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A block of mass M is attached to a massless spring of stiffness constant k fixed to a rigid wall. The block oscillates without friction on a horizontal surface with small amplitude A about the equilibrium position. Consider two scenarios: (i) a particle of mass m (m < M) is gently placed on the block when the block is at the equilibrium position x0, and (ii) the particle is gently placed when the block is at the extreme position x0 + A. In both cases the particle sticks to the block. Which of the following statements about the subsequent motion are correct?

1. The amplitude of oscillation in case (i) changes by a factor of sqrt(M / (m + M)), while in case (ii) it remains unchanged
2. The time period of oscillation is the same in both cases after the mass m is added
3. The total mechanical energy decreases in both cases
4. The instantaneous speed of the combined mass is less than the original block speed immediately after m is placed, in both cases

Correct answer: The amplitude of oscillation in case (i) changes by a factor of sqrt(M / (m + M)), while in case (ii) it remains unchanged

Solution

At the equilibrium in case (i), the block moves at maximum speed; placing m causes a perfectly inelastic collision reducing speed, so the new amplitude is A*sqrt(M/(M+m)). At the extreme in case (ii), velocity is zero, so the mass is simply added at rest with no change in amplitude. The time period in both cases becomes 2*pi*sqrt((M+m)/k), which is the same. Total energy decreases in case (i) due to the inelastic collision, but in case (ii) no energy is lost.

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