A non-uniform rod AB has a mass `M` ad length 2l. The mass per unit length of the rod is mx 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 (a) through A (b) through the mid-point of AB.

Explanation

`M=int_(0)^(2l)dM=int_(0)^(2l)(mx).dx`
`=2ml^(2)`
`thereforem=(M)/(2l^(2))`
(a). `I_(A)=int_(0)^(2l)(dM)X^(2)=int_(0)^(2l)(mXdX)X^(2)`
`=int_(0)^(2l)((M)/(2l^(2))X^(3)dX)=2Ml^(2)`
(b). `I_(C)=int_(0)^(2l)(dM)(X-l)^(2)`

`=int_(0)^(2l)(mXdX)(X-l)^(2)`
`=int_(0)^(2l)((M)/(2l^(2)))(X)(X-l)^(2)dX`
`=(1)/(3)Ml^(2)`