A rod of length 5L is bent at a right angle, keeping one side len… — Solution | JEE Physics

MCQEasyJEE 2025Centre of Mass

A rod of length $5L$ is bent at a right angle, keeping one side length as $2L$ . The position of the centre of mass of the system (Consider $L = 10 \, \text{cm}$ ):

A
$2\hat{i} + 3\hat{j}$
B
$3\hat{i} + 7\hat{j}$
C
$5\hat{i} + 8\hat{j}$
D
$4\hat{i} + 9\hat{j}$

Answer

Correct answer:D

Step-by-step solution

Standard Method

Given: A rod of total length $5L$ is bent at a right angle into two segments of lengths $2L$ and $3L$ . Take the joint as the origin. The $2L$ segment lies along the $x$ -axis and the $3L$ segment lies along the $y$ -axis. Also, $L = 10 \, \text{cm}$ .

Find: The position vector of the centre of mass.

The centre of mass of the $2L$ segment is at $\left(L,0\right)$ and the centre of mass of the $3L$ segment is at $\left(0,1.5L\right)$ .

Using the centre of mass formula,

x_{\text{cm}} = \frac{2L \cdot L + 3L \cdot 0}{5L} = \frac{2}{5}L

and

y_{\text{cm}} = \frac{2L \cdot 0 + 3L \cdot 1.5L}{5L} = \frac{9}{10}L

Substitution and Final Position

Now substitute $L = 10 \, \text{cm}$ :

x_{\text{cm}} = \frac{2}{5} \cdot 10 = 4 \, \text{cm}

y_{\text{cm}} = \frac{9}{10} \cdot 10 = 9 \, \text{cm}

Treat each arm by length-weighting

Since the rod is uniform, mass is proportional to length. Therefore, the coordinates of the centre of mass can be found by taking the length-weighted average of the midpoints of the two perpendicular arms. This gives the position vector

\vec{r}_{\text{cm}} = 4\hat{i} + 9\hat{j}

Therefore, the correct option is D.

Common mistakes

Taking the centre of mass of each segment at its endpoint is incorrect because a uniform straight segment has its centre of mass at its midpoint. Use $\left(L,0\right)$ for the $2L$ arm and $\left(0,1.5L\right)$ for the $3L$ arm.
Using equal weights for the two arms is wrong because the masses of the two parts are not equal. For a uniform rod, mass is proportional to length, so the weights must be in the ratio $2L:3L$ .
Forgetting to place the bend at the origin leads to wrong coordinates. First assign a clear coordinate system, then locate the midpoints of the two perpendicular segments in that frame.

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Step-by-step solution

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