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Moment of inertia about z-axis in spherical coordinates

  1. Apr 3, 2010 #1
    1. The problem statement, all variables and given/known data

    Use spherical coordinates to find the moment of inertia about the z-axis of a solid of uniform density bounded by the hemisphere [tex]\rho=cos\varphi[/tex], [tex]\pi/4\leq\varphi\leq\pi/2[/tex], and the cone [tex]\varphi=4[/tex].

    2. Relevant equations

    [tex]I_{z} = \int\int\int(x^{2}+y^{2})\rho(x, y, z) dV[/tex]

    3. The attempt at a solution

    I tried to convert that equation to cylindrical coordinates and got this (k representing density because it's uniform)

    [tex]I_{z} = k \int^{2\pi}_{0}\int^{\pi/2}_{\pi/4}\int^{1}_{0}\rho^2 sin^{2}\varphi*d\rho*d\varphi*d\theta[/tex]

    Plugged that into my calculator and got:

    [tex]\frac{k\pi(\pi+2)}{12}[/tex]

    The book answer is:
    [tex]\frac{k\pi}{192}[/tex]

    What am I doing wrong?
     
  2. jcsd
  3. Apr 3, 2010 #2

    LCKurtz

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    You mean [itex]\varphi = \pi / 4[/itex] for the cone.

    That doesn't look like cylindrical coordinates to me. But then, why would you want cylindrical coordinates anyway? Assuming that the [itex]\rho^2\sin^2(\phi)[/itex]
    is the moment arm, check your spherical coordinate dV.

    Aside from getting the dV wrong, using a calculator?

    [Edit] Looking closer your limits for [itex]\rho[/itex] are also wrong.
     
    Last edited: Apr 3, 2010
  4. Apr 3, 2010 #3
    Yes, sorry, typo.



    Sorry, typo, I meant spherical coordinates.

    I checked and dV should be

    [itex]\rho^2\sin^2(\phi)d\rho*d\varphi*d\theta[/itex]

    So that should change the integral to:

    [tex]I_{z} = k \int^{2\pi}_{0}\int^{\pi/2}_{\pi/4}\int^{1}_{0}(\rho^2 sin^{2}\varphi)^{2}*d\rho*d\varphi*d\theta[/tex]

    I usually use my calculator to check my setup, then once I know that is right, I go back and evaluate it by hand.

    I'm not sure what to do for [itex]\rho[/itex], I thought since the radius of the hemisphere was 1, then [itex]\rho[/itex] would go from 0 to 1.
     
  5. Apr 3, 2010 #4

    LCKurtz

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    The sine should not be squared.

    Have you drawn a picture of the desired volume? Your sphere is not centered at the origin and its equation isn't [itex]\rho[/itex] = 1.
     
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