Answer to Question #145436 in Classical Mechanics for Sridhar

Question

A straight rod of length L is made of a material having mass per unit length m(x)=λ|x| where x is measured from the centre of rod.The moment of inertia about an axis perpendicular to the rod and passing through one end of the rod will be? L=1m and λ=16kg/m^(2)
Ans (32/3) kg/m²

Answer 1

Moment of inertia of a point mass at a distance x from the axis of rotation (assume it is y-axis) is

$$\text{d}I=m(x)\text{d}x^2=\lambda(|x|)\text{d}x^2.$$

The total moment of inertia is

$$I=\int^{0.5}_{-0.5}\lambda(|x|)\text{d}x^2=\\\space\\ =16\int^{0.5}_{-0.5}|x|\text{d}x^2=8x^2\Big|^{0.5}_{-0.5}$$