Q. Moment of Solid Sphere About Axis?

The instant of enertia for a homogeneous solid sphere about an axis can be found by integrating spherical shells by integrating disks or by solving the equations for the moment of an element of mass throughout the volume. We will utilize the latter.

The instant of a mass element is I = ∫_V l²dm, where dm = ρ dV l is the distance of the element from the axis and ρ is the mass density. Note that the distance to an element from the centre of the sphere is r and l = r sin θ where θ is the angle between the axis and the radius, The volume element dV is

dV = (2π l)(dr)(r dθ) = 2πr² sinθ dr dθ.

ρ =m/V =m/(4/3(π R³))

And dm = ρdv the moment of inertia is        

I = ∫l2dm = ∫ (r sin θ)²ρdV.

It is convenient to alter the variable of integration from θ to - cos θ.

Let x = - cos θ Then the limits of the above integral become

The corresponding integral is

Finally,

I =2/5(mR²).