# Taking pivot about an accelerating point

Given this question:

A small ball of mass $m$ and radius $r$ rolls without slipping on the inside surface of a fixed hemispherical bowl of radius $R>r$. What is the frequency of small oscillations?

The standard solution is to write Newton's second law for the ball and then take the centre of mass of the ball to be the pivot and write

$$\tau = I \alpha.$$

Only the frictional force contributes to the torque in this case. From Newton's second law, I can express the frictional force in terms of the gravitational force and therefore the frictional force can be eliminated in the equation for torque. I then make small angle approximation and get the equation to be of the form

$$k\theta=-I\ddot{\theta}$$

from which I can find the frequency.

Another approach uses the point of contact of the ball with the sphere as the pivot. It has the advantage that the frictional force adds no torque. Both approaches give the same result.

My question is since both pivots that we have chosen are accelerating, why are not fictitious forces considered? In the first place, can the pivots that we choose when writing

$$\tau = I \alpha$$

be accelerating?

• Forces through the c.m. play no role in the angular equations of motion. Commented Feb 27, 2014 at 18:20