A particle is executing SHM of periodic time T. The time taken by a particle in moving from mean

Step 1: We know that the displacement in simple harmonic motion is given by $x = A sin(omega t)$

A particle is executing SHM of periodic time T. The time...

A particle is executing SHM of periodic time $\mathrm{T}$. The time taken by a particle in moving from mean position to half the maximum displacement is $\left(\sin 30^{\circ}=\right.$ $0.5):$
(a) $\mathrm{T} / 2$
(b) $\mathrm{T} / 4$
(c) $\mathrm{T} / 8$
(d) $\mathrm{T} / 12$

Key Concepts

Simple Harmonic Motion (SHM)

SHM is a type of periodic motion where the restoring force is directly proportional to the displacement from the equilibrium position and acts in the opposite direction. This leads to sinusoidal oscillations in time, which means that the displacement, velocity, and acceleration of the particle vary sinusoidally with time. Understanding SHM is crucial for analyzing systems that exhibit regular periodic behavior, such as springs and pendulums under small oscillations.

Angular Frequency and Period

The angular frequency (?) characterizes how rapidly the oscillation occurs and is directly related to the period (T) of the motion by the relation ? = 2?/T. This connection allows one to convert between the time domain description of the motion and its representation in terms of phase angles, which is essential when analyzing the timing of motion within a cycle.

Phase Angle and Displacement Relation

In SHM, the displacement of a particle can be expressed as x = A sin(?), where A is the amplitude and ? is the phase angle related to time by ? = ?t. This relationship shows that for any given displacement as a fraction of the amplitude, there exists a corresponding phase angle. Using inverse trigonometric functions, one can determine the time required for the particle to reach a specific displacement from the mean position.

Inverse Trigonometric Analysis in SHM Timing

When asked about the time it takes for a particle to move from the mean position to a specific fraction of the maximum displacement, the inverse sine function is used to determine the corresponding phase angle. This phase angle, when divided by the angular frequency, yields the time elapsed. Understanding this application of inverse trigonometric functions is key to solving time interval problems in SHM.

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