Questions

# A hollow sphere is filled with water through a small hole in it. It is then hung by a long thread and made to oscillate. As the water slowly flows out of the hole at the bottom, the period of oscillation will

a
Continuously decrease
b
Continuously increase
c
First decrease and then increase to original value
d
First increase and then decrease to original value

detailed solution

Correct option is D

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 T∝l ). After some time weight of water left in sphere become less than the weight of sphere itself, so the resultant C.G. gets clear the C.G. of sphere itself i.e. l decreases and hence T increases. 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.

A simple pendulum has time period T1. The point of suspension is now moved upward according to equation $\mathrm{y}={\mathrm{kt}}^{2}$  where $\mathrm{k}=1\text{\hspace{0.17em}}\mathrm{m}/{\mathrm{sec}}^{2}$. If new time period is T2 then ratio $\frac{{\mathrm{T}}_{1}^{2}}{{\mathrm{T}}_{2}^{2}}$  will be