Question

A rod of length L has a mass density given by λ = λo(1 – x/L). What is the rod’s rotational inertia measured about the end where x = 0? Write your answer in terms of its total mass, M, and length, L.

Answer 1

Answer:

$I = \frac{\lambda_o L^3}{12}$

Explanation:

Rotational inertia of the rod about its one end is given as

$I = \int dm x^2$

here we know that

$dm = \lambda dx$

so we will have

$I = \int (\lambda dx) x^2$

$I = \int \lambda_o(1 - x/L) x^2 dx$

so we have

$I = \lambda_o(\frac{L^3}{3} - \frac{L^3}{4})$

$I = \frac{\lambda_o L^3}{12}$