Question

The equation for displacement of a particle at time \(t\) is given by the equation \(\mathrm{y}=3 \cos 2 \mathrm{t}+4 \sin 2 \mathrm{t}\). The periodic time of oscillation is \(\ldots \ldots \ldots \ldots\) (A) \(2 \mathrm{~s}\) (B) \(\pi \mathrm{s}\) (C) \((\pi / 2) \mathrm{s}\) (D) \(2 \pi \mathrm{s}\)

Short answer

The periodic time of oscillation for the given displacement equation is \(\pi\) seconds.

Step-by-step solution

Identify Angular Frequency

Let's first write the given equation: \[y = 3 \cos(2t) + 4 \sin(2t)\] Notice that in both terms of the equation, the argument of the cosine and sine functions is 2t, which means they oscillate at the same frequency. This is a very crucial observation because it tells us the angular frequency involved in the given equation. The angular frequency is represented by the coefficient of t, which in this case is 2.

Find the Period of Oscillation

Now that we have identified the angular frequency as 2, we can use it to find the period of oscillation. The formula to convert angular frequency (ω) and the period (T) is given by: \[\omega = \frac{2 \pi}{T}\] In this problem, we have \(\omega = 2\). So, we can write the equation as: \[2 = \frac{2 \pi}{T}\]

Solve for T

Now, let's solve for T: \[T = \frac{2 \pi}{2}\] \[T = \pi\] The periodic time of oscillation is \(\pi\) seconds.

Determine the Correct Answer

Now that we have calculated the period to be \(\pi\) seconds, we can compare it to the given options. The correct answer is: (B) \(\pi s\)

Key concepts

Angular Frequency

Angular frequency is a key concept when dealing with oscillatory motion. It tells us how fast an object moves through its cycle in radians per unit of time. In this exercise, the angular frequency is crucial for determining the period of oscillation.
To find the angular frequency, look at the coefficient of time (t) in the trigonometric functions within the displacement equation. Here, the equation is \(y = 3 \cos (2t) + 4 \sin (2t)\). The coefficient of \(t\) in both the cosine and sine functions is 2, indicating that the angular frequency \(\omega\) is 2 radians per second.
Understanding angular frequency helps in predicting how often the oscillation completes one cycle, which is further used to calculate the oscillation period.

Trigonometric Functions

Trigonometric functions such as cosine and sine are fundamental in representing oscillatory motions. These functions describe the smooth, periodic fluctuations that occur in oscillations. In the given equation\(y = 3 \cos (2t) + 4 \sin (2t)\), both are used to model the displacement of a particle over time.
The coefficients (3 and 4 in this case) modify the amplitude of oscillation for each component. The cosine and sine functions here depend on \(2t\), showing the periodic nature at a frequency determined by the angular frequency, which is 2. Trigonometric functions, due to their cyclical nature, are perfect for modeling periodic phenomena such as the motion of pendulums or springs.

Displacement Equation

The displacement equation contains all we need to know about the movement of the particle at any given time \(t\). For the particle described here, the equation is \(y = 3 \cos (2t) + 4 \sin (2t)\). This equation gives us the position of the particle as time progresses.
With both a cosine and a sine term, this equation indicates a combination of two oscillatory motions, each with the same frequency but potentially different phases and amplitudes. By calculating the resultant amplitude and phase, you can determine the full motion of the oscillation. This form of the equation can be rewritten using trigonometric identities to express a single oscillation with a specific amplitude and phase shift, showcasing the versatility of trigonometric functions in describing oscillations.

Oscillation Period

The period of oscillation refers to the time taken for one complete cycle of oscillation. It's an essential characteristic because it defines how frequently an oscillating system repeats its motion.
Here, knowing the angular frequency \(\omega\) allows us to find the period \(T\) using the formula: \[\omega = \frac{2\pi}{T}\]. Since \(\omega = 2\), we rearrange the formula to find \(T\): \[T = \frac{2\pi}{2} = \pi\]. Thus, the oscillation period of the system is \(\pi\) seconds, meaning the pattern repeats itself every \(\pi\) seconds.
Understanding the oscillation period is crucial in applications where timing and synchronization matter, such as in waves and signal processing.