Question 42: Oscillations

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 x0. Consider two cases:
(i) when the block is at x0; and
(ii) when the block is at x = x0 + A.
In both cases, a particle with mass m( < M) is softly placed on the block after which they stick on each other. Which of the following statement(s) is(are) true about the motion after the mass m is placed on the mass M?

Accepted Answer

Case - I
Case - II
In case 1,
$$M{v_1} = (M + m){v_2} \Rightarrow {v_2} = \left( {{M \over {M + m}}} \right){v_1}$$
$$\sqrt {{k \over {M + m}}} {A_2} = \left( {{M \over {M + m}}} \right)\sqrt {{k \over M}} {A_1}$$
$${A_2} = \sqrt {{k \over {M + m}}} {A_1}$$
In case - 2, $${A_2} = {A_1}$$
$$T = 2\pi \sqrt {{{M + m} \over k}} $$ in both cases.
Total energy decreases in first case whereas remain same in 2nd case. Instantaneous speed at x
0
decreases in both cases.

Answer

The amplitude of oscillation in the first case changes by a factor of $$\sqrt {{M \over {m + M}}} $$, whereas in the second case it remains unchanged.

Answer

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

Answer

The total energy decreases in both the cases

Answer

The instantaneous speed at x0 of the combined masses decreases in both the cases

Oscillations

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