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Chapter 15: Problem 14

A ping-pong ball is floating in the middle of a lake and waves begin to propagate on the surface. Can you think of a situation in which the ball remains stationary? Can you think of a situation involving a single wave on the lake in which the ball remains stationary?

Short Answer

Expert verified
Explain the general and specific situation involving multiple waves and a single wave. Answer: It is possible for a ping-pong ball floating on a lake with propagating waves to remain stationary if there are multiple waves interfering with each other, causing constructive and destructive interferences to perfectly cancel each other out, leading to a zero net disturbance at the position of the ball. However, in the specific situation where there is only a single propagating wave, the ball cannot remain stationary, as it will move upward during a wave crest and downward during a trough. For the ball to remain stationary, multiple waves are necessary to create a balanced disturbance.

Step by step solution

01

General Situation: Multiple Waves

Consider a scenario where waves travel outward from different points on the lake and their amplitudes, frequencies, and directions are different. It is possible that these waves may interfere with each other, causing the ping-pong ball to remain stationary. This can happen when constructive and destructive interferences cancel each other perfectly, leading to a zero net disturbance at the position of the ball. In this case, the ball remains stationary.
02

Specific Situation: Single Wave

Now, let's consider the case where there's only a single wave propagating on the surface of the lake. Since a ping-pong ball is light enough, it will follow the wave motion without any resistance. If a wave crest reaches the ball, it will move upward, and when a trough reaches the ball, it will move downward. In this case, it's impossible for the ball to remain stationary with only a single propagating wave. For the ball to remain stationary, we need multiple waves interfering with each other to create a balanced disturbance, as discussed in the general situation.

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Most popular questions from this chapter

a) Starting from the general wave equation (equation 15.9 ), prove through direct derivation that the Gaussian wave packet described by the equation \(y(x, t)=(5.00 m) e^{-0.1(x-5 t)^{2}}\) is indeed a traveling wave (that it satisfies the differential wave equation). b) If \(x\) is specified in meters and \(t\) in seconds, determine the speed of this wave. On a single graph, plot this wave as a function of \(x\) at \(t=0, t=1.00 \mathrm{~s}, t=2.00 \mathrm{~s},\) and \(t=3.00 \mathrm{~s}\) c) More generally, prove that any function \(f(x, t)\) that depends on \(x\) and \(t\) through a combined variable \(x \pm v t\) is a solution of the wave equation, irrespective of the specific form of the function \(f\)

A wave traveling on a string has the equation of motion \(y(x, t)=0.02 \sin (5.00 x-8.00 t)\) a) Calculate the wavelength and the frequency of the wave. b) Calculate its velocity. c) If the linear mass density of the string is \(\mu=0.10 \mathrm{~kg} / \mathrm{m}\), what is the tension on the string?

A string is \(35.0 \mathrm{~cm}\) long and has a mass per unit length of \(5.51 \cdot 10^{-4} \mathrm{~kg} / \mathrm{m}\). What tension must be applied to the string so that it vibrates at the fundamental frequency of \(660 \mathrm{Hz?}\)

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 \(t=0\).

The tension in a 2.7 -m-long, 1.0 -cm-diameter steel cable \(\left(\rho=7800 \mathrm{~kg} / \mathrm{m}^{3}\right)\) is \(840 \mathrm{~N}\). What is the fundamental frequency of vibration of the cable?
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