Question

A rod of uniform thickness is placed along x-axis with one end at origin. If length of rod is L
and its linear mass density is proportional to x, then find distance of its centre of mass from
origin​

Answer 1

Answer:

L/2

Explanation:

because rod is uniform so it's COM will be L/2

Answer 2

Given that,

Length of rod = L

If length of rod is L  and its linear mass density is proportional to x,

$\lambda=Ax$

Where, A = constant

As rod is kept along x-axis

So, $Y_{cm} = 0$, $Z_{cm} =0$

The COM of the element has coordinates (x, 0, 0).

Mass of element dx situated at x = x is

$dm=\lambda dx$

$dm=Ax dx$

We need to calculate the X- coordinate of center of mass of the rod

Using formula of center of mass

$X_{cm}=\dfrac{\int_{0}^{L}{x dm}}{\int_{0}^{L}{dm}}$

Put the value into the formula

$X_{cm}=\dfrac{\int_{0}^{L}{x Ax dx}}{\int_{0}^{L}{Ax dx}}$

$X_{cm}=\dfrac{A\int_{0}^{L}{x^2 dx}}{A\int_{0}^{L}{x dx }}$

$X_{cm}=\dfrac{(\dfrac{x^3}{3})_{0}^{L}}{(\dfrac{x^2}{2})_{0}^{L}}$

$X_{cm}=\dfrac{2L}{3}$

Hence, The coordinate of center of mass of the rod from the origin is $(\dfrac{2L}{3},0,0)$