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Chapter 10: Problem 1427

Equation for a harmonic progressive wave is given by \(\mathrm{y}=\mathrm{A}\) \(\sin (15 \pi t+10 \pi x+\pi / 3)\) where \(x\) is in meter and \(t\) is in seconds. This wave is \(\ldots \ldots\) (A) Travelling along the positive \(\mathrm{x}\) direction with a speed of $1.5 \mathrm{~ms}^{-1}$ (B) Travelling along the negative \(\mathrm{x}\) direction with a speed of $1.5 \mathrm{~ms}^{-1} .$ (C) Has a wavelength of \(1.5 \mathrm{~m}\) along the \(-\mathrm{x}\) direction. (D) Has a wavelength of \(1.5 \mathrm{~m}\) along the positive \(\mathrm{x}\) - direction.

Short Answer

Expert verified
The correct answer is (B) Travelling along the negative x direction with a speed of \(1.5~ms^{-1}\).

Step by step solution

01

Identify the general wave equation

A general equation for a harmonic wave traveling in the x-direction is given as: \[y(x,t) = A \sin(kx \pm \omega t + \phi)\] Here, `A` is the amplitude, `k` is the wave number (\(\frac{2\pi}{\lambda}\)), `\(\omega\)` is the angular frequency (\(2\pi f\)), and `\(\phi\)` is the phase difference.
02

Compare the given equation with the general wave equation

Given wave equation is: \[\mathrm{y}=\mathrm{A} \sin (15 \pi t + 10 \pi x + \pi / 3)\] Comparing it with the general wave equation, we can identify the terms in the expression: - \(k = 10\pi\) - \(\omega = 15\pi\) - \(\phi = \pi/3\)
03

Calculate the wavelength, frequency, and wave speed

Recall that \(k = \frac{2\pi}{\lambda}\) and \(\omega = "@/pi" \times f\). So, - Wavelength, \(\lambda = \frac{2\pi}{10\pi} = 0.2\;m\) - Frequency, \(f = \frac{15\pi}{2\pi} = 7.5\;Hz\) - Wave speed (v) can be calculated by multiplying the wavelength and frequency: \(v =\lambda \times f = 0.2 \times 7.5 = 1.5~ms^{-1}\)
04

Determine the direction of wave propagation

The wave is traveling in the positive x-direction if the sign between kx and \(\omega\)t is negative, and it's traveling in the negative x-direction if the sign is positive. In our case, it's a positive sign so the wave is traveling in the negative x-direction.
05

Analyze the answer choices

Based on our calculated parameters and identified direction, the correct answer is: (B) Travelling along the negative x direction with a speed of \(1.5~ms^{-1}\).

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Most popular questions from this chapter

A particle executing S.H.M. has an amplitude \(\mathrm{A}\) and periodic time \(\mathrm{T}\). The minimum time required by the particle to get displaced by \((\mathrm{A} / \sqrt{2})\) from its equilibrium position is $\ldots \ldots \ldots \mathrm{s}$. (A) \(\mathrm{T}\) (B) \(\mathrm{T} / 4\) (C) \(\mathrm{T} / 8\) (D) \(\mathrm{T} / 16\)

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) \(\mathrm{b}\) (C) \(\mathrm{c}\) (D) \(\mathrm{d}\) Statement \(-1:\) The periodic time of a S.H.O. depends on its amplitude and force constant. Statement \(-2:\) The elasticity and inertia decides the frequency of S.H.O. (A) a (B) \(b\) (C) c (D) \(\mathrm{d}\)

One end of a mass less spring having force constant \(\mathrm{k}\) and length \(50 \mathrm{~cm}\) is attached at the upper end of a plane inclined at an angle \(e=30^{\circ} .\) When a body of mass \(m=1.5 \mathrm{~kg}\) is attached at the lower end of the spring, the length of the spring increases by $2.5 \mathrm{~cm}$. Now, if the mass is displaced by a small amount and released, the amplitude of the resultant oscillation is ......... (A) \((\pi / 7)\) (B) \((2 \pi / 7)\) (C) \((\pi / 5)\) (D) \((2 \pi / 5)\)

An open organ pipe has fundamental frequency \(100 \mathrm{~Hz}\). What frequency will be produced if its one end is closed? (A) \(100,200,300, \ldots\) (B) \(50,150,250 \ldots .\) (C) \(50,100,200,300 \ldots \ldots\) (D) \(50,100,150,200 \ldots \ldots\)

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) \((\pi / 3)\)
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