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Chapter 15: Problem 50

Write the equation for a sinusoidal wave propagating in the negative \(x\) -direction with a speed of \(120 . \mathrm{m} / \mathrm{s}\), if a particle in the medium in which the wave is moving is observed to swing back and forth through a \(6.00-\mathrm{cm}\) range in \(4.00 \mathrm{~s}\). Assume that \(t=0\) is taken to be the instant when the particle is at \(y=0\) and that the particle moves in the positive \(y\) -direction immediately after \(t=0\).

Short Answer

Expert verified
Question: A sinusoidal wave is observed to travel in the negative x-direction with a wave speed of 120 m/s. The oscillating particle is observed to swing back and forth through a 6.00 cm range and take 4.00 seconds to complete an oscillation. Find the equation of the wave. Answer: The equation of the wave is given by y(x,t) = 0.030 * cos(π/240 * x + 0.5π * t - π/2), where x is the position, and t is the time.

Step by step solution

01

Find the amplitude of the wave

The amplitude (A) is the maximum displacement of the particles from their mean position. In this problem, we are given that a particle is observed to swing back and forth through a 6.00 cm range. Therefore, the amplitude of the wave is half of this range. A = 6.00 cm / 2 = 3.00 cm = 0.030 m
02

Find the frequency and angular frequency of the wave

The time taken for a complete oscillation of the particle is given as 4.00 seconds. This time is called the time period (T) of the wave. The frequency (f) is the inverse of the time period. f = 1 / T = 1 / 4.00 s = 0.25 Hz The angular frequency (ω) is related to the frequency by the following formula: ω = 2πf ω = 2π(0.25 Hz) = 0.5π rad/s
03

Determine the wave number (k) of the wave

The wave speed (v) is given as 120 m/s. The wave number (k) is related to the wave speed and angular frequency by the following formula: v = ω / k Rearrange the equation and solve for k: k = ω / v = (0.5π rad/s) / (120 m/s) = π/240 rad/m
04

Write the equation for the sinusoidal wave

Since the wave is propagating in the negative x-direction and particles move in the positive y-direction immediately after t=0, the equation for the sinusoidal wave can be represented as: y(x,t) = A * cos(kx + ωt + φ) Here, φ is the phase constant. As we are assuming that at t=0, the particle is at y=0 and moves in the positive y-direction immediately after t=0, this means the wave is at its mean position and hence has a phase constant of -π/2. φ = -π/2 Now, substitute the values of A, k, ω, and φ in the equation: y(x,t) = 0.030 * cos(π/240 * x + 0.5π * t - π/2)

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Most popular questions from this chapter

The largest tension that can be sustained by a stretched string of linear mass density \(\mu\), even in principle, is given by \(\tau=\mu c^{2},\) where \(c\) is the speed of light in vacuum. (This is an enormous value. The breaking tensions of all ordinary materials are about 12 orders of magnitude less than this.) a) What is the speed of a traveling wave on a string under such tension? b) If a \(1.000-\mathrm{m}\) -long guitar string, stretched between anchored ends, were made of this hypothetical material, what frequency would its first harmonic have? c) If that guitar string were plucked at its midpoint and given a displacement of \(2.00 \mathrm{~mm}\) there to produce the fundamental frequency, what would be the maximum speed attained by the midpoint of the string?

You and a friend are holding the two ends of a Slinky stretched out between you. How would you move your end of the Slinky to create (a) transverse waves or (b) longitudinal waves?

Students in a lab produce standing waves on stretched strings connected to vibration generators. One such wave is described by the wave function \(y(x, t)=(2.00 \mathrm{~cm}) \sin \left[\left(20.0 \mathrm{~m}^{-1}\right) x\right] \cos \left[\left(150 . \mathrm{s}^{-1}\right) t\right],\) where \(y\) is the transverse displacement of the string, \(x\) is the position along the string, and \(t\) is time. Rewrite this wave function in the form for a right- moving and a left-moving wave: \(y(x, t)=\) \(f(x-v t)+g(x+v t)\); that is, find the functions \(f\) and \(g\) and the speed, \(v\)

A sinusoidal wave on a string is described by the equation \(y=(0.100 \mathrm{~m}) \sin (0.75 x-40 t),\) where \(x\) and \(y\) are in meters and \(t\) is in seconds. If the linear mass density of the string is \(10 \mathrm{~g} / \mathrm{m}\), determine (a) the phase constant, (b) the phase of the wave at \(x=2.00 \mathrm{~cm}\) and \(t=0.100 \mathrm{~s}\) (c) the speed of the wave, (d) the wavelength, (e) the frequency, and (f) the power transmitted by the wave.

A string with linear mass density \(\mu=0.0250 \mathrm{~kg} / \mathrm{m}\) under a tension of \(T=250 . \mathrm{N}\) is oriented in the \(x\) -direction. Two transverse waves of equal amplitude and with a phase angle of zero (at \(t=0\) ) but with different frequencies \((\omega=3000\). rad/s and \(\omega / 3=1000 . \mathrm{rad} / \mathrm{s}\) ) are created in the string by an oscillator located at \(x=0 .\) The resulting waves, which travel in the positive \(x\) -direction, are reflected at a distant point, so there is a similar pair of waves traveling in the negative \(x\) -direction. Find the values of \(x\) at which the first two nodes in the standing wave are produced by these four waves.
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