# Non-uniform inertia

by LASmith
Tags: non uniform inertia
 P: 21 1. The problem statement, all variables and given/known data 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 2. Relevant equations I=$\int$r$^{2}$.dm volume = 2$\pi$rL.dr 3. 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 $\rho_{0}$
Mentor
P: 41,489
 Quote by LASmith However, I do not understand how to express this without using the term $\rho_{0}$
Find an expression for M in terms of ρ0.
P: 21
 Quote by Doc Al 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.

Mentor
P: 41,489
Non-uniform inertia

 Quote by LASmith 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.

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