Moment of inertia of a cylinder of mass m , length L and radius R about an axis passing through its centre and perpendicular to the axis of the cylinder is I = M R 2 4 + L 2 12 . If such a cylinder is to be made for a given mass of a material, the ratio L R for it to have minimum possible I is:

Step-by-step solution

Let a cylinder of mass m 1 , length L and radius R then take elementary disc of radius R and trickiness dx at distance of x from axis O O ' then moment of inertia about O O ' as this element dl = dmR 2 4 + dmx 2 I = ∫ dl = ∫ dmR 2 4 + ∫ n = L / 2 n = - L / 2 M L dx × x 2 ⇒ I = MR 2 4 + ML 2 12 ⇒ I = M 4 × V πL + ML 2 12 dI dL = - m V 4 πL 2 + M × 2 L 12 = 0 ⇒         V = 2 3 πL 3         ⇒