Question

A composite system of two uniform rods of same metal are shown in the figure. Lengths of rods are as shown in figure. The co-ordinates of the centre of mass of the system of rods are

• A. \(\left(\frac{L}{6},\frac{4L}{3}\right)\)
• B. \(\left(\frac{L}{6},\frac{2L}{3}\right)\)
• C. \(\left(\frac{L}{3},\frac{2L}{3}\right)\)
• D. \(\left(\frac{L}{4},\frac{2L}{3}\right)\)

Hint:

Mass is proportional to length for a uniform rod; take the weighted average of the rod centres.

Answer 1

The Correct Option is B

Step 1: Formula-first approach
$x_{cm}=\dfrac{m_1x_1+m_2x_2}{m_1+m_2}$ and similarly for y, with $m_1=2M$, $m_2=4M$.

Step 2: Compute
$x_{cm}=\dfrac{2M(L/2)}{6M}=\dfrac{L}{6}$, $y_{cm}=\dfrac{4M(L)}{6M}=\dfrac{2L}{3}$.

Step 3: Compare the options
Only $\left(\dfrac{L}{6},\dfrac{2L}{3}\right)$ matches both coordinates.

Final Answer:
The centre of mass is at (L/6, 2L/3). \[ \boxed{\text{(B) }\left(\dfrac{L}{6},\dfrac{2L}{3}\right)} \]