Question

The moment of inertia of solid cylinder about its own axis is the same as its moment of inertia about an axis passing through its centre of gravity and perpendicular to its length. The relation between its length and radius r is

Answer 1

hope it helps you........

Answer 2

Answer:

The relation between length and the radious of the cylinder is $L=\sqrt{3} R$

Explanation:

The mass of the cylinder is M

The radius of the cylinder is R

The length of the cylinder is L

The moment of inertia of cylinder about its own axis is

$\dfrac{MR^2}{2}$                              (1)

The moment of inertia about an axis passing through the center of mass of the cylinder and perpendicular to the length of the cylinder is

$\dfrac{1}{12}ML^2+\dfrac{1}{4}MR^2$                                (2)

As given that both the moments are same so equating equation 1 and 2 we get,

$\dfrac{MR^2}{2}=\dfrac{1}{12}ML^2+\dfrac{1}{4}MR^2\\MR^2=\dfrac{1}{6}ML^2+\dfrac{1}{2}MR^2\\MR^2-\dfrac{1}{2}MR^2=\dfrac{1}{6}ML^2\\\dfrac{1}{2}MR^2=\dfrac{1}{6}ML^2\\R^2=\dfrac{1}{3}L^2\\L=\sqrt{3}R$