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Chapter 11: Problem 19

What is the speed of the wave represented by \(y(x, t)=\) $A \sin (k x-\omega t),\( where \)k=6.0 \mathrm{rad} / \mathrm{cm}\( and \)\omega=5.0 \mathrm{rad} / \mathrm{s} ?$

Short Answer

Expert verified
Answer: The speed of the wave is \(\frac{1}{120}~\frac{\mathrm{m}}{\mathrm{s}}\).

Step by step solution

01

Identify the given values in the problem

The given values in the problem are: - Angular frequency, \(\omega=5.0 \mathrm{rad} / \mathrm{s}\) - Wave number, \(k=6.0 \mathrm{rad} / \mathrm{cm}\)
02

Convert the wave number to SI units

To make our calculations more consistent, we will convert the wave number from rad/cm to rad/m. To do this, we need to multiply the wave number by the conversion factor of 100 cm/m. \(k = 6.0 \mathrm{rad} / \mathrm{cm} \times 100 \mathrm{cm} / \mathrm{m} = 600 \mathrm{rad} / \mathrm{m}\)
03

Calculate the speed of the wave using the formula

Now that we have the wave number in SI units and the angular frequency, we will use the wave speed formula to determine the speed of the wave. \(v = \frac{\omega}{k} = \frac{5.0 \mathrm{rad} / \mathrm{s}}{600 \mathrm{rad} / \mathrm{m}}\)
04

Simplify the expression and solve for the speed

Simplify the expression and solve for \(v\): \(v = \frac{5.0}{600} \frac{\mathrm{rad} / \mathrm{s}}{\mathrm{rad} / \mathrm{m}} = \frac{1}{120} \frac{\mathrm{m}}{\mathrm{s}}\)
05

Write down the final answer

The speed of the wave represented by the given equation is: \(v = \frac{1}{120} \frac{\mathrm{m}}{\mathrm{s}}\)

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Most popular questions from this chapter

Deep-water waves are dispersive (their wave speed depends on the wavelength). The restoring force is provided by gravity. Using dimensional analysis, find out how the speed of deep-water waves depends on wavelength \(\lambda\), assuming that \(\lambda\) and \(g\) are the only relevant quantities. (Mass density does not enter into the expression because the restoring force, arising from the weight of the water, is itself proportional to the mass density.)
What is the wavelength of the radio waves transmitted by an FM station at 90 MHz? (Radio waves travel at \(3.0 \times 10^{8} \mathrm{m} / \mathrm{s} .\)
A cord of length \(1.5 \mathrm{m}\) is fixed at both ends. Its mass per unit length is \(1.2 \mathrm{g} / \mathrm{m}\) and the tension is \(12 \mathrm{N} .\) (a) What is the frequency of the fundamental oscillation? (b) What tension is required if the \(n=3\) mode has a frequency of \(0.50 \mathrm{kHz} ?\)
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 \(\sqrt{F L / m}.\)
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