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Chapter 15: Problem 64

Two waves traveling in opposite directions along a string fixed at both ends create a standing wave described by \(y(x, t)=1.00 \cdot 10^{-2} \sin (25 x) \cos (1200 t) .\) The string has a linear mass density of \(0.01 \mathrm{~kg} / \mathrm{m},\) and the tension in the string is supplied by a mass hanging from one end. If the string vibrates in its third harmonic, calculate (a) the length of the string, (b) the velocity of the waves, and (c) the mass of the hanging mass.

Short Answer

Expert verified
Answer: The length of the string is \(L = 0.377\,\mathrm{m}\), the velocity of the waves is \(v = 48\,\mathrm{m/s}\), and the mass of the hanging mass is \(m = 11.83\,\mathrm{kg}\).

Step by step solution

01

Calculate the Angular Wave Number (k)

For a given standing wave, the angular wave number (k) can be calculated from the equation: \(y(x, t) = A\sin(kx)\cos(\omega t)\) Comparing this to the given equation: \(y(x, t) = 1.00 \times 10^{-2} \sin (25 x) \cos (1200 t)\) We have \(k = 25 \frac{rad}{m}\).
02

Calculate the length of the string (L)

In a standing wave vibrating in its nth harmonic, the length of the string (L) can be found using the following formula: \(L = \frac{n\pi}{k}\) Substituting the values for n and k: \(L = \frac{3\pi}{25} = 0.377 \,\mathrm{m}\)
03

Calculate the Angular Frequency (\(\omega\))

From the given standing wave equation, we can obtain the angular frequency \(\omega\): \(y(x, t) = 1.00 \times 10^{-2} \sin (25 x) \cos (1200 t)\) \(\omega = 1200\,\mathrm{rad/s}\)
04

Calculate the velocity of the waves (v)

The relationship between the angular wave number (k) and the angular frequency (\(\omega\)) is given by: \(v = \frac{\omega}{k}\) Plugging in the values for \(\omega\) and k, we get: \(v = \frac{1200\,\mathrm{rad/s}}{25\,\mathrm{rad/m}} = 48\,\mathrm{m/s}\)
05

Calculate the mass of the hanging mass (m)

The tension (T) in the string is the force exerted by the hanging mass (m), which is given by: \(T = mg\) The wave speed v can also be calculated using the formula: \(v = \sqrt{\frac{T}{\mu}}\) Solving for m, we get: \(m = \frac{\mu v^2}{g}\) Substituting the known values: \(m = \frac{0.01\,\mathrm{kg/m} \times (48\,\mathrm{m/s})^2}{9.81\,\mathrm{m/s^2}} = 11.83\,\mathrm{kg}\) Therefore, the solutions to the problem are: (a) The length of the string is \(L = 0.377\,\mathrm{m}\). (b) The velocity of the waves is \(v = 48\,\mathrm{m/s}\). (c) The mass of the hanging mass is \(m = 11.83\,\mathrm{kg}\).

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Most popular questions from this chapter

The displacement from equilibrium caused by a wave on a string is given by \(y(x, t)=(-0.00200 \mathrm{~m}) \sin \left[\left(40.0 \mathrm{~m}^{-1}\right) x-\right.\) \(\left.\left(800 . \mathrm{s}^{-1}\right) t\right] .\) For this wave, what are the (a) amplitude, (b) number of waves in \(1.00 \mathrm{~m},\) (c) number of complete cycles in \(1.00 \mathrm{~s},\) (d) wavelength, and (e) speed?

A rubber band of mass \(0.21 \mathrm{~g}\) is stretched between two fingers, putting it under a tension of \(2.8 \mathrm{~N}\). The overall stretched length of the band is \(21.3 \mathrm{~cm} .\) One side of the band is plucked, setting up a vibration in \(8.7 \mathrm{~cm}\) of the band's stretched length. What is the lowest frequency of vibration that can be set up on this part of the rubber band? Assume that the band stretches uniformly.

A traveling wave propagating on a string is described by the following equation: $$ y(x, t)=(5.00 \mathrm{~mm}) \sin \left(\left(157.08 \mathrm{~m}^{-1}\right) x-\left(314.16 \mathrm{~s}^{-1}\right) t+0.7854\right) $$ a) Determine the minimum separation, \(\Delta x_{\min }\), between two points on the string that oscillate in perfect opposition of phases (move in opposite directions at all times). b) Determine the separation, \(\Delta x_{A B}\), between two points \(A\) and \(B\) on the string, if point \(B\) oscillates with a phase difference of 0.7854 rad compared to point \(A\). c) Find the number of crests of the wave that pass through point \(A\) in a time interval \(\Delta t=10.0 \mathrm{~s}\) and the number of troughs that pass through point \(B\) in the same interval. d) At what point along its trajectory should a linear driver connected to one end of the string at \(x=0\) start its oscillation to generate this sinusoidal traveling wave on the string?

A \(50.0-\mathrm{cm}\) -long wire with a mass of \(10.0 \mathrm{~g}\) is under a tension of \(50.0 \mathrm{~N}\). Both ends of the wire are held rigidly while it is plucked. a) What is the speed of the waves on the wire? b) What is the fundamental frequency of the standing wave? c) What is the frequency of the third harmonic?

Consider a monochromatic wave on a string, with amplitude \(A\) and wavelength \(\lambda\), traveling in one direction. Find the relationship between the maximum speed of any portion of string, \(v_{\max },\) and the wave speed, \(v\)
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