# Moment of Inertia for Earth by superposition

This was a 2 part problem...
PART A: calculate moment of inertia of a uniform sphere of mass M and radius R by using the information provided:
the moment of inertia of a thin spherical shell at radius R with mass m spinning about its axis is 2/3mR2.

I did this by integrating over thin shells with density, p, from radius -R to R.
Knowing that dI=1/2y2dm=1/2y2pdV. So I integrated 1/2y2ppiy2dz and substituting density for p=M/(4/3)piR3

the answer I got was 2/5piR2

PART B: (this was where I started having difficulty understanding..) you can calculate the moment of inertia of a layered Earth model by superposing the results for a uniform sphere. The moment of inertia of a 2 layer Earth model with a core of mass M1 and an outer shell mass M2 separated at radius R12 and a total radius of R is given by:

EQUATION #1--->
I=2/5M1R212+2/5(pm4/3piR3)R2-2/5(pm4/3piR312)R23

where pm=density of mantle.

(below is copy-pasted equation from hw sheet...not sure which one is easier to read)
I =
2
5
M1R2
12 +
2
5
(m
4
3
R3)R2 −
2
5
(m
4
3
R3
12)R2
12

It tells me to derive equation #1(above) by superposition

I have no idea where to get started on this part...

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I=2/5M1R212+2/5(pm4/3piR3)R2-2/5(pm4/3piR312)R23

this equation above has a few errors:

the first R is supposed to be R-squared subscript 12
the second to last R is supposed to be R-cubed subscript 12
the last R is supposed to be R-squared subscript 12

Please learn to use the tex feature on this site. Also, by superposition, they mean the total moment of inertia will just be a sum of moment of inertias for the different bodies that make up the system.

So you will have the moment of inertia for the core. For the mantle, you can imagine it as a sphere with the same density as the mantle and same radius, with a smaller sphere with the radius of the core and negative mass. The negative mass will cancel out the mass of the larger sphere, so it will look like a shell.

That leaves you with 3 spheres. So you will have 3 different moments of inertia to sum together.