Question

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 \(2 \pi / \omega\)

Short answer

The given function \(\sin^2(\omega t)\) represents a periodic motion (not a Simple Harmonic Motion) with a period of \(2 \pi / \omega\). Therefore, the correct answer is (D) A periodic motion with period \(2 \pi / \omega\).

Step-by-step solution

Recognize the waveform

The given function is \(\sin^2(\omega t)\), which is a trigonometric function squared. It represents a non-negative, periodic motion. However, it is not a Simple Harmonic Motion (SHM) as its shape is not sinusoidal. So options (A) and (B) can be discarded.

Evaluate the period of the given function

To find the period of the motion, we need to find the value of \(t\) for which the function repeats its cycle. Let's look for the period of the motion given by \(\sin^2(\omega t)\): We know that the sine function has a period of \(2\pi\), which means: \[\sin(\omega t) = \sin(\omega t + 2 \pi n)\] where \(n\) is an integer. Simply square both sides to find the period of the given function: \[\sin^2(\omega t) = \sin^2(\omega t + 2 \pi n)\] Thus, \(\sin^2(\omega t)\) has the same period as the sine function, which is \(2 \pi\). To find the period concerning \(t\), we need to divide by the coefficient of \(t\) in the argument, which is \(\omega\): Period of \(\sin^2(\omega t) = \dfrac{2 \pi}{\omega}\)

Conclusion

Now we know that the given function \(\sin^2(\omega t)\) describes a periodic motion (not an SHM) with period \(2 \pi / \omega\). So the correct answer is: (D) A periodic motion with period \(2 \pi / \omega\).

Key concepts

Simple Harmonic Motion

Simple Harmonic Motion (SHM) is a type of periodic motion where an object moves back and forth over the same path. This kind of motion is very predictable and characterized by its sinusoidal waveforms. The most famous examples of SHM include the oscillations of a pendulum or the vibrations of a spring-mass system. What makes SHM unique is that the restoring force is directly proportional to the displacement from the equilibrium position, described by the equation: \[ F = -kx \]where

• \( F \) is the restoring force,
• \( k \) is the positive constant known as the spring constant, and
• \( x \) is the displacement.

SHM has a distinct sinusoidal pattern, normally represented by sine or cosine functions, such as \( x(t) = A \sin(\omega t + \phi) \). Here, the graph is continuous and symmetrical, making it ideal for illustrating perfect periodic cycles. Understanding SHM is useful for mastering other forms of oscillations.

Sine Function

The sine function, denoted as \( \sin(x) \), is a fundamental building block in trigonometry and periodic motion analysis. Essentially, it's used to model oscillatory behavior. The function maps the angle (in radians) to the ratio of lengths within a right triangle. Its graph is a perfect wave, repeating every \( 2\pi \) units along the x-axis.
Key properties of the sine function include:

• Amplitude: The height from the center line to the peak, usually 1 in \( \sin(x) \).
• Period: The distance (or time) it takes for the wave to complete one full cycle, which is \( 2\pi \) for sine.
• Frequency: The number of cycles the wave completes in a unit interval, and is the reciprocal of the period.

The sine function is widely applicable, from describing simple waves in physics to solving equations in engineering. Understanding this function is crucial when analyzing signals, as seen in sound waves or alternating electrical currents.

Periodicity

Periodicity refers to the characteristic of any function or motion that repeats at regular intervals. In mathematics and physics, periodicity is an essential concept for understanding how systems behave over time. A function is considered periodic if there is a positive length \( T \) such that: \[ f(x) = f(x + T) \] for all values of\( x \). This means the function repeats its values at intervals of \( T \), known as the period.
For instance, the simple sine function \( \sin(x) \) is periodic with a period of \( 2\pi \). This periodicity is pivotal in analyzing how waves behave, determining how signals cycle, and predicting future behavior in periodic systems. For motion described by \( \sin^2(\omega t) \), its period would be found by scaling according to \( \omega \), leading us to conclude the period is \( \frac{2\pi}{\omega} \). Understanding periodicity enables the prediction and analysis of repeated patterns in nature and industry alike.

Trigonometric Functions

Trigonometric functions are mathematical functions that relate angles to ratios of triangle sides. These are essential in studying phenomena involving cycles and waves because they are inherently periodic. The most common trigonometric functions are sine \( \sin \), cosine \( \cos \), and tangent \( \tan \), each with unique properties and applications.

• Sine and Cosine: Both functions repeat every \( 2\pi \) radians, ideal for modeling oscillatory phenomena.
• Tangent: This function has a different periodicity of \( \pi \) but is crucial for defining angles and slopes.

These functions help express relationships in systems with rotational dynamics, cyclical events, or wave mechanics. In physics, they are key in the analysis of vectors, forces, and motions. For the function \( \sin^2(\omega t) \), squaring a sine function results in a different pattern and periodicity of \( \frac{2\pi}{\omega} \), showcasing how alterations to trigonometric functions can yield significant changes in behavior and applications. Mastery of trigonometric functions allows better understanding and prediction of real-world cyclic patterns and complex systems.