Question

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 $ cm):

• A. \( 2\hat{i} + 3\hat{j} \)
• B. \( 3\hat{i} + 7\hat{j} \)
• C. \( 5\hat{i} + 8\hat{j} \)
• D. \( 4\hat{i} + 9\hat{j} \)

Hint:

To find the center of mass of a system of objects, calculate the weighted average of the positions of each object. The weights are the masses of the objects in the system.

Answer 1

The Correct Option is D

The rod is bent at a right angle into two segments of lengths \( 2L \) and \( 3L \). Assume the corner (joint) is at the origin. Let the \( 2L \) segment lie along the x-axis and the \( 3L \) segment along the y-axis.

• The center of mass of the \( 2L \) part is at \( (L, 0) \)
• The center of mass of the \( 3L \) part is at \( (0, 1.5L) \)

Using the formula for center of mass: \[ x_{\text{cm}} = \frac{2L \cdot L + 3L \cdot 0}{5L} = \frac{2}{5}L, \quad y_{\text{cm}} = \frac{2L \cdot 0 + 3L \cdot 1.5L}{5L} = \frac{9}{10}L \] Substitute \( L = 10 \) cm: \[ x_{\text{cm}} = \frac{2}{5} \cdot 10 = 4 \text{ cm}, \quad y_{\text{cm}} = \frac{9}{10} \cdot 10 = 9 \text{ cm} \] \[ \Rightarrow \vec{r}_{\text{cm}} = 4\hat{i} + 9\hat{j} \]