A block having mass M is placed on a horizontal frictionless...

A block having mass $\mathrm{M}$ is placed on a horizontal frictionless surface. This mass is attached to one end of a spring having force constant $\mathrm{k}$. The other end of the spring is attached to a rigid wall. This system consisting of spring and mass $\mathrm{M}$ is executing SHM with amplitude $\mathrm{A}$ and frequency $\mathrm{f}$. When the block is passing through the mid-point of its path of motion, a body of mass $\mathrm{m}$ is placed on top of it, as a result of which its amplitude and frequency changes to $\mathrm{A}^{\prime}$ and $\mathrm{f}$. If the velocity before putting the mass and after putting it is $\mathrm{v}$ and $\mathrm{v}^{1}$ respectively, then $\left(\mathrm{v}^{1} / \mathrm{v}\right)=\ldots \ldots \ldots \ldots$ (A) $\{\mathrm{M} /(\mathrm{m}+\mathrm{M})\}$ (B) $\{(\mathrm{M}+\mathrm{m}) / \mathrm{M}\}$ (C) $\{(M+m) /(M-m)\}\left(A^{1} / A\right)$ (D) $\{(M-m) /(M+m)\}\left(A^{1} / A\right)$

A block having mass $\mathrm{M}$ is placed on a horizontal frictionless surface. This mass is attached to one end of a spring having force constant $\mathrm{k}$. The other end of the spring is attached to a rigid wall. This system consisting of spring and mass $\mathrm{M}$ is executing SHM with amplitude $\mathrm{A}$ and frequency $\mathrm{f}$. When the block is passing through the mid-point of its path of motion, a body of mass $\mathrm{m}$ is placed on top of it, as a result of which its amplitude and frequency changes to $\mathrm{A}^{\prime}$ and $\mathrm{f}$.

If the velocity before putting the mass and after putting it is $\mathrm{v}$ and $\mathrm{v}^{1}$ respectively, then $\left(\mathrm{v}^{1} / \mathrm{v}\right)=\ldots \ldots \ldots \ldots$
(A) $\{\mathrm{M} /(\mathrm{m}+\mathrm{M})\}$
(B) $\{(\mathrm{M}+\mathrm{m}) / \mathrm{M}\}$
(C) $\{(M+m) /(M-m)\}\left(A^{1} / A\right)$
(D) $\{(M-m) /(M+m)\}\left(A^{1} / A\right)$

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Key Concepts

Simple Harmonic Motion (SHM)

This concept describes the periodic motion of a system where the restoring force is proportional to displacement from equilibrium. In SHM, key parameters include amplitude (the maximum displacement), frequency (the number of oscillations per unit time), and phase. One important characteristic is that the velocity is maximum when the object passes through the equilibrium position, where the displacement is zero.

Conservation of Linear Momentum

This principle states that in a closed system with no external forces, the total linear momentum remains constant before and after an interaction. In the context of this problem, when an additional mass is added to a moving body (at the equilibrium position of SHM), momentum conservation allows us to predict the new velocity of the combined mass immediately after the collision.

Impact of Mass Change on Oscillatory Systems

In systems executing SHM, altering the mass affects the dynamics, such as the frequency and amplitude of oscillation. When mass is added during motion (especially at the equilibrium point), the immediate kinetic energy is redistributed, and the effective mass of the oscillating system changes. This change must be accounted for when determining subsequent motion characteristics, like the new amplitude and the modified maximum velocity after the collision.

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