A particle of mass m is attached to three identical springs...

A particle of mass $m$ is attached to three identical springs $A, B$ and $C$ each of force constant $k$ as shown in figure. If the particle of mass $m$ is pushed slightly against the spring $A$ and released, then the time period of oscillation is (a) $2 \pi \sqrt{\frac{2 m}{k}}$ (b) $2 \pi \sqrt{\frac{m}{2 k}}$ (c) $2 \pi \sqrt{\frac{m}{k}}$ (d) $2 \pi \sqrt{\frac{m}{3 k}}$

A particle of mass $m$ is attached to three identical springs $A, B$ and $C$ each of force constant $k$ as shown in figure. If the particle of mass $m$ is pushed slightly against the spring $A$ and released, then the time period of oscillation is
(a) $2 \pi \sqrt{\frac{2 m}{k}}$
(b) $2 \pi \sqrt{\frac{m}{2 k}}$
(c) $2 \pi \sqrt{\frac{m}{k}}$
(d) $2 \pi \sqrt{\frac{m}{3 k}}$

Key Concepts

Simple Harmonic Motion

Simple harmonic motion (SHM) is a type of periodic oscillatory motion where the restoring force is directly proportional to the displacement, but in the opposite direction. This concept forms the basis for understanding oscillations in systems like the mass attached to springs, and it provides the framework for analyzing the motion when the particle is displaced from its equilibrium position.

Hooke's Law

Hooke's Law states that the force exerted by a spring is proportional to the displacement from its rest position, with the proportionality constant being the spring constant (k). This foundational principle is used to determine the restoring force acting on the mass in any spring-mass system, making it essential for analyzing oscillatory behavior.

Effective Spring Constant in Multi-Spring Systems

When multiple springs are connected to a mass, their individual forces combine to yield an effective spring constant in the direction of displacement. Understanding how to calculate this effective stiffness is key in solving problems where several springs are attached, as it determines the overall restoring force acting on the system.

Oscillation Period Formula

For a mass-spring system exhibiting simple harmonic motion, the time period of oscillation is given by T = 2??(m/k_eff), where m is the mass and k_eff is the effective spring constant. This formula is fundamental in determining how the mass and the system’s stiffness together affect the speed of the oscillations.

Small Oscillation Approximation

The small oscillation approximation assumes that the displacement from equilibrium is small enough that the restoring forces remain linear with respect to displacement. This approximation simplifies the analysis by ensuring that the motion can be accurately described using the principles of simple harmonic motion and linear superposition of forces.

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