The function sin^2(ωt) represents (A) A SHM with periodic time π/ ω(B) A SHM with a periodic tim

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The function sin^2(ωt) represents (A) A SHM with periodic...

The function $\sin ^{2}(\omega t)$ represents
(A) A SHM with periodic time $\pi / \omega$
(B) A SHM with a periodic time $2 \pi / \omega$
(C) A periodic motion with periodic time $\pi / \omega$
(D) A periodic motion with period $2 \pi / \omega$

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Key Concepts

Simple Harmonic Motion

Simple harmonic motion (SHM) is a type of periodic motion characterized by smooth, sinusoidal oscillations about an equilibrium position, where the restoring force is directly proportional to the displacement. While all SHM is periodic, not all periodic functions, such as sin²(?t), exhibit the properties that define SHM.

Double-Angle Identity

The double-angle identity for sine squared, specifically sin²(?) = (1 - cos(2?)) / 2, is a trigonometric identity that expresses the square of sine in terms of cosine with twice the frequency. This identity is crucial for analyzing the periodic behavior of sin²(?t) by revealing that its period is influenced by the double-frequency cosine term.

Periodic Motion

Periodic motion refers to any motion or function that repeats itself at regular time intervals, known as the period. This concept highlights that the behavior of the motion recurs in a predictable pattern after a fixed duration.

Period of a Trigonometric Function

The period of a trigonometric function is the smallest positive interval over which the function completes one full cycle and then repeats identically. The period is essential in understanding how quickly the function oscillates and is determined by the function's argument.

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