# A hollow sphere is filled with water through a small hole in it. It is hung by a long thread and, as the water flows out of the hole at the bottom, one finds that the period of oscillation first increases and then decreases. Explain.

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A hollow sphere is filled with water through a small hole in it. It is hung by a long thread and, as the water flows out of the hole at the bottom, one finds that the period of oscillation first increases and then decreases. Explain.

## 3 Answers

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 = mk

^{2}/mgd (Since, I = mk

^{2})

= k

^{2}/gd

So, k

^{2}/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 k

^{2}/d will decrease. Therefore the period of oscillation first increases and then decreases

^{2}/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 k

^{2}/gdSo, k

^{2}) = k

^{2}/mgd (Since, I = mk

^{2}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 = mk