Non-uniform inertia

  1. 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]
     
  2. jcsd
  3. Doc Al

    Staff: Mentor

    Find an expression for M in terms of ρ0.
     
  4. I realise this, however as the density is not constant, I am unsure of how to do this.
     
  5. Doc Al

    Staff: Mentor

    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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