# Moment of Inertia / Mass Element proof

1. Dec 27, 2006

### raintrek

1. The problem statement, all variables and given/known data

A thin square plate of side a has one corner at the origin and two sides along the positive x and y axes. If the density of the plate is given by p(x,y) = xy show that its mass is M=(1/4)a^4
If the distance of the mass element dM = pdS from the origin is r the moment of inertia of the plate is I= integral [r^2 (pdS)] where S is the surface of the plate. Prove that I=Ma^2

2. Relevant equations

3. The attempt at a solution

I've proved the first half by showing

M = pdV and taking a small mass element, then integrating

(0->a)int(dx) (0->a)int(dy) xy = (a^2)/2 * (a^2)/2 = (1/4)a^4

However I can't seem to get started with the second part...

2. Dec 27, 2006

### Galileo

Just use the given definition:

$$I=\int_0^a \int_0^a r^2\rho dxdy$$.

What is r^2 in terms of x and y?

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