A block with mass M is connected by a massless spring with stiffness constant to a rigid wall and moves without friction on a horizontal surface. The block oscillates with small amplitude A about an equilibrium position . Consider two cases: (i) when the

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 $m (m < M)$ 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 $M$ ? [2016]

Smart Achievers · Accepted Answer

(1, 2, 4)

Case (i) : Applying principle of conservation of linear momentum.

$x_{0}$

$M (A_{1} \times ω_{1}) = (M + m) (A_{2} \times ω_{2})$

$∴$ $M A_{1} \times \sqrt{\frac{K}{M}} = (M + m) A_{2} \times \sqrt{\frac{K}{M + m}}$

$∴$ $A_{2} = \sqrt{\frac{M}{M + m}} A_{1} \Rightarrow \frac{A_{2}}{A_{1}} = \sqrt{\frac{M}{M + m}}$

Also, $E_{1} = \frac{1}{2} M V_{1}^{2}$

and $E_{2} = \frac{1}{2} (M + m) V_{2}^{2}$ $(∵ V_{2} = (\frac{M}{M + m}) V_{1} from eq (i))$

$\Rightarrow E_{2} = \frac{1}{2} (M + m) {(\frac{M}{M + m})}^{2} V_{1}^{2}$

$= \frac{1}{2} (\frac{M}{M + m}) V_{1}^{2}$

Clearly, $E_{1} > E_{2}$

The new time period $T_{2} = 2 \sqrt{\frac{m + M}{K}}$

Instantaneous speed at $X_{0}$ of the combined masses $V_{2} = \frac{M V_{1}}{M + m} < V_{1}$

Case (ii) : The new time period $T_{2} = 2 \sqrt{\frac{m + M}{K}}$

Also, $A_{2} = A_{1} and E_{2} = E_{1}$

In this case also, instantaneous speed at $X_{0}$ of the combined masses decreases.

Solution

1 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.

2 The final time period of oscillation in both the cases is same.

3 The total energy decreases in both the cases.

4 The instantaneous speed at $x_{0}$ of the combined masses decreases in both the cases.

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