Question about rotation w/ an additioanl object.

[hakujin]

First, while this is "homework" related I'm not seeking any direct answers. I'm stuck on a concept that I'm not sure even exists.

In short I've got a rotating disk with a weight at the edge with "an initial" ω. If that object was to decrease the distance towards the center, would the ω change and if so why?

The phrase "an initial" led me to believe that there was a starting force or torque which would remain constant which was required to not only rotate the mass of the disk but compensate for the mass at the edge of the disk. If that's true, given the mass at the edge has any factor at all on the needed initial energy/force/torque, I couldn't figure out how to calculate it. I check over my book and am completely missing it.

I assumed the mass at the edge would have be a factor since it's basically a m*g*d Newton force down (since the disk is horizontal in this situation), but I don't know how to calculate the needed force to create the initial ω.

I figured it would be something to the effect of... g[(mass of the disk)+(mass of the object*distance)] but that's the best I've got since mainly what we've done is bridge based problems an rotation around a fixed point.

If anything, a jumping off point on terminology to look up would be a big help.

 

[Simon Bridge]

If a rigid mass distribution is rotated about anywhere other than the center of mass, then it will wobble unless there is a balancing force applied. In 3D the disk+mass will also try to precess. This is how wheel-balancing machines (you know: when you get a new tire fitted?) work.

It would be an easier problem if there were two small masses on the rim, diametrically opposite each other, and they were both drawn closer to the center at the same time... so I suspect that you are overthinking things.

I suppose you need to decide if energy is being supplied to the configuration.

 

[mfb]

If that object was to decrease the distance towards the center, would the ω change and if so why?

Assuming your disk is fixed to rotate in the horizontal plane around its geometric center, without external torques, ω would increase as the moment of inertia reduces and the product of both is conserved.

 

[Simon Bridge]

That was my take - it's the ice-skater demo all over again.
When I was a kid there was this huge round-a-bout that we'd push up to speed then jump on. We'd start close to the edge and walk all over it ... I'm picturing something like that with a kid as the small mass.

 

[PJC01]

Thank you for reaching out with your question. It sounds like you are trying to understand the concept of rotational motion and how it relates to the distribution of mass in a rotating object. To answer your first question, yes, if the object at the edge of the rotating disk were to decrease its distance towards the center, the angular velocity (ω) of the disk would change. This is because the moment of inertia of the system would change, and according to the law of conservation of angular momentum, the angular velocity must also change to maintain the same total angular momentum.

In terms of the initial force or torque required to rotate the disk, it would depend on the moment of inertia of the system and the initial angular velocity you are trying to achieve. The moment of inertia takes into account not only the mass of the object, but also its distribution around the axis of rotation. So, in your example, the mass at the edge of the disk would have a greater effect on the moment of inertia compared to the mass at the center.

To calculate the moment of inertia for a disk with a mass at the edge, you can use the equation I = 1/2 * M * R^2 + M * d^2, where M is the mass of the disk, R is the radius of the disk, and d is the distance of the weight from the center of the disk. This equation takes into account the mass of the disk and the weight at the edge, as well as their respective distances from the axis of rotation.

I hope this helps to clarify some of the concepts you are struggling with. As for terminology, some key terms to look into would be moment of inertia, angular velocity, and conservation of angular momentum. Good luck with your studies!