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Mass and Moment of Inertia of a Planet

  1. Jan 5, 2009 #1
    1. The problem statement, all variables and given/known data
    There is a gaseous spherical planet with a nonconstant density rho(r) = rho_o (1 - r / R), where rho_0 is the maximum density attained at the planet's core, R is the radius of the planet, and r is the distance from the center of the planet.
    Use calculus to find the total mass of the planet in terms of rho_0 and R. Then find the moment of inertia of the planet in terms of its total mass M and R.

    2. Relevant equations

    3. The attempt at a solution

    I found the Mass of the planet as (rho_o * pi *R3)/3

    I integrated thin spherical shells to find the total mass using the integral - 4piR2dr*density

    Then I found average density as rho_o/4

    Now, I am using the integration of thin solid disks to find teh moment of inertia of the sphere...which comes out to be 8/15 * density*pi R5

    If we substitute the averahe density in terms of M in the above equation, the answer comes out to be 2/5 * M*R2 but I think the correct answer is 4/15*M*R2

    Can anyone tell me if I am correct or am I doing something wrong
  2. jcsd
  3. Jan 5, 2009 #2


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    You can't use the average density to find the moment of inertia. rho_0*pi*R^3/3 is the mass all right. To get the moment of inertia, I used that the moment of inertia of a hollow sphere is (2/3)*m(r)*r^2. m(r)=rho(r)*4*pi*r^2*dr. Integrating the whole thing does give (4/15)*M*R^2. It would be tough to use disks, since they don't have uniform density.
  4. Jan 6, 2009 #3

    I got it. I guess I was too impatient to think clearly :)
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