Question

The moment of inertia of a solid cone of mass m and base radius r about its vertical axis is

Answer 1

Here in this case the density of the sphere D= M/V
Where M = mass of the sphere and 
V = volume of the sphere = 43π2R3-R3=43π7R3
So the density is , D = 3M4π(7R3)
Now consider a hollow sphere of thickness dx at a distance x from the centre ,where x is the distance between R and 2R .
Now the mass of the sphere = dm = d4πx2dx=3Mx2dx7R3
Now the moment of inertia of this elemental hollow sphere is dI .
Now , for a hollow sphere moment of inertia is I=23MR2Therefore ,dI=23dmx2⇒dI=27MR3x4dxNow integrating from R to 2R we get ,I=∫dI=27MR3∫R2Rx4dx=27MR3x55R2R=235MR332R5-R5=6235MR2
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Hope this helps you ...