A hollow sphere is completely filled with water through a small hole in it. It is hung by a long thread and made to oscillate. Calculate the time period of oscillation as the water slowly flows out of the hole at the bottom. 

The given system is like a simple pendulum, whose effective length (l) is equal to the distance between point of suspension and C.G. (Centre of Gravity) of the hanging body.

When water slowly flows out the sphere, the C.G. of the system is lowered, and hence l increases, which in turn increases time period (as $\mathrm{T}\propto \sqrt{l}).$ 
After some time weight of water left in sphere becomes less than the weight of sphere itself then the resultant C.G. stops lowering further and it moves back o C.G. of sphere itself, i.e., l decreases and hence T decreases.

Finally when the sphere becomes empty, the resulting C.G. is the C.G. of sphere, i.e., length becomes equal to the original length and hence the time period becomes equal to the same value as when it was full of water.