# C.O.M and Inertia of a sphere; top half twice as dense as lower half

1. Apr 22, 2012

### Kneemar

1. The problem statement, all variables and given/known data
A sphere of radius L is made up of an upper hemisphere of uniform mass density σ = 2σ0 and a lower hemisphere of uniform mass density σ = σ0. Origin of co-ordinate system lies at centre of sphere with the denser hemisphere above the xy plane. Using spherical polar co-ordinates:

2. Relevant equations
(i) Find the mass of the sphere

(ii) Show the the centre of mass lies at zCOM = L/8

(iii) The moment of inertia for rotation of the sphere about the z-axis is defined as Iz = ∫∫∫lz2dM, where lz is the perpendicular distance of a mass element dM = σdV from the z-axis. Show that Iz = (4/5)(pi)σ0L5

3. The attempt at a solution

(i) M = 4/3(pi)L3(1.5σ0) = 2(pi)L3σ0

(ii) zCOM = (∫zdM)/∫dM = (1/(2(pi)L3σ0))*∫zdM

∫zdM = ∫zσdV
z = rcosθ
dV = r2sinθdrdθd$\varphi$
∫zdM = σ∫r3sinθcosθdpdθd$\varphi$

I could be doing okay up to this point, but not convinced.

Now what do I do considering I have the 2 different density hemispheres?

(iii) No attempt worth writing down

Cheers :)

ps. Sorry if my input style is horrible (particularly equations might be messy). It's my first post.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution