Moment of Inertia of a Thin Spherical Shell

To find the moment of inertia of a thin spherical shell, we can consider the shell to be made up of infinitesimally thin concentric spherical shells. Each of these shells can be considered as a thin ring with radius r and width dr.

The moment of inertia of each thin ring can be calculated using the formula:

dI = dm * r²

where dI is the moment of inertia of the thin ring, dm is the mass of the thin ring, and r is the radius of the thin ring.

The mass of the thin ring can be calculated as the product of its density, ρ, its volume, dV, and its thickness, dr:

dm = ρ * dV = ρ * 4πr² * dr

Substituting this expression for dm into the equation for dI, we get:

dI = (ρ * 4πr² * dr) * r²
= 4πρr⁴dr

To find the total moment of inertia of the thin spherical shell, we need to integrate this expression over the entire volume of the shell. Since the shell is thin, the integration can be done with respect to r.

∫dI = ∫4πρr⁴dr

Integrating both sides of the equation, we get:

I = 4πρ * (1/5)r⁵

where I is the moment of inertia of the thin spherical shell.

Since the mass of the spherical shell is given as M, we can substitute M for ρ * V, where V is the volume of the shell.

M = ρ * V = ρ * (4/3)πR³

Simplifying, we find ρ = (3M)/(4πR³).

Substituting this expression for ρ into the equation for I, we get:

I = 4π * [(3M)/(4πR³)] * (1/5)R⁵
= (3/5)MR²

Thus, the moment of inertia of a thin spherical shell of mass M and radius R about its diameter is (2/3)MR², which corresponds to option D.