Find the moment of inertia of solid sphere of mass M about a diameter as shown in Fig.

Explanation

The sphere can be considered as made up of many concentric thin shells. From previous example, a spherical subshell of radius r has moment of inertia:
`dI=2/3dMr^(2)`.
Calculations: We know that
`dM=M/(4/3piR^(3))xx4pir^(2)dr`
= `(3M)/R^(3)xxr^(2)dr`.
Substituting for dM in the equation for moment of inertia, we get
`dI=2/3xx(3M)/R^(3)r^(2)(dr)r^(2)`.
Integrating over the whole volume, we get
`I=(2M)/(R^(3))int_(0)^(R)r^(2)dr`
= `(2MR^(2))/5`.