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Two strings on a musical instrument are tuned to play at 392 $\mathrm{Hz}(\mathrm{G})$ and 494 $\mathrm{Hz}(\mathrm{B}) .(a)$ What are the frequencies of the first two overtones for each string? (b) If the two strings have the same length and are under the same tension, what must be the ratio of their masses $\left(m_{\mathrm{G}} / m_{\mathrm{B}}\right) ?(c)$ If the strings, instead, have the same mass per unit length and are under the same tension, what is the ratio of their lengths $\left(\ell_{G} / \ell_{B}\right) ?(d)$ If their masses and lengths are the same, what must be the ratio of the tensions in the two strings?

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Physics 101 Mechanics

Physics 102 Electricity and Magnetism

Chapter 15

Wave Motion

Periodic Motion

Mechanical Waves

Electromagnetic Waves

Cornell University

Rutgers, The State University of New Jersey

University of Washington

McMaster University

Lectures

03:40

In physics, electromagnetic radiation (EM radiation or EMR) refers to the waves (or their quanta, photons) of the electromagnetic field, propagating (radiating) through space, carrying electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) light, ultraviolet, X-rays, and gamma rays. Electromagnetic waves of different frequency are called by different names since they have different sources and effects on matter. In order of increasing frequency and decreasing wavelength these are: radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays and gamma rays.

10:59

In physics, Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. They underpin all electric, optical and radio such electromagnetic technologies as power generation, electric motors, wireless communication, cameras, televisions, computers, and radar. Maxwell's equations describe how electric and magnetic fields are generated by charges, currents, and changes of these fields. The equations have two major variants. The microscopic Maxwell equations have universal applicability but are unwieldy for common calculations. They relate the electric and magnetic fields to total charge and total current, including the complicated charges and currents in materials at the atomic scale. The macroscopic Maxwell equations define two new auxiliary fields that describe the large-scale behaviour of matter without having to consider atomic scale details. The equations were published by Maxwell in his 1864 paper "A Dynamical Theory of the Electromagnetic Field". In the original paper Maxwell fully derived them from the Lorentz force law (without using the Lorentz transformation) and also from the conservation of energy and momentum.

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so the overtones are given by f N equals to end times F one, which is the fundamental pregnancy and n is toe three for and so on support. So all the anti jizz. So now for G? Um, it's F too, which is equal to twice off the fundamental frequency. 3 90 toe hurts and that's 7 84 hurts and, ah F three here is so that at me. I need a little bit off space here. So if T here is three times 3 92 herds, which is 1176 hurts, we can do the same thing for B. So for B F two is twice off for 94 hearts, which is equal to 9 88 herds on DA three. Here is three times 4 40 Herds, which is 14 82 hurts now for part B. If the two strings have the same length, they have the same wavelength, and the frequency difference is then, due to the difference in the whip speed caused by different masses of the spring. Sorry is master the strings and the mark. US enters the problem through the mass per unit length so you'll see in a minute. What I'm talking about. So we have f frequency JJ off G over frequency off here, which is velocity off G by reeling, divided by velocity off e over whaling. Now the mass enters here by mass per unit length calculations. So we have tension in the string divided by mass per unit length off G, divided by tension in the string eroded by Massport per unit length. And as we mentioned here that, um, the force on the strings are saying so from there we can, right, We can expand this mass per unit length expressions we have ft over MG divided by and divided by square root off FT. Divided by and a buy and and l are same for both. So from there we can get it off Ft and l. So all we have here is square root off AM a divided by mg. So if we take square on both sides, we have m a over M G. That's equal to F G over F A. Holds great. So that's 4 94 divided by 3 92 Who's squared, and that's equal to 1.5 tonight. Now, for part, see if the two strings have the same mass per unit length and the same tension than their waves, speed on both strings is the same. No, the frequency difference is due to the difference in we've linked only and, ah for the fundamental. The wavelength is basically twice the length off the stream, so f d over f B is equal. Do velocity over Lambda Gee, where, as we mentioned that, um, the wave speed or velocity is basically wave speeds. So we've speed a same onboard the strings. And a reason for that is they had the same mass per unit length and the same tension. So we have the over Lambda B. So from there we can write Lambda B over Lambda G. That's equal to twice off Lambda twice off L B times twice off L G. And this is because we know that four fundamental. The wavelength is basically twice the length off the street. So that's why we have to l B and two l g coming in the picture. So from there we can write l d over l G hopes equal do f B over f G because Stuart has matter here, which is 4 94 by 3 92 herds, and that's equal to 1.26 Now, finally, in party, if those two strings have the same length, they will have the same. So if they have the same length, they will have the same wavelength. And they also have the same mass per unit length, which is new basically. So the frequency differences then, due to the difference in we've speed caused by different tensions in the string. So the different thing here is Ft. So this is different and these are saying for both. So we have f B over F A. That's equal toe freebie over lander since Lambda I seem for both. So we have to be a over lambda, which gives us V B by V A. And from there we can write that in terms off tension and mass per unit length. So we have FT B bye mule, which is mastering it link with the same for both. So we have square root off f t a buy me, which gives us square root off f d be by f t eight on dhe. From there, if we take square on both sides. We have f T B by F d a. That's equal Do small left B by small f a a. And then there's a square. So now we can use Ah, the frequencies that's given So we have FT be by if looks f d a. Has equal do 3 92 That's F D by f averages for 94. Then there's a square. So that gives us 0.630 All right, Thanks for watching.

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