The acceleration d^2 x / d t^2 of a particle varies with...

The acceleration $d^{2} x / d t^{2}$ of a particle varies with displacement $x$ as $\frac{d^{2} x}{d t^{2}}=-k x$
where $k$ is a constant of the motion. The time period $T$ of the motion is equal to
(a) $2 \pi k$
(b) $2 \pi \sqrt{k}$
(c) $2 \pi / \sqrt{k}$
(d) $2 \pi / k$

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Key Concepts

Time Period of Oscillatory Motion

The time period is the duration required to complete one full cycle of oscillation. It is mathematically related to the angular frequency by the formula T = 2?/?. When applied to an SHM system with ? = ?k, the period is T = 2?/?k, showing how the oscillation period depends on the system’s intrinsic properties.

Angular Frequency

Angular frequency represents the rate at which the phase of an oscillation changes and is fundamental in the analysis of periodic motions. For a system where d²x/dt² = -kx, the angular frequency is given by ? = ?k, linking the system's dynamics to its oscillatory behavior.

Simple Harmonic Motion (SHM)

Simple harmonic motion is a type of periodic motion where the restoring force is directly proportional to the negative of the displacement from equilibrium. This leads to oscillations that are sinusoidal in nature, meaning the motion can be described with sine or cosine functions over time.

Differential Equations in Oscillatory Systems

In the study of mechanical oscillations, differential equations are used to model how acceleration, velocity, and displacement are related. For SHM, the differential equation d²x/dt² = -kx characterizes the system, with the constant k indicating the strength of the restoring force.

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