Question

Two identical blocks A and B of mass m each are connected to each other by spring of spring constant k. Block B is initially shifted to a small distance $x_0$ from natural length of spring to the left and then released. Choose the correct statements for this problem, after the spring attains it's natural lenth.

• A. Velocity of centre of mass of the system is $\frac{1}{2}\sqrt{\frac{k}{m}}x_0$
• B. Maximum elongation in spring during the subsequent motion is $\frac{x_0}{\sqrt{2}}$
• C. Maximum elongation in spring during the subsequent motion is $x_0$
• D. Maximum speed of block A during subsequent motion be $\sqrt{\frac{K}{m}}x_0$

Answer 1

Answer: (C)

Maximum elongation in spring during subsequent motion is $x_0$

Answer 2

We analyze the motion in steps:

1. Relative Motion: The relative extension of the spring oscillates as a simple harmonic oscillator with amplitude $x_0$ and angular frequency $\omega_{\rm rel}=\sqrt{\frac{2k}{m}}$.

2. Block Speeds: The center-of-mass (COM) momentum is conserved, implying zero COM speed after the blocks separate from the wall. At the natural length, the speeds of the blocks are related by $v_B-v_A=x_0\sqrt{\frac{2k}{m}}$, and since $v_A+v_B=0$, we find $|v_A|=|v_B|=\frac{x_0}{\sqrt{2}}\sqrt{\frac{k}{m}}$.

3. Option Analysis:

   • (A) COM speed = 0, so this is false.
   • (B) Spring oscillation amplitude is $x_0$, not $\frac{x_0}{\sqrt{2}}$, so this is false.
   • (C) The maximum elongation is indeed $x_0$, which is true.
   • (D) The maximum speed of block A is $\frac{x_0}{\sqrt{2}}\sqrt{\frac{k}{m}}$, not $\sqrt{\frac{k}{m}}x_0$, so this is false.

Therefore, only option (C) is correct.