Angular momentum: Grooved cone with a mass sliding down the groove

[madafo3435]

Homework Statement

A cone of height h and base radius R is free to rotate around a fixed vertical axis. It has a thin groove cut in its surface. The cone is set rotating freely with angular speed ω0, and a small block of mass m is released in the top of the frictionless groove and allowed to slide under gravity. Assume that the block stays in the groove. Take the moment of inertia of the cone around the vertical axis to be I0. What is the angular speed of the cone when the block reaches the bottom?

Relevant Equations

I have considered the cone-block system, it seems to me the most sensible system, but I have problems in analyzing the angular momentum. In order for the cone to maintain its position, the axis of rotation must exert some torque and this confuses me. For example, the weight of the block generates a torque that forces the cone to oscillate, so the axis must do some torque to preserve the position of the cone, but I feel that the problem is too complicated with these considerations ...

.

 

[haruspex]

In order for the cone to maintain its position, the axis of rotation must exert some torque

If you think of the torque the axle exerts as a pair of forces, what plane are they in? What torque do they have about the axis?

 

[madafo3435]

If you think of the torque the axle exerts as a pair of forces, what plane are they in? What torque do they have about the axis?

I suppose that in the vertical plane that moves with the weight of the cube ... could I then ignore the torque exerted by the weight, because I am only interested in rotation about the axis through the cone? In this case I find that for this axis angular momentum is conserved because normal forces are internal and there with the conservation of momentum in this way I can find a simple solution

 

[haruspex]

I suppose that in the vertical plane that moves with the weight of the cube ... could I then ignore the torque exerted by the weight, because I am only interested in rotation about the axis through the cone? In this case I find that for this axis angular momentum is conserved because normal forces are internal and there with the conservation of momentum in this way I can find a simple solution

Yes.

 

[madafo3435]

Yes.

Thank you very much for your comment!