Question

If the equation for a particle performing S.H.M. is given by \(\mathrm{y}=\sin 2 \mathrm{t}+\sqrt{3} \cos 2 \mathrm{t}\), its periodic time will be \(\ldots \ldots .\) s. (A) 21 (B) \(\pi\) (C) \(2 \pi\) (D) \(4 \pi\).

Short answer

The periodic time of the given S.H.M. is \(2\pi\) seconds. The correct answer is (C) \(2\pi\).

Step-by-step solution

Determine the period of each term

In the given equation, y(t) = sin(2t) + √3 * cos(2t). The two individual trigonometric functions are sin(2t) and cos(2t). Both of these functions have a period of \(2\pi\), because any multiple of their argument will give the same value and they only repeat at those values. For example: sin(x) = sin(x+2π) cos(x) = cos(x+2π)

Determine the period of their sum

To find the period of the sum, we're looking for the smallest positive value of t for which both sin(2t) and cos(2t) repeat. Since both functions have a period of \(2\pi\), their sum will also have the same period. So, the period of the given equation y(t) = sin(2t) + √3 * cos(2t) is \(2\pi\).

Convert the period to periodic time

The periodic time is the time taken to complete one oscillation or cycle. It is represented by the variable T. As per the given equation, when the period of the S.H.M. is \(2\pi\), the periodic time T is also \(2\pi\). So the periodic time of the given S.H.M. is \(2\pi\) seconds. The correct answer is (C) \(2\pi\).

Key concepts

Trigonometric Functions in Simple Harmonic Motion

Trigonometric functions, such as sine and cosine, play a crucial role in describing simple harmonic motion (S.H.M.). They are periodic functions often used to model oscillations, like those found in pendulums or springs.

In the context of S.H.M., each function represents an oscillating position, velocity, or other aspect of motion over time. Consider the function in the problem, \[ y(t) = \sin(2t) + \sqrt{3} \cos(2t) \]This equation combines two basic trigonometric functions: sine, \(\sin(2t)\), and cosine, \(\cos(2t)\).

• Sine \(\sin(2t)\): This component represents oscillations starting from zero, moving upwards initially.
• Cosine \(\cos(2t)\): This component starts the oscillation at its maximum value.

These individual functions combine to form a new oscillatory pattern with its unique characteristics. Understanding these functions helps in predicting how the system behaves over time.

Periodicity in Trigonometric Functions

Periodicity is a key feature of trigonometric functions and is essential to understanding simple harmonic motion. This means that the values of sine and cosine functions repeat at regular intervals.

For standard sine and cosine functions, the period is \(2\pi\). This means:

• \(\sin(x) = \sin(x + 2\pi n)\)
• \(\cos(x) = \cos(x + 2\pi n)\)

where \(n\) is an integer. In S.H.M., applying transformations like \(\sin(2t)\) or \(\cos(2t)\) compresses the period by a factor of the coefficient, which is 2 in this case.

Thus, both \(\sin(2t)\) and \(\cos(2t)\) have a period of \(\pi\). However, when they are combined in the equation given in the exercise, the resulting function retains the period of \(2\pi\). This is because both components have the same period and start anew after every \(2\pi\).
Understanding periodicity helps identify the repeating nature of cyclical phenomena like sound waves, tides, and even day-night cycles.

Oscillation Period in Simple Harmonic Motion

The oscillation period, often represented by \(T\), is a fundamental property of simple harmonic motion. It defines the duration of time needed to complete one full cycle of movement.

In the given exercise, the function representing the particle's motion is \[ y(t) = \sin(2t) + \sqrt{3} \cos(2t) \]The goal is to determine how long it takes for this scenario to return to its initial state, thus finding its oscillation period.

Since both sinusoidal components \(\sin(2t)\) and \(\cos(2t)\) have a synchronized period of \(2\pi\), the oscillation period of the entire function is also \(2\pi\) seconds.

An understanding of oscillation periods is vital, as it provides insights into the timing and rhythm of natural and engineered systems, like clocks, circuits, and musical instruments.