Oscillations with fluid and pendulum

[LCSphysicist]

Homework Statement

A pendulum is constructed as a light thin-walled sphere of
radius R filled up with water and suspended at the point 0 from a
light rigid rod (Fig. 4.1.1). The distance between the point 0 and the
centre of the sphere is equal to 1. How many times will the small
oscillations of such a pendulum change after the water freezes? The
viscosity of water and the change of its volume on freezing are to
be neglected

Relevant Equations

All below

That's a good question, i am not sure how the water in liquid state will influence in the motion, but i imagine that can not exert any torque, i would say in the first case:

Hollow sphere inertia moment: 2mr²/3 + ml²
(2mr²/3 + ml²)θ'' = -mglθ (1)

In the second case, otherwise, we will have
2(m+m')r²/5 + (m+m')l² = -(m+m')glθ

Anyway, i am certain that have some error in (1), something say to me that the water can indeed exert a torque although it doesn't contribute to inertia moment, but, as i don't know the mechanics of a fluid, i am not sure.

 

[haruspex]

Hollow sphere inertia

The shell is light and can be ignored.
The water does not exert a torque about its centre, but it still exerts a torque about the point of suspension. What does the parallel axis theorem give for its MoI about that point?

 

[Merlin3189]

I think there was a recent question like this https://www.physicsforums.com/threa...ndulum-consisting-of-a-disk-of-mass-m.989467/