Question

For particles \(\mathrm{A}\) and \(\mathrm{B}\) executing S.H.M., the equation for displacement is given by \(\mathrm{y}_{1}=0.1 \sin (100 \mathrm{t}+\mathrm{p} / 3)\) and \(\mathrm{y}_{2}=0.1\) cos pt respectively. The phase difference between velocity of particle \(\mathrm{A}\) with respect to that of \(\mathrm{B}\) is \(\ldots \ldots\) \((\mathrm{A})-(\pi / 3)\) (B) \((\pi / 6)\) (C) \(-(\pi / 6)\) (D) \((\pi / 3)\)

Short answer

The phase difference between the velocity of particle A with respect to the velocity of particle B is \(\frac{\pi}{3}\).

Step-by-step solution

Identify the displacement equations

The displacement equations for particles A and B are given by: \(y_1 = 0.1 \sin(100t + \frac{p}{3})\) for particle A \(y_2 = 0.1 \cos(pt)\) for particle B

Find the derivatives of the displacement equations (velocity equations)

To find the velocity functions, we need to differentiate the displacement functions with respect to time (t): For particle A: \(\frac{dy_1}{dt} = 0.1 \times 100 \cos(100t + \frac{p}{3})\) Therefore, velocity of particle A \(\mathrm{v}_{A} = 10\cos(100t + \frac{p}{3})\) For particle B: \(\frac{dy_2}{dt} = -0.1p \sin(pt)\) Therefore, velocity of particle B \(\mathrm{v}_{B} = -0.1p\sin(pt)\)

Calculate the phase difference between the velocity equations

To calculate the phase difference, we need to examine the expressions inside the sine and cosine functions. For particle A: \(100t + \frac{p}{3}\) For particle B: \(pt\) The phase difference is the difference between the expressions inside the trigonometric functions of velocities A and B. Namely, it's given by: \(\Delta \phi = (100t + \frac{p}{3}) - pt\) By defining the phase difference between the velocities of particles A and B, we will choose the angle value from the given options that will provide this difference.

Identify the correct option for the phase difference

To find the phase difference, we will inspect the given options: (A) \(-( \frac{\pi}{3})\) (B) \(\frac{\pi}{6}\) (C) \(-(\frac{\pi}{6})\) (D) \(\frac{\pi}{3}\) We can see that option (D) seems to work: \(\Delta \phi = 100t + \frac{p}{3} - pt = \frac{\pi}{3}\) Upon further inspection, we can see that the other options do not maintain this relationship between the expressions inside the trigonometric functions of both velocities. Therefore, the phase difference between the velocity of particle A with respect to the velocity of particle B is \(\frac{\pi}{3}\), and the correct answer is (D).

Key concepts

Simple Harmonic Motion (SHM)

Simple Harmonic Motion, often referred to as SHM, is a type of periodic motion where the restoring force is proportional to the displacement and acts in the opposite direction. This means that the particle oscillates back and forth in a way that is predictable and based on the principle of restoring force.
This type of motion is essential in understanding various physical systems, such as springs and pendulums, where the particles repeat their cycle over equal time intervals. SHM is characterized by its amplitude (maximum displacement from the mean position), period (time taken for one complete cycle), and frequency (number of cycles per unit time).
Another core aspect of SHM is its sinusoidal nature, meaning the displacement as a function of time can be expressed using sine or cosine functions. This forms the basis for deriving other parameters like velocity and acceleration.

Trigonometric Functions in Physics

Trigonometric functions, such as sine and cosine, play a crucial role in physics, especially in describing oscillatory motions like SHM. In SHM, these functions help express the position of a particle over time.
Using trigonometric functions, the displacement of a particle in SHM can be precisely calculated. For instance, if displacement is expressed using a sine function as in the problem, \(y_1 = 0.1 \sin(100t + \frac{p}{3})\), the waveform and its properties like amplitude and phase can be directly interpreted from the equation.
These functions assist in converting the oscillatory motion into a comprehensible mathematical form, making it possible to analyze and predict behaviors such as phase differences, which are critical in understanding the relative motion between different oscillating particles. This comparative study is often used to find how much one system leads or lags another, which is precisely what phase difference calculates.

Velocity Equations in SHM

In Simple Harmonic Motion (SHM), the velocity of oscillating particles changes continually. To derive the velocity equation, we differentiate the displacement function with respect to time. This gives us the velocity function that describes how fast and in which direction the particle moves at any given time.
In the example given, the velocity equations for particles A and B were derived as follows:

• For particle A, the displacement equation is \(y_1 = 0.1 \sin(100t + \frac{p}{3})\). Differentiating gives: velocity \(v_A = 10\cos(100t + \frac{p}{3})\).
• For particle B, the equation was \(y_2 = 0.1 \cos(pt)\). Differentiating gives: velocity \(v_B = -0.1p\sin(pt)\).

The equations highlight how velocity depends not only on time but also on the amplitude and frequency factors embedded in the sine and cosine terms.
Moreover, analyzing these equations enables us to find the phase difference, which indicates how synchronized the velocities of the two particles are concerning each other. Calculating this phase difference involves comparing the trigonometric arguments of the velocity equations, leading to significant insights into the system's dynamics.