A weightless spring which has a force constant k oscillates...

A weightless spring which has a force constant $k$ oscillates with frequency $n$ when a mass $m$ is suspended from it. The spring is cut into two equal halves and a mass $2 \mathrm{~m}$ is suspended from one part of spring. The frequency of oscillation will now become (a) $n$ (b) $2 n$ (c) $\frac{n}{\sqrt{2}}$ (d) $n(2)^{1 / 2}$

A weightless spring which has a force constant $k$ oscillates with frequency $n$ when a mass $m$ is suspended from it. The spring is cut into two equal halves and a mass $2 \mathrm{~m}$ is suspended from one part of spring. The frequency of oscillation will now become
(a) $n$
(b) $2 n$
(c) $\frac{n}{\sqrt{2}}$
(d) $n(2)^{1 / 2}$

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Key Concepts

Simple Harmonic Motion

Simple harmonic motion (SHM) describes a type of periodic oscillatory movement where the restoring force is directly proportional to the displacement from an equilibrium position. In mass-spring systems, SHM is exhibited when a mass attached to a spring oscillates, and its behaviour is accurately predicted by Hooke's law.

Effective Spring Constant

The effective spring constant is a measure of a spring's stiffness, indicating the ratio of the restoring force to the displacement. When a spring is cut into shorter sections, each section acts as a stiffer spring; specifically, the spring constant is inversely proportional to the length of the spring segment.

Frequency Dependence on Mass and Spring Constant

The oscillation frequency of a mass-spring system is determined by the relationship f = (1/2?)?(k/m), where k represents the spring constant and m represents the mass. This formula demonstrates how the frequency increases with a stiffer spring (larger k) and decreases with a greater mass, illustrating the balance between mass and spring stiffness in dynamic systems.

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