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 motion of the particle is \(\ldots \ldots\) (A) Damped motion (B) Periodic motion (C) Rotational motion (D) S.H.M.

Short answer

The motion of the particle is (B) Periodic motion, as determined by analyzing the equation \(y = 3\cos2t + 4\sin2t\) and rewriting it as \(y(t) = 5\cos(2t-\tan^{-1}(\frac{4}{3}))\), which represents a periodic cosine function.

Step-by-step solution

Rewrite the displacement equation in terms of sine and cosine functions

Let's rewrite the equation as: \(y(t) = A\cos(2t - \phi)\) Where A is the amplitude and \(\phi\) is the phase angle. To do this, we need to find the values of A and \(\phi\).

Calculate A and \(\phi\)

From the given equation \(y = 3\cos2t + 4\sin2t\), we have a set of simultaneous equations: \(A\cos\phi = 3\) \(A\sin\phi = 4\) To find the amplitude, A, take the square root of the sum of the squares: \(A = \sqrt{(3^2+4^2)} = 5\) To find the phase angle \(\phi\), take the inverse tangent of the ratio: \(\phi = \tan^{-1}(\frac{4}{3})\)

Rewrite the displacement equation with A and \(\phi\)

Now, we can rewrite the displacement equation with the calculated values of A and \(\phi\): \(y(t) = 5\cos(2t-\tan^{-1}(\frac{4}{3}))\)

Identify the motion type

The displacement equation represents a cosine function, which is a periodic function. Therefore, the motion of the particle is periodic. The correct answer is (B) Periodic motion.

Key concepts

Displacement Equation

The displacement equation for a particle's motion over time helps track the particle's position based on specific mathematical functions. Given an equation like \( y(t) = 3\cos(2t) + 4\sin(2t) \), we analyze it to reveal the characteristics of the motion. When breaking down the equation, you can express it as a single cosine function, \( y(t) = A\cos(2t - \phi) \), where \( A \) represents the amplitude and \( \phi \) the phase angle. These components are essential to understand the motion's nature without having to consider two separate sine and cosine terms.
The combined amplitude \( A \) scales the function's height, and the phase angle \( \phi \) shifts the graph along the time axis. This expression helps identify the motion's regularity and periodicity, crucial for understanding physical systems.

Amplitude and Phase Angle

Amplitude and phase angle are vital elements of the displacement equation, giving deep insights into the behavior of periodic motion.

• **Amplitude (A)** represents the maximum extent of a particle's movement from its mean position. It is calculated using the Pythagorean expression: \( A = \sqrt{3^2 + 4^2} = 5 \). This value quantifies how far the particle deviates from equilibrium.
• **Phase Angle (\(\phi\))** determines the initial state of the motion. Its calculation, \( \phi = \tan^{-1}\left(\frac{4}{3}\right) \), informs us about how the wave function is shifted in time. The arctangent function \( \tan^{-1} \) resolves the ambiguity in direction, ensuring that the phase angle is accurately positioned within the coordinate plane.

Together, amplitude and phase angle define a unique waveform, allowing us to predict systematic oscillations over time. By incorporating them into the displacement equation, understanding shifts and oscillation magnitudes in waves becomes simpler.

Trigonometric Functions

Trigonometric functions like sine and cosine are key to expressing periodic motions. They provide a mathematical framework for modeling oscillations and waves.

• **Cosine Function**: Regularly used to model repetition in cycles, the cosine function indicates a wave with consistent periods and amplitudes. It helps denote the position of the particle along its path.
• **Sine Function**: Similar to the cosine, the sine function indicates cyclical movement but is typically shifted by a phase. In the displacement equation, it complements the cosine to cover rotational and shifted waveforms.

These functions are periodic, repeating every full cycle (usually denoted by \(2\pi\) for radians). Analyzing their role in the displacement equation \( y(t) = 3\cos(2t) + 4\sin(2t) \) helps determine wave properties like frequency, phase shift, and amplitude.
Ultimately, understanding trigonometric functions allows for the prediction of future positions of particles in periodic motion. They are fundamental tools in fields needing wave and oscillation analysis.