Question

A hollow sphere is filled with water through a small hole in it

Answer 1

Hollow sphere acts as a physical pendulum as the sphere is filled with water. The time period T for a physical pendulum is defined as,

T = 2π√I/mgd

Here, I is the rotational inertia of the sphere, m is the mass of the sphere which is filled with water, g is the acceleration due to gravity and d is the distance from the pivot to the center of mass.

In the above equation T = 2π√I/mgd,

I/mgd = mk2/mgd (Since, I = mk2)

= k2/gd

So, k2/d is the deciding factor here.

As the water flow out from the distance of center of mass from axis, that increases up to a certain limit and radius of gyration increases in square. So there is net increase in period. But after certain limit d decreases, because, center of mass shifted toward heavy portion of sphere, and rate in decrease in d is increases. But the increment in radius of gyration remains constant. Thus after a certain limit k2/d will decrease. Therefore the period of oscillation first increases and then decreases