A particle is performing simple harmonic motion along x-axis...

A particle is performing simple harmonic motion along $x$-axis with amplitude $4 \mathrm{~cm}$ and time period $1.2$ s. The minimum time taken by the particle to move from $x=+2$ to $x=4 \mathrm{~cm}$ and back again is given by (a) $0.4 \mathrm{~s}$
(b) $0.35$
(c) $0.25$
(d) $0.65$

Key Concepts

Angular Frequency and Time Period

Angular frequency is fundamentally linked to the time period of the oscillation. It quantifies how fast the phase of the oscillatory motion is changing and is calculated as ? = 2?/T, where T is the time period. This connection allows one to translate differences in phase angles into differences in time, which is crucial for timing the motion between specific displacements.

Phase Angle and Time Calculation

In the context of SHM, the phase angle is the argument of the trigonometric function used to describe the displacement. By relating the displacement to a particular cosine (or sine) value, one can use the inverse cosine (or sine) function to determine the corresponding phase angle. The elapsed time between positions is then determined by the difference in phase angles divided by the angular frequency. This concept is key when calculating the time interval for the particle to move from one displacement to another and back.

Simple Harmonic Motion

Simple Harmonic Motion (SHM) is a type of periodic oscillatory motion where the restoring force is directly proportional to the negative of the displacement from an equilibrium position. This concept forms the basis for understanding many physical systems and their time-dependent behavior, including the prediction of positions and velocities at different times.

Amplitude and Displacement

The amplitude in SHM is the maximum displacement from the equilibrium position and represents the extent of the motion. The displacement at any given time can be expressed using trigonometric functions. Understanding how the amplitude sets the bounds for the motion is essential for determining when and where the oscillating particle will be at any point in time.

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