# Non-uniform inertia

## Homework Statement

A cylinder with radius R and mass M has density that increases linearly with radial distance r from the cylinder axis, ie. $\rho$=$\rho$$_{0}$(r/R), where $\rho$$_{0}$ is a positive constant. Show that the moment of inertia of this cylinder about a longitudinal axis through the centre is given by I=(3MR$^{3}$)/5

## Homework Equations

I=$\int$r$^{2}$.dm
volume = 2$\pi$rL.dr

## The Attempt at a Solution

I=$\int$r$^{2}$$\rho$.dv
=$\int$(r$^{3}$$\rho$$_{0}$/R.)dv
=$\int$(r$^{3}$$\rho$$_{0}$/R.)(2$\pi$rL).dr

integrate between 0 and R to obtain
2$\rho_{0}$$\pi$R$^{4}$L/5

However, I do not understand how to express this without using the term $\rho_{0}$

## Answers and Replies

Doc Al
Mentor
However, I do not understand how to express this without using the term $\rho_{0}$
Find an expression for M in terms of ρ0.

Find an expression for M in terms of ρ0.

I realise this, however as the density is not constant, I am unsure of how to do this.

Doc Al
Mentor
I realise this, however as the density is not constant, I am unsure of how to do this.
Set up an integral to solve for the total mass, just like you set one up for the rotational inertia.

Once you get M in terms of ρ0, you can rewrite your answer in terms of M instead of ρ0.