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]
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 ρ_{0}, you can rewrite your answer in terms of M instead of ρ_{0}.