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Chapter 15: Problem 41

A \(3.00-\mathrm{m}\) -long string, fixed at both ends, has a mass of \(6.00 \mathrm{~g}\). If you want to set up a standing wave in this string having a frequency of \(300 . \mathrm{Hz}\) and three antinodes, what tension should you put the string under?

Short Answer

Expert verified
To find the tension required in the string, follow these steps: 1. Calculate the linear mass density: 渭 = 0.00200 kg/m. 2. Calculate the period: T = 0.00333 s. 3. Determine the wavelength: 位 = 2.00 m. 4. Calculate the wave velocity: v = 600 m/s. 5. Calculate the tension: T = 720 N. The tension required is 720 N.

Step by step solution

01

Determine the linear mass density and period

First, we'll calculate the linear mass density \(渭\) by dividing the mass of the string by its length: $$ 渭 = \frac{m}{L} = \frac{6.00\,\text{g}}{3.00\,\text{m}} = 2.00\,\text{g/m} = 0.00200\,\text{kg/m} $$ Next, we'll find the period \(T\) using the given frequency \(f\): $$ T = \frac{1}{f} = \frac{1}{300\,\text{Hz}} = 0.00333\,\text{s} $$
02

Find the wavelength

Since we have 3 antinodes in the string, it means we have \(1.5\) complete wavelengths over the length of the string. Therefore, we can find the wavelength \(位\) by dividing the length of the string by \(1.5\): $$ 位 = \frac{L}{1.5} = \frac{3.00\,\text{m}}{1.5} = 2.00\,\text{m} $$
03

Calculate the wave velocity

Now we have the wavelength \(位\) and the period \(T\), we can find the wave velocity \(v\) using the wave equation: $$ v = \frac{位}{T} = \frac{2.00\,\text{m}}{0.00333\,\text{s}} = 600\,\text{m/s} $$
04

Calculate the tension

Finally, we can use the formula \(v = \sqrt{\frac{T}{渭}}\) to determine the required tension in the string to create the desired standing wave: $$ T = 渭v^2 = (0.00200\,\text{kg/m})(600\,\text{m/s})^2 = 720\,\text{N} $$ The tension required to create the desired standing wave with a frequency of \(300\,\text{Hz}\) and three antinodes in the string is \(720\,\text{N}\).

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Most popular questions from this chapter

Suppose that the tension is doubled for a string on which a standing wave is propagated. How will the velocity of the standing wave change? a) It will double. c) It will be multiplied by \(\sqrt{2}\). b) It will quadruple. d) It will be multiplied by \(\frac{1}{2}\).

What is the wave speed along a brass wire with a radius of \(0.500 \mathrm{~mm}\) stretched at a tension of \(125 \mathrm{~N}\) ? The density of brass is \(8.60 \cdot 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\).

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?

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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