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A non-uniform rod A B has a mass M and length 21 . The mass per unit length of the rod is m x at a point of the rod distant x from A. Find the moment of inertia of this rod about an axis perpendicular to the rod
Through A.

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(1) $2 {MI}^{2}$

(2) $5 {MI}^{2}$

(3) $8 {MI}^{2}$

(4) $3 {MI}^{2}$

answer is A.

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Detailed Solution

$M = \int_{0}^{2} dM = \int_{0}^{2 l} (mx) \cdot dx$

$= 2 {ml}^{2}$

$m = \frac{M}{2 l^{2}} I_{A} = \int_{0}^{2 I} (dM) X^{2}$

$= \int_{0}^{2} (mXdX) X^{2}$

$= \int_{0}^{2 l} (\frac{M}{2 i^{2}} X^{3} dX) = 2 {MI}^{2}$

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