Moment of inertia of spherical shell problem

[karnten07]

Homework Statement

Show that the moment of inertia of a uniform thin spherical shell of mass m and radius a about a diameter is 2/3 ma^2

Homework Equations

Moment of inertia = Sum m x R^2
volume of sphere = 4/3 pi r^3

The Attempt at a Solution

I think i have to integrate the moment of inertia euation for this spherical shell, but I am not sure I am understanding my notes here.
The inertia is equal along all axes, so by adding the inertia along each axis gives me the total inertia?

i have some rough sketchy notes i took in a class, so i will write them here to see if anyone can decipher it.

x^2 + y^2 + z^2 = a^2 where a is the radius of the sphere

Any ideas would be much appreciated, thankyou

 

[learningphysics]

The challenge is setting up the integral... I recommend using spherical or cylindrical coordinates... You need to choose an axis through the center... anyone is fine... z-axis.

I = \int\int\int \sigma*r^2dV

\sigma is the density of the sphere.

r = \sqrt{x^2 + y^2}

so

I = \int\int\int \sigma*(x^2+y^2)dV

try to set up this integral over the volume of the sphere using spherical or cylindrical coordinates... you can use cartesian too but it's a little more tedious...

 

[Mindscrape]

I know lp is trying to pedagogical, but I strongly advise against any coordinates but spherical. You don't integrate cubes in spherical coordinates, and likewise you don't integrate spheres in cartesian coordinates. (Another Hint: you will need to rederive, or look-up, the cartesian to spherical formulas.)