Moment of Inertia of Hollow Cylinder Derivation

[BrainSalad]

For a uniform, hollow cylinder, why is this derivation wrong?

M = mass of whole solid cylinder
m = mass of missing cylindrical piece
R = radius of whole cylinder
r = radius of missing cylindrical piece

moment of inertia = moment of inertia of whole cylinder - moment of inertia of missing cylindrical piece

I = MR²/2 - mr²/2

m/M = pi*r²*h/pi*R²*h = r²/R²

m = M*r²/R²

I = MR²/2 - M*r⁴/2R²

I = MR⁴/2R² - M*r⁴/2R²

I = M/2R²*(R⁴ - r⁴)

 

[Delta2]

It seems correct, it is only that the result contains in its expression the mass M of the whole cylinder and not the mass of the corresponding hollow cylinder. If you substitute M=\frac{M_{h}R^2}{R^2-r^2} you ll get the usual expression for the inertia of a hollow cylinder I=\frac{1}{2}M_h(R^2+r^2).

 

[olivermsun]

It's fine as written. If you want to express $I$ using the mass of the outer shell only, call it $M^\prime,$ then you have to use $M^\prime = M - m = M (1 - r^2/R^2)$ to get the usual form for $I$.

 

[BrainSalad]

Thanks guys. Just a matter of confusion due to the specific application of the formula.