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 frequency of the particle is \(\ldots \ldots \mathrm{s}^{-1}\). (A) \((1 / \pi)\) (B) \(\pi\) (C) \((1 / 2 \pi)\) (D) \((\pi / 2)\)

Short answer

The frequency of the particle is \(\boxed{(A) \, \frac{1}{\pi} \, \text{s}^{-1}}\).

Step-by-step solution

Rewrite the displacement equation using the cosine identity

Let us use the identity \(\cos A \cos B + \sin A \sin B = \cos(A - B)\). Let \(A=2t\) and \(B=\alpha\). We want to find \(\alpha\) such that \(3\cos 2t + 4\sin 2t = R\cos(2t-\alpha)\), where \(R\geq 0\) The angle \(\alpha\) is given by: \(\tan \alpha = \frac{\text{coefficient of }\sin2t}{\text{coefficient of }\cos2t}\), which in our case means \(\tan \alpha = \frac{4}{3}\). Now we can find R: \(R = \sqrt{(3\cos 2t)^2 + (4\sin 2t)^2} = \sqrt{9\cos^2 2t + 16\sin^2 2t} = 5\) Thus, the displacement equation can be rewritten as: \(y = 5\cos(2t - \alpha)\)

Compare the simplified equation to the general harmonic equation

Now, let's compare our simplified equation to the general harmonic equation \(y = A \cos(2 \pi ft + \phi)\): \(5\cos(2t - \alpha) \, = A \cos(2 \pi ft + \phi)\)

Identify the frequency

Since our equation is \(y = 5\cos(2t - \alpha)\), the coefficient of \(t\) is 2. In the general harmonic equation, this is equal to \(2\pi f\): \(2 = 2 \pi f\) Solving for \(f\), we get: \(f = \frac{1}{\pi}\) Looking at the provided options, the frequency of the particle is \(\boxed{(A) \, \frac{1}{\pi} \, \text{s}^{-1}}\).

Key concepts

Frequency

Frequency refers to the number of complete cycles or oscillations that a wave undergoes in a specific amount of time, typically a second. It is a key characteristic of oscillating systems, such as waves and vibrations, and is measured in Hertz (Hz), or cycles per second. In the context of our exercise, frequency determines how fast the harmonic motion is occurring. To calculate frequency from the given displacement equation, we identify the term with time variable, which in this case is multiplied by 2. In mathematical terms, this corresponds to the equation structure \(y = A \cos(2 \pi ft + \phi)\). By comparing the equations, we understand that the frequency \(f\) is directly linked to the coefficient of \(t\). Knowing that 2 is equal to \(2 \pi f\), we solve for \(f\), yielding \(f = \frac{1}{\pi}\) Hz.

Displacement Equation

The displacement equation describes how the position of a particle changes over time in harmonic motion. It typically involves trigonometric functions like sine and cosine, which relate to periodic movements such as waves.In this exercise, the displacement equation is presented as \(y = 3 \cos 2t + 4 \sin 2t\). This equation builds on the understanding that both the cosine and sine functions can describe oscillatory motion. Here, the \(3\) and \(4\) are amplitudes for the cosine and sine components, showing how far the particle moves from its mean position.By transforming these traditional trigonometric components into a simplified form \(y = R \cos(2t - \alpha)\), we make it easier to analyze by consolidating both harmonics into a single term. This transformation relies on finding the resultant amplitude \(R\) and phase \(\alpha\), which then closely resembles the format of the standard harmonic equation.

Trigonometric Identity

Trigonometric identities are fundamental tools in understanding wave equations and their transformations. They help simplify complex trigonometric expressions by using known equalities between different functions.In this exercise, we utilize the identity: \(\cos A \cos B + \sin A \sin B = \cos(A - B)\). Applying this allows us to rewrite a sum of sine and cosine terms as a single cosine or sine term with an adjusted angle. Specifically, the displacement equation \(y = 3 \cos 2t + 4 \sin 2t\) is rewritten into \(y = R \cos(2t - \alpha)\) based on this identity.This simplification is essential to convert the given equation into a form that matches the general harmonic equation, facilitating the determination of key elements like frequency and phase shift.

Harmonic Equation

The harmonic equation is a standard representation of oscillating systems, typically taking the form \(y = A \cos(2 \pi ft + \phi)\). Here, \(A\) is the amplitude, \(f\) the frequency, and \(\phi\) the phase angle.In this exercise, we compare the simplified displacement equation \(y = 5 \cos(2t - \alpha)\) to the harmonic equation. This comparison reveals important characteristics:

• The amplitude \(A\) is 5, showing the maximum displacement.
• The term \(2t\) suggests a frequency format where \(2 = 2 \pi f\), leading to \(f = \frac{1}{\pi}\).
• The phase shift \(\alpha\) determines the initial positional offset in its cycle.

Understanding this format allows for a straightforward interpretation of the particle's motion, making elements like frequency easy to extract from the equation directly.