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=\intr^{2}.dm
volume = 2\pirL.dr

The Attempt at a Solution

I=\intr^{2}\rho.dv
=\int(r^{3}\rho_{0}/R.)dv
=\int(r^{3}\rho_{0}/R.)(2\pirL).dr

integrate between 0 and R to obtain
2\rho_{0}\piR^{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.