Ball performing small oscillations within a hollow cylinder

[peripatein]

I am not sure why only the second term counts/is used. I could make an educated guess, or try to at least. Is it because around the point of stable equilibrium the first derivative is zero as the potential obtains its minimum and the first term is subtracted from the potential to obtain the "effective" potential? Am I close?
The second part of that problem, with the hollow cylinder and the ball, asks for the angular frequency of the small oscillations supposing that this time the ball slips and doesn't roll.

 

[voko]

I am not sure why only the second term counts/is used. I could make an educated guess, or try to at least. Is it because around the point of stable equilibrium the first derivative is zero as the potential obtains its minimum and the first term is subtracted from the potential to obtain the "effective" potential? Am I close?

This is not just close, this is exactly it.

The second part of that problem, with the hollow cylinder and the ball, asks for the angular frequency of the small oscillations supposing that this time the ball slips and doesn't roll.

If the problem says "does not roll", then there no kinetic energy due to rolling. Even though I am not sure how one could achieve that in practice.

 

[peripatein]

It says "slips without rolling". Does that mean that I am solely left with the translational kinetic energy whereas the rotational equals zero, and all my expressions remain the same?

 

[voko]

I believe this is what is meant. But, as I said, I do not like this. Even if there is no friction, a freely moving ball will still rotate inside a cylinder. This is seen from considering the torques due to the weight of the cylinder and due to the normal force. I would bring this up to to your professor.