Question

A uniform solid right circular cone of base radius R has mass M. Prove that the moment of inertia of the cone about its central symmetry axis is \(\frac{3}{10}\)MR².

Answer 1

Consider a uniform solid right circular cone of mass M, base radius R and height h. The axis of rotation passes through its centre and the vertex, Its constant mass density is 

p = \(\frac m v\) = \(\frac{M}{\frac{1}{3}\pi R^2h}\)

We consider an elemental disc of mass dm, radius r and thickness dz. If the distance of each mass element from the axis is given by the variable z,

The moment of inertia of an element disc about the axis of rotation is

Since the cone extend from z = 0 to x = h, the MI of the cone about its central symmetry axis is as required