1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Moment of Inertia of a cylinder with varying density

  1. Apr 4, 2012 #1
    A cylinder with radius R and mass M has density that increases linearly with distance r from the cylinder axis, ρ = αr, where α is a positive constant.
    Calculate the moment of inertia of the cylinder about a longitudinal axis through its center in terms of M and R.

    I have I = ∫r^2dm
    and ρ = αr = m/V, I solved for m and took the derivative to get dm = α(dV = dr).
    Putting this into the Inertia equation, I get I = α∫r^2 (dV + dr). Using the equation for volume, i took the derivative and got dV = L * 2∏r dr. Plugging this into the original equation, and integrating, I get I = α(Lr^2)/2 + α(r^3)/2 with bounds of R2 and R1. Not sure where to go from here...
    Last edited: Apr 4, 2012
  2. jcsd
  3. Apr 4, 2012 #2


    User Avatar
    Science Advisor

    dm=2πr L ρ dr - Because you can break the disk into thin rings distance r from center, each of which has volume 2πr L dr


    Try using these.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook