A non-uniform rod AB has a mass M and length 2l.The mass per unit lengthof 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, a)through A,b) through the mid point of AB. Ans) let a element of length dx at adistance x from the point A of the rod.mass of this element will be dm = mxdxso moment of inertia of this element about point A is dI = dm * x^{2}for moment of inertia of whole rod perform intigrationI= ∫dI = _{o}∫^{2L}dm * x^{2} =_{o}∫^{2L}m * x3 dx =4mL^{4}and we know M =_{o}∫^{2L}dm = _{o}∫^{2L}mxdx =2mL^{2}so moment of inerta I =2ML^{2}