When a mass m is suspended from the free end of a massless...

When a mass $\mathrm{m}$ is suspended from the free end of a massless spring having force constant $\mathrm{k}$, its oscillates with frequency $\mathrm{f}$. Now if the spring is divided into two equal parts and a mass $2 \mathrm{~m}$ is suspended from the end of anyone of them, it will oscillate with a frequency equal to......... (A) $\mathrm{f}$ (B) $2 \mathrm{f}$ (C) $(\mathrm{f} / \sqrt{2})$ (D) $\sqrt{2 f}$

When a mass $\mathrm{m}$ is suspended from the free end of a massless spring having force constant $\mathrm{k}$, its oscillates with frequency $\mathrm{f}$. Now if the spring is divided into two equal parts and a mass $2 \mathrm{~m}$ is suspended from the end of anyone of them, it will oscillate with a frequency equal to.........
(A) $\mathrm{f}$
(B) $2 \mathrm{f}$
(C) $(\mathrm{f} / \sqrt{2})$
(D) $\sqrt{2 f}$

Key Concepts

Oscillation Frequency of a Mass-Spring System

The angular frequency and hence the oscillation frequency of a mass attached to a spring in simple harmonic motion is given by the square root of the ratio of the effective spring constant to the mass. Changes in either the spring constant or the mass will directly affect the oscillation frequency, providing a direct link between the physical properties of the system and its dynamic behavior.

Simple Harmonic Motion (SHM)

This concept involves motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that displacement. In the context of a mass-spring system, SHM describes the periodic oscillations around an equilibrium position, with the system's dynamics entirely determined by its mass and the properties of the spring.

Hooke's Law and Spring Constant

Hooke's Law states that the force exerted by a spring is proportional to its displacement from equilibrium, with the spring constant (k) serving as the proportionality factor. This law is fundamental in understanding how springs behave under load and is a key principle in analyzing the dynamics of oscillatory systems.

Effective Spring Constant for Spring Segments

When a uniform spring is divided into segments, the effective spring constant of each segment changes inversely with its length. For instance, halving the length of a spring (assuming uniform material and properties) effectively doubles its spring constant. This concept is crucial when comparing the oscillatory behavior of a full spring to that of a segment of the same spring.

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