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 = mxdx

so moment of inertia of this element about point A is

  dI = dm * x²

for moment of inertia of whole rod perform intigration

I=  ∫dI = _o∫^2Ldm * x²

            =_o∫^2Lm * x3 dx

            =4mL⁴

and we know M =_o∫^2Ldm

                      = _o∫^2Lmxdx

                      =2mL²

so moment of inerta  I =2ML²