Moment of Inertia of Spherical Shell


can u tell me how to find the moment of inertia of a hollow sphere? please do send me the mathematical derivation as well. thanx


The moment of inertia for a spherical shell is 2/3*M*R2. You might imagine the spherical shell to be made up of a series of tiny mass elements the mass of each being its volume times its density r. The volume would be the thickness of the shell times the difference in latitude covered by the mass element times the difference in longitude covered by the mass element. Lets call the differential longitude dq and the differential latitude dl. Then the mass of each element is r*dq*dl. The moment of inertia of the whole shell is the sum of the moments of inertia of each of the little mass elements. The moment of inertia, i, for a mass particle is m*r2 where m is the particle mass and r is its distance from the axis or rotation. In our case i=r*dq*dl*R*cos(l) where R is the radius of the spherical shell and l is the latitude measured from the equator of the sphere. To sum up all the moments of inertia i we must integrate over all q from 0 to 2*p and all l from -p to p. Carrying out this integration gives us the result.