Question of Physics

Question

A thin rod of length L has a non-uniform mass distribution. The linear mass density λ is given by λ= Ax² where A is a positive constant.

a) Find the total mass of the rod.

b) Find x_cm (the x coordinate of the center of mass of the rod) in terms of L .

Answer 1

The mass is the integral of density over the length of the rod.

Let's first do the indefinite integral

∫Ax²dx = Ax³/3

a)

The definite integral from 0 to L is AL³/3

b)

First note that the mass of any section between x-values a and b

is the definite integral from a to b, i.e.

(b³-a³)A/3

The center of gravity is the value x_cm, such that the mass to the left is the same as the mass to the right. That is,

(x_cm³-0³)A/3 = (L³-x_cm³)A/3

x_cm = L / 2^1/3