/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 28 A transverse wave on a string is... [FREE SOLUTION] | 91影视

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Chapter 11: Problem 28

A transverse wave on a string is described by $y(x, t)=(1.2 \mathrm{mm}) \sin [(2.0 \pi \mathrm{rad} / \mathrm{s}) t-(0.50 \pi \mathrm{rad} / \mathrm{m}) x]$ Plot the displacement \(y\) and the velocity \(v_{y}\) versus \(t\) for one complete cycle of the point \(x=0\) on the string.

Short Answer

Expert verified
Also, calculate the period of one complete cycle and provide the expressions for the displacement and velocity of the point as a function of time.

Step by step solution

01

Identify the amplitude, frequency, and wave number

From the given wave equation, we can identify the following parameters: Amplitude: \(A = 1.2 \,\text{mm}\) Frequency: \(\omega = 2.0 \,\pi\, \text{rad/s}\) Wave number: \(k = 0.50 \,\pi\, \text{rad/m}\) Since we are asked to analyze the point \(x=0\), the wave equation simplifies to: \(y(t) = A \sin(\omega t)\)
02

Calculate the period of one complete cycle

The period \(T\) of the wave can be calculated as the inverse of its frequency: \(T = \frac{2 \pi}{\omega}\) Substitute the value of \(\omega\) and solve for \(T\): \(T = \frac{2 \pi}{2.0 \,\pi\, \text{rad/s}} = 1\, \text{s}\) So, one complete cycle takes 1 second.
03

Derive the expression for velocity \(v_y\)

To find the expression for \(v_y\), we take the time derivative of the displacement \(y(t)\): \(v_y(t) = \frac{dy(t)}{dt}\) Differentiate \(y(t) = A \sin(\omega t)\) with respect to time: \(v_y(t) = A \omega \cos(\omega t)\) Substitute the values for \(A\) and \(\omega\): \(v_y(t) = (1.2 \,\text{mm}) (2.0 \,\pi\, \text{rad/s}) \cos[(2.0 \,\pi\, \text{rad/s}) t]\)
04

Plot the displacement and velocity as a function of time

Now we can plot the displacement \(y(t)\) and the velocity \(v_y(t)\) as a function of time for one complete cycle, which takes 1 second. The displacement and velocity functions are: \(y(t) = (1.2 \,\text{mm}) \sin[(2.0 \,\pi\, \text{rad/s}) t]\) \(v_y(t) = (1.2 \,\text{mm}) (2.0 \,\pi\, \text{rad/s}) \cos[(2.0 \,\pi\, \text{rad/s}) t]\) These graphs will be sinusoidal in nature, with the displacement graph being a sine wave and the velocity graph being a cosine wave. The displacement will vary between +1.2 mm and -1.2 mm, while the velocity will vary between \((1.2 \,\text{mm}) (2.0 \,\pi\, \text{rad/s})\) and \(-(1.2 \,\text{mm}) (2.0 \,\pi\, \text{rad/s})\). The period for both graphs will be 1 second.

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Most popular questions from this chapter

The formula for the speed of transverse waves on a spring is the same as for a string. (a) A spring is stretched to a length much greater than its relaxed length. Explain why the tension in the spring is approximately proportional to the length. (b) A wave takes 4.00 s to travel from one end of such a spring to the other. Then the length is increased \(10.0 \% .\) Now how long does a wave take to travel the length of the spring? [Hint: Is the mass per unit length constant?]

The tension in a guitar string is increased by \(15 \% .\) What happens to the fundamental frequency of the string?
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 \(x=0.\)

(a) Use a graphing calculator or computer graphing program to plot \(y\) versus \(x\) for the function $$ y(x, t)=(5.0 \mathrm{cm})[\sin (k x-\omega t)+\sin (k x+\omega t)] $$ for the times \(t=0,1.0 \mathrm{s},\) and \(2.0 \mathrm{s}\). Use the values \(k=\pi /(5.0 \mathrm{cm})\) and \(\omega=(\pi / 6.0) \mathrm{rad} / \mathrm{s} .\) (b) Is this a traveling wave? If not, what kind of wave is it?
A metal guitar string has a linear mass density of $\mu=3.20 \mathrm{g} / \mathrm{m} .$ What is the speed of transverse waves on this string when its tension is \(90.0 \mathrm{N} ?\)
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