What is the Moment of Inertia of a Sphere with Different Densities and Radii?

[cholo1968]

1. Sphere inner core radius 3490km and density 13000kg/m³. Outer radius 6400km and density 4000kg/m³. Show that the moment of inertia is approximately 8 x 10³⁷kg m².



2. I know that the moment of inertia of a sphere is 2/5xmr².



3. I have calculated the volume for each part of the sphere using 4/3∏r³. Then used the equation in part 2 to calculate the moment of inertia for the inner and outer parts of the sphere. I have read that for more complex shapes the moments of inertia need to be added together to give the total moment of inertia, however this only gives 7.2 x 10³⁷kg m².
Is there some formula I am missing because I have tried calculating each mass and adding them together and using the outer radius for the calculation and then used a mean radius for the calculation but nothing remotely close to the required answer of 8 x 10³⁷. Any guidance would be appreciated as I can do the maths. Just think I am missing something vital.

 

[OldEngr63]

After you calculate the volumes, I presume you have calculated the respective masses. In calculating the volumes, did you account for the fact that the outer volume is hollow, so you have to subtract the inner volume from it?

The expression I = (2/5) m r^2 applies for a solid sphere, so in trying to apply it to the outer (hollow) section, you have to first apply it to a solid sphere (of appropriate mass, which you have to calculate), get that MMOI, then apply it to a smaller sphere the size of the inner sphere with density equal to the outer sphere, and subtract that MMOI from the on just found in order to get the MMOI for the hollow outer sphere. The answer given is correct.

This is a very good problem! I shall have to remember it!

 

[cholo1968]

Thankyou OldEngr63. After looking at the problem again and using your guidance I achieved the correct answer to within 0.01. I was not looking far enough in front of me at the whole problem. Thakyou again.