(adsbygoogle = window.adsbygoogle || []).push({}); 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. [itex]\rho[/itex]=[itex]\rho[/itex][itex]_{0}[/itex](r/R), where [itex]\rho[/itex][itex]_{0}[/itex] 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[itex]^{3}[/itex])/5

2. Relevant equations

I=[itex]\int[/itex]r[itex]^{2}[/itex].dm

volume = 2[itex]\pi[/itex]rL.dr

3. The attempt at a solution

I=[itex]\int[/itex]r[itex]^{2}[/itex][itex]\rho[/itex].dv

=[itex]\int[/itex](r[itex]^{3}[/itex][itex]\rho[/itex][itex]_{0}[/itex]/R.)dv

=[itex]\int[/itex](r[itex]^{3}[/itex][itex]\rho[/itex][itex]_{0}[/itex]/R.)(2[itex]\pi[/itex]rL).dr

integrate between 0 and R to obtain

2[itex]\rho_{0}[/itex][itex]\pi[/itex]R[itex]^{4}[/itex]L/5

However, I do not understand how to express this without using the term [itex]\rho_{0}[/itex]

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# Non-uniform inertia

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