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Chapter 11: Problem 24

Write the equation for a transverse sinusoidal wave with a maximum amplitude of \(2.50 \mathrm{cm}\) and an angular frequency of 2.90 rad/s that is moving along the positive \(x\) -direction with a wave speed that is 5.00 times as fast as the maximum speed of a point on the string. Assume that at time \(t=0,\) the point \(x=0\) is at \(y=0\) and then moves in the \(-y\) -direction in the next instant of time.

Short Answer

Expert verified
\(\lambda = 10\pi A\) So the wavelength of the transverse sinusoidal wave is equal to 10 times the amplitude multiplied by pi.

Step by step solution

01

Find the wavelength#

Since the wave speed is 5.00 times the maximum speed of a point on the string, we can use this information to find the wavelength. The maximum speed of a point on the string is given by the product of amplitude and angular frequency: \(v_{max} = A \cdot \omega\) So the wave speed, v, is related to the maximum speed as: \(v = 5.00 \cdot v_{max} = 5.00 \cdot (A \cdot \omega)\) The wave speed is also related to the wavelength and angular frequency through the equation: \(v=\frac{\omega \cdot \lambda}{2\pi}\) We can now solve for the wavelength: \(\lambda = \frac{2\pi\cdot v}{\omega} = \frac{2\pi\cdot 5.00 \cdot (A \cdot \omega)}{\omega}\)

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Most popular questions from this chapter

While testing speakers for a concert. Tomás sets up two speakers to produce sound waves at the same frequency, which is between \(100 \mathrm{Hz}\) and $150 \mathrm{Hz}$. The two speakers vibrate in phase with one another. He notices that when he listens at certain locations, the sound is very soft (a minimum intensity compared to nearby points). One such point is \(25.8 \mathrm{m}\) from one speaker and \(37.1 \mathrm{m}\) from the other. What are the possible frequencies of the sound waves coming from the speakers? (The speed of sound in air is \(343 \mathrm{m} / \mathrm{s} .\) )
A seismic wave is described by the equation $y(x, t)=(7.00 \mathrm{cm}) \cos [(6.00 \pi \mathrm{rad} / \mathrm{cm}) x+(20.0 \pi \mathrm{rad} / \mathrm{s}) t]\( The wave travels through a uniform medium in the \)x$ -direction. (a) Is this wave moving right ( \(+x\) -direction) or left (-x-direction)? (b) How far from their equilibrium positions do the particles in the medium move? (c) What is the frequency of this wave? (d) What is the wavelength of this wave? (e) What is the wave speed? (f) Describe the motion of a particle that is at \(y=7.00 \mathrm{cm}\) and \(x=0\) when \(t=0 .\) (g) Is this wave transverse or longitudinal?
Suppose that a string of length \(L\) and mass \(m\) is under tension \(F\). (a) Show that \(\sqrt{F L} m\) has units of speed. (b) Show that there is no other combination of \(L, m,\) and \(F\) with units of speed. [Hint: Of the dimensions of the three quantities $L, m, \text { and } F, \text { only } F \text { includes time. }]$ Thus, the speed of transverse waves on the string can only be some dimensionless constant times \(\sqrt{F L / m}.\)
(a) Sketch graphs of \(y\) versus \(x\) for the function $$ y(x, t)=(0.80 \mathrm{mm}) \sin (k x-\omega t) $$ for the times \(t=0,0.96 \mathrm{s},\) and \(1.92 \mathrm{s} .\) Make all three graphs of the same axes, using a solid line for the first, a dashed line for the second, and a dotted line for the third. Use the values \(k=\pi /(5.0 \mathrm{cm})\) and $\omega=(\pi / 6.0) \mathrm{rad} / \mathrm{s}$ (b) Repeat part (a) for the function $$ y(x, t)=(0.50 \mathrm{mm}) \sin (k x+\omega t) $$ (c) Which function represents a wave traveling in the \(-x\) direction and which represents a wave traveling in the \(+x\) -direction?
A transverse wave on a string is described by $y(x, t)=(1.2 \mathrm{cm}) \sin [(0.50 \pi \mathrm{rad} / \mathrm{s}) t-(1.00 \pi \mathrm{rad} / \mathrm{m}) x]$ Find the maximum velocity and the maximum acceleration of a point on the string. Plot graphs for displacement \(y\) versus \(t\), velocity \(v_{y}\) versus \(t\), and acceleration \(a_{y}\) versus \(t\) at \(x=0.\)
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