What will be the displacement of a particle in SHM when its...

What will be the displacement of a particle in SHM when its velocity is half the maximum velocity $(A=$ amplitude)
(A) $\frac{3}{\sqrt{2}} A$
(B) $\sqrt{2} A$
(C) $\frac{3}{4} A$
(D) $\frac{\sqrt{3}}{2} A$

Key Concepts

Velocity-Displacement Relationship in SHM

In SHM, the instantaneous velocity and displacement of the particle are connected through the relationship derived from energy conservation, often expressed as v^2 = ?^2(A^2 – x^2), where v is the instantaneous velocity, ? is the angular frequency, A is the amplitude, and x is the displacement. This relationship allows one to determine the displacement when given a particular value of the velocity relative to its maximum value.

Maximum Velocity in SHM

The maximum velocity in SHM occurs when the particle passes through the equilibrium position and is given by v_max = ?A. This concept is useful for comparing the instantaneous velocity at any displacement to its maximum value, providing insights into the energy distribution between kinetic and potential forms within the oscillating system.

Simple Harmonic Motion (SHM)

SHM is a type of periodic motion where the restoring force is directly proportional to the negative of the displacement from an equilibrium position. This results in oscillatory motion that can be described by sinusoidal functions. SHM is fundamental in understanding various physical systems, from mass-spring systems to pendulums, under small displacements.

Amplitude

Amplitude is the maximum displacement of a particle from its equilibrium position in SHM. It represents the extent of the oscillation and is a critical parameter in defining the energy and dynamics of the system. The amplitude remains constant in ideal SHM where there is no damping.

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