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 $\mathrm{f}$. The ratio of amplitudes $\left(\mathrm{A}^{1} / \mathrm{A}\right)=\ldots \ldots \ldots$ (A) $\sqrt{\{}(\mathrm{M}+\mathrm{m}) / \mathrm{m}\}$ (B) $\sqrt{\{m} /(\mathrm{M}+\mathrm{m})\}$ (C) $\sqrt{\{} \mathrm{M} /(\mathrm{M}+\mathrm{m})\}$ (D) $\sqrt{\{}(\mathrm{M}+\mathrm{m}) / \mathrm{M}\}$

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 $\mathrm{f}$.
The ratio of amplitudes $\left(\mathrm{A}^{1} / \mathrm{A}\right)=\ldots \ldots \ldots$
(A) $\sqrt{\{}(\mathrm{M}+\mathrm{m}) / \mathrm{m}\}$
(B) $\sqrt{\{m} /(\mathrm{M}+\mathrm{m})\}$
(C) $\sqrt{\{} \mathrm{M} /(\mathrm{M}+\mathrm{m})\}$
(D) $\sqrt{\{}(\mathrm{M}+\mathrm{m}) / \mathrm{M}\}$

Key Concepts

Simple Harmonic Motion (SHM)

SHM is a type of periodic motion in which the restoring force is directly proportional to the displacement from the equilibrium position and acts in the opposite direction. This motion is characterized by sinusoidal oscillations, well?defined amplitude, and a specific angular frequency determined by system parameters such as mass and spring constant.

Angular Frequency and Amplitude Relationship

In SHM, the maximum speed of the oscillating object is related to the amplitude and angular frequency by the relation v_max = ?A. This means that when the amplitude or the mass (which affects the angular frequency) changes, the maximum speed, and hence the characteristics of the motion, are also altered.

Momentum Conservation in Inelastic Collisions

When an additional mass is added to a moving system in an inelastic manner, such as a mass being placed on a moving block, the horizontal component of momentum is conserved immediately after the mass is added. This conservation principle is used to determine the new velocity of the combined mass system, which directly affects the kinetic energy available for subsequent oscillations.

Effect of Mass Variation on SHM Properties

The oscillatory properties of a system, including both amplitude and angular frequency, depend on the mass involved in the oscillation. Adding a mass to an oscillating system increases the total mass and thereby decreases the angular frequency. If the addition of mass occurs while the system is in motion, the same velocity is not maintained relative to the new total mass; instead, the transformed energy results in a reduced amplitude, reflecting the interplay of momentum conservation and energy redistribution.

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