Question

a uniform disc and hollow right circular cone have the same formula for their moment of inertia when rotating about the central axis why is it so?

Answer 1

Answer:

This is as a result that about the central axis a collapsed hollow cone is equivalent to a uniform disc

Explanation:

The integration of the differential mass of the hollow right circular cone yields

$I=\int\limits dmr^2 = \int\limits^a_b {\frac{2Mxr^2}{R^2 +H^2} } \, dx = \frac{2MR^2dx}{(R^2 +H^2)^2} \frac{(R^2 +H^2)^2}{4} = \frac{1}{2}MR^2$

and for a uniform disc

I = 1/2πρtr⁴ = 1/2Mr².