Question

A uniform rod of length $5\text{m}$ is placed along $x$ - axis as shown in the figure. The position of centre of mass of the rod from the origin is

• A. $2.5\text{m}$
• B. $5\text{m}$
• C. $7.5\text{m}$
• D. $10\text{m}$

Answer 1

The correct option is C $7.5\text{m}$

Let the mass of uniform rod be $M$
Mass per unit length of the rod $=\left(\frac{M}{L}\right)$ is constant since the rod is uniform.

Let $dm$ be the mass of an element of length $dx$ at a distance $x$ from origin
$\therefore dm=\left(\frac{M}{L}\right)dx=\left(\frac{M}{5}\right)dx$

Position of centre of mass, $x_{com}=\frac{\int xdm}{\int dm}$
$x_{com}=\frac{{\int }_5^{10}x\left(\frac{M}{L}\right)dx}{{\int }_5^{10}\frac{M}{L}dx}=\frac{{\int }_5^{10}xdx}{{\int }_5^{10}dx}$
$=\frac{1}{5}{\left(\frac{x^2}{2}\right)}_5^{10}$
$=\frac{1}{5}\left(\frac{{10}^2-5^2}{2}\right)$
$=\frac{75}{10}=7.5\text{m}$

Alternate:
As from symmetry we can say that the COM of the rod will lie at $\frac{L}{2}$, since rod is uniform
So, from origin COM will be at $5+\frac{5}{2}=7.5\text{m}$