Moment of Inertia for a hollow Sphere

[Xyius]

I am confused about one thing on this derivation. Okay so the guide I am 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..

$$I=\frac{2}{3}MR^2$$​

 

[tiny-tim]

Welcome to PF!

Hi Xyius! Welcome to PF!

… It seems as though the lower case "r" was used to cancel out one of the "R's" on the bottom …

It's a misprint … there is no r ! …

read it as R. ​

 

[Xyius]

It's a misprint … there is no r ! …

read it as R. ​

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

 

[tiny-tim]

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

 

[Xyius]

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

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

 

[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