For a particle executing S . H M ., when the potential energy of the oscillator becomes 1 / 8 th

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For a particle executing S . H M ., when the potential...

For a particle executing $\mathrm{S} . \mathrm{H} \mathrm{M} .$, when the potential energy of the oscillator becomes $1 / 8$ the maximum potential energy, the displacement of the oscillator in terms of amplitude A will be........... (A) $(\mathrm{A} / \sqrt{2})$ (B) $\{\mathrm{A} /(2 \sqrt{2})\}$ (C) $(\mathrm{A} / 2)$ (D) $\{\mathrm{A} /(3 \sqrt{2})\}$

For a particle executing $\mathrm{S} . \mathrm{H} \mathrm{M} .$, when the potential energy of the oscillator becomes $1 / 8$ the maximum potential energy, the displacement of the oscillator in terms of amplitude A will be...........
(A) $(\mathrm{A} / \sqrt{2})$
(B) $\{\mathrm{A} /(2 \sqrt{2})\}$
(C) $(\mathrm{A} / 2)$
(D) $\{\mathrm{A} /(3 \sqrt{2})\}$

Key Concepts

Energy-Displacement Relationship

Due to the quadratic dependence of potential energy on the displacement in SHM, any fraction of the maximum potential energy corresponds to a square of a fractional amplitude. By equating a given portion of the maximum potential energy to (1/2)kx², one can solve for x in terms of A, thus linking the energy fraction directly to the corresponding displacement.

Maximum Potential Energy and Amplitude

The maximum potential energy in a simple harmonic oscillator occurs when the displacement is at its maximum value, the amplitude A. At this point, the potential energy reaches (1/2)kA². This maximum energy serves as a benchmark for comparing the energy at other positions within the oscillation.

Simple Harmonic Motion

Simple harmonic motion (SHM) is a type of periodic motion where the restoring force acting on a particle is directly proportional to its displacement from its equilibrium position and is directed toward that equilibrium. This kind of motion is typically described using sine or cosine functions and is characterized by properties such as amplitude, period, frequency, and phase.

Potential Energy in SHM

In SHM, the potential energy stored in the system is given by the expression (1/2)kx², where k is the force constant and x is the displacement from equilibrium. This quadratic relation shows that the potential energy depends on the square of the displacement, meaning that it increases rapidly as the particle moves away from the equilibrium position.

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