What is wrong with my derivation for the moment of inertia of a sphere?

In summary, the conversation revolves around deriving the moment of inertia for a sphere using infinitesimally thin spherical shells. However, the original dI equation used was incorrect and after correcting it with the correct moment of inertia for a thin shell, the final answer came out to be (2/3)MR^2 instead of (3/5)MR^2. The error was identified and corrected with the help of a missing factor of 2/3.
  • #1
eg2333
6
0
Ok, so I thought about a derivation for the moment of inertia, but my answer comes out to (3/5)MR^2

Basically, what I did was I considered the sphere as a sum of infinitesimally thin spherical shells.

The moment of inertia for one shell is dI=(r^2)*dm

where dm=(M/V)*4*pi*r^2*dr

where V=(4/3)*pi*R^3

so the equation dI=3*pi*M*r^4*dr when simplified.

Integrating this from 0 to R (Summing up the spherical shells from the center to the edge of the big sphere) gives me (3/5)*M*R^2. The process clearly yields the wrong answer, so I need help seeing where the fault is.
 
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  • #2
The moment of inertia of a thin shell is (2/3)MR^2, not MR^2, so your original dI should be (2/3)r^2 dm - there's your missing factor of 2/3.
 
  • #3
Ohhhhh, I see. I was treating it as a point.
 

FAQ: What is wrong with my derivation for the moment of inertia of a sphere?

What is the equation for calculating the moment of inertia of a sphere?

The equation for calculating the moment of inertia of a sphere is I = (2/5)mr^2, where m is the mass of the sphere and r is the radius.

Why doesn't my derivation for the moment of inertia of a sphere match the expected result?

There could be several reasons why your derivation for the moment of inertia of a sphere does not match the expected result. Some common mistakes include incorrect integration, using the wrong equation, or not considering the distribution of mass within the sphere.

Can I use the moment of inertia equation for a solid sphere to calculate the moment of inertia for a hollow sphere?

No, the moment of inertia equation for a solid sphere cannot be used to calculate the moment of inertia for a hollow sphere. This is because the distribution of mass is different for a hollow sphere, and thus the equation should be adjusted accordingly.

How can I check if my derivation for the moment of inertia of a sphere is correct?

You can check the correctness of your derivation by comparing it to the known and accepted equation for the moment of inertia of a sphere. You can also plug in different values for mass and radius to see if your derivation produces the expected result.

Are there any assumptions that need to be made in order to derive the moment of inertia of a sphere?

Yes, there are a few assumptions that are typically made when deriving the moment of inertia of a sphere. These include assuming the sphere is a perfect, uniform shape with a constant density, and neglecting any external forces or rotations.

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