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 $$I=MR24+L212$$. If such a cylinder is to be made for a given mass of material, the ratio $$L/R$$ for it to have minimum possible I is :

Question

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  $$I=MR24+L212$$. If such a cylinder is to be made for a given mass of material, the ratio $$L/R$$ for it to have minimum possible I is :

Infinity Learn · Accepted Answer

$$I=MR24+ML212$$​Volume $$=V=$$π$$R2L$$​⇒$$R2=V$$$$π$$L∴$$I=MV4$$$$π$$$$L+ML212$$​⇒$$dIdL=MV4$$π−$$1L2+M2L12For$$ minima, $$dIdL=0$$​⇒π$$R2L4$$$$π$$$$L2=2L12$$​⇒$$RL2=23$$​⇒$$LR=32$$

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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=MR24+L212. If such a cylinder is to be made for a given mass of material, the ratio L/R for it to have minimum possible I is :

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