Non-uniform Inertia of a Cylinder: I = (3MR^3)/5

[LASmith]

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 $\rho_{0}$

 

[Doc Al]

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

Find an expression for M in terms of ρ₀.

 

[LASmith]

Find an expression for M in terms of ρ₀.

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

 

[Doc Al]

I realize 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.