Question

Calculate the frequency with which a solid hemisphere of constant density oscillates, when it lies inverted on a flat horizontal surface, in the presence of a vertical gravitational field. HINT: ?You will have to find the location of the center of mass of the hemisphere first.

Answer 1

Center of mass of solid sphere = 3r/8 from flat surface

Here circular surface is in contact with ground so center of mass from point of contact with ground d = 5r/8

If we displace the sphere with angle

Then torque on hemisphere is - mgsin.d

If is small then torque = - mg.d = -5mgr/8

As torque = I×

I = moment of inertia = 2mr²/5

(2mr²/5) = - 5mgr/8

= -(25g/16r)

² = 25g/18r ....... By comparing equation of SHM

= (25g/18r)^0.5

Frequency f = /2? = (1/2?)(25g/18r)^0.5