Question

A non-uniform rod AB has a mass M and length 2L the mass per unit length of the rod is `mx` at a point of the rod distant `x` from A. Then what is the moment of inertia of this rod about an axis perpendicular to the rod?

1)through A
2)through the midpoint of AB

Answer 1

See below.

Explanation:

From `A`

`I_A=int_0^(2L)m(x)x^2 dx = m int_0^(2L)x^3 dx = m/4(2L)^4=4mL^4`

From the midpoint of `[AB]`

Using the Huygens–Steiner theorem we have

`I_([AB]/2) = I_A-M L^2`

Here `M = int_0^(2L)mx dx = 1/2m(2L)^2 = 2mL^2`

and then

`I_([AB]/2) =4mL^4-2mL^2 xx L^2 = 2mL^4`