A particle executing S.H.M. has an amplitude A and periodic time T. The minimum time required by

step 1 Hello and welcome to this video solution of numerate. So here we are given that a particle execu

A particle executing S.H.M. has an amplitude A and periodic...

A particle executing S.H.M. has an amplitude $\mathrm{A}$ and periodic time $\mathrm{T}$. The minimum time required by the particle to get displaced by $(\mathrm{A} / \sqrt{2})$ from its equilibrium position is $\ldots \ldots \ldots \mathrm{s}$. (A) $\mathrm{T}$ (B) $\mathrm{T} / 4$ (C) $\mathrm{T} / 8$ (D) $\mathrm{T} / 16$

A particle executing S.H.M. has an amplitude $\mathrm{A}$ and periodic time $\mathrm{T}$. The minimum time required by the particle to get displaced by $(\mathrm{A} / \sqrt{2})$ from its equilibrium position is $\ldots \ldots \ldots \mathrm{s}$.
(A) $\mathrm{T}$
(B) $\mathrm{T} / 4$
(C) $\mathrm{T} / 8$
(D) $\mathrm{T} / 16$

Key Concepts

Time Displacement Calculation

In SHM, calculating the time required for the particle to reach a specific displacement involves solving the displacement equation A cos(?t + ?) for t. By setting the displacement to a fraction of the amplitude and using inverse trigonometric functions, one can determine the corresponding time; this process is key for analyzing the temporal aspects of oscillatory motion.

Phase Angle

The phase angle in SHM indicates the initial state of the oscillatory motion and determines the position and velocity of the particle at any given time. It is used to relate the displacement function, typically written as A cos(?t + ?), to the observed behavior of the motion.

Period and Angular Frequency

The period (T) is the time taken to complete one full cycle of the motion, and it is related to the angular frequency (?) by the relation ? = 2?/T. Understanding this relationship is crucial for relating time-dependent variables such as displacement and phase in SHM.

Amplitude

Amplitude in SHM is the maximum displacement of the oscillating particle from the equilibrium position. It is a measure of the energy contained in the oscillation and remains constant for ideal, undamped motion.

Simple Harmonic Motion (SHM)

SHM is a type of periodic oscillatory motion where the restoring force acting on the particle is directly proportional to its displacement from the equilibrium position and directed towards it. This fundamental concept serves as a basis for understanding many physical systems, including springs and pendulums, and is characterized by sinusoidal motion.

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