JEE Main 2025 Apr 7 Shift 1, Physics Q6: Centre of Mass

Treat each straight segment as a uniform rod. With $L=10$ cm the two legs are of lengths $2L=20$ cm (horizontal) and $3L=30$ cm (vertical). Let the linear mass density be uniform.

Mass proportions: horizontal leg has mass $\propto 2$ (call it $2m$); vertical leg has mass $\propto 3$ (call it $3m$).

Centres of mass of each leg (in cm):

• Horizontal leg midpoint: $(10,0)$
• Vertical leg midpoint: $(0,15)$

Compute the composite COM:

$x_{\text{COM}}=\frac{2m\cdot 10+3m\cdot 0}{5m}=4\text{ cm}$

$y_{\text{COM}}=\frac{2m\cdot 0+3m\cdot 15}{5m}=9\text{ cm}$

So ${\vec{r}}_{\text{COM}}=4\hat{i}+9\hat{j}$ (in cm).

$\therefore$ Option (4).