Physics Challenge Question: Stability on a Cylinder

[IDValour]

Homework Statement

A solid cube of side $l = r*pi/2$ and of uniform density is placed on the highest point of a cylinder of radius $r$ as shown in the attached figure. If the cylinder is sufficiently rough that no sliding occurs, calculate the full range of the angle through which the block and swing (or wobble) without tipping off. (You can assume this range of equilibrium positions is stable).

Homework Equations

None that I've been made aware of.

The Attempt at a Solution

I'm trying to consider this in terms of a point at which the center of mass is directly above the axis of rotation but I'm struggling from there to be honest. Stability isn't something we've covered in my spec, so I'm not sure I have the knowledge to tackle this. I tried drawing a couple diagrams but I didn't get far.

 

[brainpushups]

I would recommend writing an expression for the potential energy of the cube as it rocks on the cylinder. You can find the stable and unstable equilibrium by differentiating the expression.

 

[IDValour]

The source of this problem claims not to need calculus of any form.

 

[andrewkirk]

calculate the full range of the angle through which the block and swing

Did you mean to type 'through which the block can swing'?

If sliding is assumed not to happen then the question becomes one of at what angle does the block topple over, when balancing on the midpoint of its lowest side (the pivot point). Think about the relationship between the centre of gravity of the block, the pivot point and the direction of the force of gravity.

 

[TSny]

I'm trying to consider this in terms of a point at which the center of mass is directly above the axis of rotation ...

Yes, I think that's the key idea.
Draw a picture corresponding to this special configuration.