the weight moves), the assumption that the string is attached to the center of the motion becomes questionable. It rapidly becomes a very complicated problem.

In a quick-n-dirty lab setup, friction between the string and the blue tube becomes an important and hard-to-model bit of the dynamics.

• Thanks for your answer, Rob. The slotted mass did oscillate and the radius did change (I did a quick experiment), however, it did not change much. So can I just assume that the radius did not change throughout the experiment and therefore the angular velocity is constant? And for the first scenario, doesn't a mass in non-uniform circular motion still travels in circular path? Commented Apr 8, 2020 at 5:42
• You've given us enough information to conclude that "uniform circular motion" wasn't a good description of the device you made. But you haven't given us enough information to decide how bad the approximation is, nor what the next better approximation might be.
– rob
Commented Apr 8, 2020 at 7:23

Without a rigid support and variable torque, I feel its quite unbelievable to rotate the mass vertically with a string. If you are certain that you may control the experiment with exact calculations then I suppose that your ball won't rotate because of other factors.

• I tried it, it works as long as the slotted mass is heavy Commented Apr 8, 2020 at 3:50
• What are the other factors? Commented Apr 8, 2020 at 4:08
• Do note that there's a mass hanging on the other end of the string.
– user258881
Commented Apr 8, 2020 at 6:22