A particle moves such that its acceleration a is given by...

A particle moves such that its acceleration $a$ is given by $a=-b x$. Where $x$ is the displacement from equilibrium position and $\mathrm{b}$ is a constant. The period of oscillation is [NCERT 1984; CPMT 1991; MP PMT 1994; MNR 1995] (a) $2 \pi \sqrt{b}$ (b) $\frac{2 \pi}{\sqrt{b}}$ (c) $\frac{2 \pi}{b}$ (d) $2 \sqrt{\frac{\pi}{b}}$

A particle moves such that its acceleration $a$ is given by $a=-b x$. Where $x$ is the displacement from equilibrium position and $\mathrm{b}$ is a constant. The period of oscillation is [NCERT 1984; CPMT 1991; MP PMT 1994; MNR 1995]
(a) $2 \pi \sqrt{b}$
(b) $\frac{2 \pi}{\sqrt{b}}$
(c) $\frac{2 \pi}{b}$
(d) $2 \sqrt{\frac{\pi}{b}}$

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Key Concepts

Simple Harmonic Motion

Simple Harmonic Motion (SHM) describes a type of oscillatory motion where the restoring force is directly proportional to the negative of the displacement from the equilibrium position. This results in sinusoidal motion and is typically represented by the equation a = -?² x, highlighting that the acceleration is always directed opposite to the displacement.

Angular Frequency

Angular frequency, denoted by ?, quantifies how rapidly an object oscillates in SHM. It is derived from the proportionality constant in the acceleration equation (a = -?² x) and is a crucial factor in determining the speed of oscillations within the system.

Oscillation Period

The oscillation period is the time taken for one complete cycle of motion in SHM. It is inversely related to the angular frequency, given by the formula T = 2?/?, which shows that as the angular frequency increases, the period of oscillation decreases. This concept is key to understanding the timing and behavior of oscillatory systems.

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