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

`I = (\lambda_o L^3)/(12)`

Step-by-step 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((L^3)/(3) - (L^3)/(4))`

`I = (\lambda_o L^3)/(12)`