# Moment of Inertia for a hollow Sphere

1. Feb 22, 2010

### Xyius

I am confused about one thing on this derivation. Okay so the guide im following goes like this..

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$$\sigma=\frac{M}{A}$$

$$dm=\sigma dA=(\frac{M}{4\pi R^2})2\pi rsin\phi Rd\phi$$

$$dm=\frac{M}{2}sin\phi d\phi$$​

This is one part that confuses me. It seems as though the lower case "r" was used to cancel out one of the "R's" on the bottom. I can't reason another way how it went away. The rest of the derivation goes like this.. (Just in case anyone needs it)

Here, radius of elemental ring about the axis is R sinθ. Moment of inertia of elemental mass is :
$$dI=R^2sin^2\phi dm=R^2sin^2\phi (\frac{M}{2}sin\phi d\phi)$$​

Therefore the total moment of inertia is..

$$\oint R^2sin^2\phi (\frac{M}{2}sin\phi d\phi)$$
$$\frac{MR^2}{2}\oint sin^3\phi d\phi = \frac{MR^2}{2}\oint (1-cos^2\phi )sin\phi d\phi = \frac{MR^2}{2}\oint sin\phi - sin\phi cos^2\phi d\phi = \frac{MR^2}{2}(-cos\phi +\frac{1}{3} cos^3\phi )$$​

The limits are 0 to $$\pi$$ hence..
$$I=\frac{2}{3}MR^2$$​

2. Feb 23, 2010

### tiny-tim

Welcome to PF!

Hi Xyius! Welcome to PF!
It's a misprint … there is no r !

3. Feb 23, 2010

### Xyius

Re: Welcome to PF!

Thanks! But since that expression came from the differential dA, why is it "R"?? Because the radius is constantly changing right?

4. Feb 23, 2010

### tiny-tim

Yup! The circumference is 2πRsinφ, and the thickness is Rdφ.

5. Feb 23, 2010

### Xyius

Ohh! Makes sense now! Cool thanks a lot! :D

6. Feb 23, 2010

### Bob S

The best way to calculate the moment of inertia (MOI) of a hollow sphere is to calculate the MOI of two solid spheres, and subtract the MOI of the smaller sphere from the MOI of the larger sphere.

The best way to calculate the MOI of a solid sphere is to use cylindrical coordinate system, r, θ, z. Using ρ as density, the basic form of the integral is

I = ∫∫∫ρ·r2·r·dr·dθ·dz

where r is the perpendicular distance from the axis of rotation, R is the radius of the sphere, and ρ = M/(4πR3/3). Do the z integration last. You will need to determine and use the appropriate integration limits. Hint: Use Phythagorean theorem.

Bob S