Eduprobe
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 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?
The instantaneous speed at x0 of the combined masses decreases in both the cases.
The amplitude of oscillation in the first case changes by a factor of √(M/(m+M)), whereas in the second case it remains unchanged
The final time period of oscillation in both cases is the same
The total energy decreases in both cases
Solution:
In case I,
From Conservation of momentum, MV1 = (M+m)V2
MV1/(M+m) = V2
√(k(M+m)A2²) = M/(M+m) √(kMA1²)
A2 = √(M/(M+m))A1
In case II,
A2 = A1
T = 2π√((M+m)/k) in both cases.
Total energy decreases in the first case whereas it remains the same in the second case. Instantaneous speed at x0 decreases in both cases.
Answer is A, B, and D.