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: Understanding the Concept
The rods are of the same metal, so mass is proportional to length. The rod along the x-axis has length \(L\) and mass \(2M\); the rod along the y-axis has length \(2L\) and mass \(4M\).

Step 2: Centres of the rods
Rod on the x-axis has its centre at \(\left(\dfrac{L}{2},0\right)\). Rod on the y-axis has its centre at \((0,L)\).

Step 3: Calculation
\(x_{cm}=\dfrac{2M\cdot\frac{L}{2}+4M\cdot0}{6M}=\dfrac{L}{6}\)
\(y_{cm}=\dfrac{2M\cdot0+4M\cdot L}{6M}=\dfrac{2L}{3}\)

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