Moment of Inertia of Spherical Shell

## Question:

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
## Answer:

The moment of inertia for a spherical shell is 2/3*M*R^{2}. 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*r^{2}
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.