Question

A block with mass $M$ is connected by a massless spring with stiffness constant $k$ to a rigid wall and moves without friction on a horizontal surface. The block oscillates with small amplitude A about an equilibrium position $x_0$. Consider two cases: (i) when the block is at $x_0$; and (ii) when the block is at $x=x_0+A$. In both the cases, a particle with mass $m(<M)$ is softly placed on the block after which they stick to each other. Which of the following statement(s) is(are) true about the motion after the mass m is placed on the mass $M$?

• A. The amplitude of oscillation in the first case changes by a factor of $\sqrt{\frac{M}{m+M}}$, whereas in the second case it remains unchanged
• B. The final time period of oscillation in both the cases is same
• C. The total energy decreases in both the cases
• D. The instantaneous speed at $x_0$ of the combined masses decreases in both the cases.

Answer 1

Answer: (A), (B), (D)

The amplitude of oscillation in the first case changes by a factor of $\sqrt{\frac{M}{m+M}}$, whereas in the second case it remains unchanged
The final time period of oscillation in both the cases is same
The instantaneous speed at $x_0$ of the combined masses decreases in both the cases.

In case I,

From Conservation of momentum,

$M{V}_1=(M+m)V_2$

$\frac{M{V}_1}{M+m}=V_2$

$\sqrt{\frac{k}{M+m}}{A}_2=\frac{M}{M+m}\sqrt{\frac{k}{M}}{A}_1$

$A_2=\sqrt{\frac{M}{M+m}}{A}_1$

In case II,

$A_2=A_1$

$T=2\pi \sqrt{\frac{M+m}{k}}$ in both the cases.

Total energy decreases in first case where as remain same in 2nd case. Instantaneous speed at $x_0$ decreases in both case.

Answer is A,B and D.