Moment of Inertia of Hollow Sphere

[christikiki22]

Homework Statement

Calculate the moment of inertia of a spherical shell (i.e. hollow sphere) of uniform surface density about an axis passing through its center.

Homework Equations

The Attempt at a Solution

Integral ( r^2 * dm)
Integral (r^2 * p * dV) ... where p=density
p * Integral (r^2 * dV)

where V= (4/3)* pi * r^3
and dV= 4 * pi* r^2
therefore,

p * Integral (r^2 *4 * pi* r^2)
4*p*pi* Integral (r^4)
4*p*pi* ((r^5)/5) from 0 to r

p=density=m/V= m/(4/3 * pi * r^3)

4*m/(4/3 * pi * r^3)*pi* ((r^5)/5

(3*m*r^2)/5

but it should be (2*m*r^2)/3.
Please help!

 

[D H]

Integral ( r^2 * dm)
...
and dV= 4 * pi* r^2

The r in the first expression represents the distance from the axis of rotation to the point in question while the r in the second expression represents the distance from the center of the sphere to the point in question. Those are two different things.

 

[vela]

A hollow sphere is a two-dimensional surface, so you need to use $dm = \sigma dA$, where $\sigma$ is the surface density.