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.

 

[PJC01]

This derivation is incorrect because it assumes that the moment of inertia of the missing cylindrical piece is equal to mr2/2, which is the moment of inertia of a solid cylinder with radius r. However, the missing cylindrical piece is not a solid cylinder, it is a hollow cylinder with a different distribution of mass. Therefore, the moment of inertia of the missing cylindrical piece should be calculated using the parallel axis theorem, taking into account the difference in distribution of mass between a solid and hollow cylinder. This results in a different value for the moment of inertia of the missing cylindrical piece, making the overall derivation incorrect.