Question

The linear mass density of a non uniform rod of
length L varies as a = 2x, where x is distance from
one end. The distance of centre of mass from this
end is

Answer 1

Answer:

See below.

Explanation:

We are given the mass density of a uniform rod varies as,

$\lambda( x ) = 2x$

To get the mass of the whole rod which has length L, we integrate the mass density w.r.t to x which is the distance from one end of the rod,

$M = \int\limits^L_0 \lambda(x) \, dx =\int\limits^L_0 2x\ dx = L^2$

Similarly, we integrate,

$C = \int\limits^L_0 {x} \lambda(x) \ dx = \frac{2L^3}{3}$

To get the position of the centre of mass,

$COM = \frac{\frac{2L^3}{3}}{L^2} = \frac{2L}{3}$

Answer 2

Answer:

2L/3

Explanation:

See the attachments.

I hope you are satisfied by my answer.