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Chapter 10: Problem 1354

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

Expert verified
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

01

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
02

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 \(\mathrm{v}_{B} = -0.1p\sin(pt)\)
03

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.
04

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).

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Most popular questions from this chapter

A small spherical steel ball is placed at a distance slightly away from the center of a concave mirror having radius of curvature \(250 \mathrm{~cm}\). If the ball is released, it will now move on the curved surface. What will be the periodic time of this motion? Ignore frictional force and take $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$. (A) \((\pi / 4) \mathrm{s}\) (B) \(\pi \mathrm{s}\) (C) \((\pi / 2) \mathrm{s}\) (D) \(2 \pi \mathrm{s}\)

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For the following questions, statement as well as the reason(s) are given. Each questions has four options. Select the correct option. (a) Statement \(-1\) is true, statement \(-2\) is true; statement \(-2\) is the correct explanation of statement \(-1\). (b) Statement \(-1\) is true, statement \(-2\) is true but statement \(-2\) is not the correct explanation of statement \(-1\) (c) Statement \(-1\) is true, statement \(-2\) is false (d) Statement \(-1\) is false, statement \(-2\) is true (A) a (B) \(b\) (C) \(\mathrm{c}\) (D) d Statement \(-1:\) For a particle executing S.H.M. with an amplitude of $0.01 \mathrm{~m}\( and frequency \)30 \mathrm{hz}\(, the maximum acceleration is \)36 \pi^{2} \mathrm{~m} / \mathrm{s}^{2}$. Statement \(-2:\) The maximum acceleration for the above particle is $\pm \omega 2 \mathrm{~A}\(, where \)\mathrm{A}$ is amplitude. (A) a (B) \(\mathrm{b}\) (C) \(c\) (D) \(\mathrm{d}\)

Sound waves propagates with a speed of \(350 \mathrm{~m} / \mathrm{s}\) through air and with a speed of \(3500 \mathrm{~m} / \mathrm{s}\) through brass. If a sound wave having frequency \(700 \mathrm{~Hz}\) passes from air to brass, then its wavelength ......... (A) decreases by a fraction of 10 (B) increases 20 times (C) increases 10 times (D) decreases by a fraction of 20
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