Mass and Moment of Inertia of a Planet

[guru_to_be]

Homework Statement

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.

Homework Equations

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 the 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

 

[Dick]

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.

 

[guru_to_be]

Thanks

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