A particle moves on the x-axis as per the equation x=x0 sin^2 ωt . The motion is simple harmonic

step 1 Dear students, in problem 148 of chapter 9, we have asked that a particle moves on x -axis as a

A particle moves on the x-axis as per the equation x=x0...

A particle moves on the $x$-axis as per the equation $x=x_{0} \sin ^{2} \omega t .$ The motion is simple harmonic
(A) With amplitude $x_{0}$
(B) With amplitude $2 x_{0}$
(C) With time period $\frac{2 \pi}{\omega}$
(D) With time period $\frac{\pi}{\omega}$

Key Concepts

Simple Harmonic Motion

Simple harmonic motion (SHM) is a type of periodic motion in which the restoring force acting on a particle is directly proportional to its displacement from a fixed equilibrium position and is directed towards that equilibrium. This characteristic leads to sinusoidal oscillations that repeat in time, and the mathematical description usually involves sine or cosine functions.

Trigonometric Transformation

Often, trigonometric identities are used to rewrite expressions into a form that reveals their oscillatory behavior. For instance, the identity sin²? = 1/2 (1 - cos2?) allows one to express a squared sine function as a cosine function with a doubled argument, which can then be recognized as a harmonic oscillation with a modified angular frequency.

Equilibrium Position and Amplitude

When analyzing oscillatory motion, it is important to distinguish between the maximum value of the displacement from the original reference and the amplitude of the oscillation about the equilibrium position. In cases where the motion is shifted from the origin, the amplitude is defined as half the difference between the maximum and minimum displacements, not simply the coefficient in the original function.

Time Period Determination

The time period of a harmonic oscillator is determined by the angular frequency of its sinusoidal component. If the function of time includes a sine or cosine term with an argument that is a multiple of the original frequency, the period is adjusted inversely with that multiplier. For example, a cosine function with an argument 2? implies a period of ?/?, since the period T is given by 2? divided by the effective angular frequency.

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