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

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

    3. The attempt at a solution


    integrate between 0 and R to obtain

    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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