Question

State and explain moment of inertia in a spherical shell

Answer 1

Answer:
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.

Answer 2

Rotational inertia is important in almost all physics problems that involve mass in rotational motion. It is used to calculate angular momentum and allows us to explain (via conservation of angular momentum) how rotational motion changes when the distribution of mass changes.