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Chapter 11: Problem 59

What is the wavelength of the radio waves transmitted by an FM station at 90 MHz? (Radio waves travel at \(3.0 \times 10^{8} \mathrm{m} / \mathrm{s} .\)

Short Answer

Expert verified
Answer: The wavelength of the radio waves transmitted by an FM station at 90 MHz is 3.33 meters.

Step by step solution

01

Write down the wave speed formula

The wave speed formula is: speed = frequency 脳 wavelength or v = f 脳 位 Where v is the speed of the radio waves, f is the frequency of the radio waves, and 位 is the wavelength of the radio waves.
02

Rearrange the formula to solve for wavelength

To find the wavelength, we need to rearrange the equation: 位 = v / f
03

Convert the given frequency to Hz

The frequency is given in MHz, which is megahertz. We need to convert it to hertz (Hz) for our calculations: 90 MHz = 90 脳 10^6 Hz
04

Plug in given values and solve for wavelength

Now, we can plug in the values for the speed of radio waves (v = 3.0 脳 10^8 m/s) and frequency (f = 90 脳 10^6 Hz) into the rearranged equation: 位 = (3.0 脳 10^8 m/s) / (90 脳 10^6 Hz)
05

Calculate the wavelength

Now, we can perform the calculation: 位 = (3.0 脳 10^8 m/s) / (90 脳 10^6 Hz) = (3.0 / 90) 脳 10^2 m = 0.033 脳 10^2 m = 3.33 m The wavelength of the radio waves transmitted by an FM station at 90 MHz is 3.33 meters.

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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?]
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 \(90.0 \mathrm{N} ?\)
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 \(\sqrt{F L / m}.\)
When the string of a guitar is pressed against a fret, the shortened string vibrates at a frequency \(5.95 \%\) higher than when the previous fret is pressed. If the length of the part of the string that is free to vibrate is \(64.8 \mathrm{cm},\) how far from one end of the string are the first three frets located?
(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?
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