Derive an expression for the moment of inertia of a thin spherical shell about a diameter.

Explanation

Consider a solid sphere. Suppose a smaller concentric sphere is removed from the solid sphere we get a hollow or a thick spherical shell (See Fig. 7.2.58). Let Mbe the mass, R the external radius and r the internal radius of the hollow sphere. Volume of the shell `=4/3 pi(R^(3)-r^(3))`

Mass per unit volume of the shell `=rho`
`rho = M/((4/3)pi(R^(3)-r^(3)))`
`=(3M)/(4pi(R^(3)-r^(3))`
M.I. of the solid sphere of radius R about a diameter
`=2/3 xx "mass" xx ("radius")^(2)`
`=2/5 xx 4/3pi R^(3) rho R^(2)`
`=8/15 pi R^(5) rho`
M.I. of the solid sphere of radius r about a diameter
`=8/15 pi r^(5) rho`
M.I. of the hollow sphere about the diameter = `l`
`=8/15 piR^(5) - 8/15 pi r^(5) rho = 8/15 pi rho (R^(5)-r^(5))`
`l=2/5 M (R^(5)-r^(5))/(R^(3)-r^(3))`