Lectures on Physics has been derived from Benjamin Crowell's Light and Matter series of free introductory textbooks on physics. See the editorial for more information....

Homework problems

If one stereo system is capable of producing 20 watts of sound power and another can put out 50 watts, how many times greater is the amplitude of the sound wave that can be created by the more powerful system? (Assume they are playing the same music.)
Many fish have an organ known as a swim bladder, an air-filled cavity whose main purpose is to control the fish's buoyancy an allow it to keep from rising or sinking without having to use its muscles. In some fish, however, the swim bladder (or a small extension of it) is linked to the ear and serves the additional purpose of amplifying sound waves. For a typical fish having such an anatomy, the bladder has a resonant frequency of 300 Hz, the bladder's Q is 3, and the maximum amplification is about a factor of 100 in energy. Over what range of frequencies would the amplification be at least a factor of 50?
As noted in section 2.4, it is only approximately true that the amplitude has its maximum at

Being more careful, we should actually define two different symbols,

and fres for the slightly different frequency at which the amplitude is a maximum, i.e. the actual resonant frequency. In this notation, the amplitude as a function of frequency is

Show that the maximum occurs not at fo but rather at the frequency

Hint: Finding the frequency that minimizes the quantity inside the square root is equivalent to, but much easier than, finding the frequency that maximizes the amplitude.

(a) Let W be the amount of work done by friction per cycle of oscillation, i.e. the amount of energy lost to heat. Find the fraction of the original energy E that remains in the oscillations after n cycles of motion.

(b) From this prove the equation (1 - W/E)Q = e - 2π (recalling that the number 535 in the definition of Q is e). (c) Use this to prove the approximation 1/Q≈ (1/2π)W/E. [Hint: Use the approximation ln(1+x)≈ x, which is valid for small values of x.]

The goal of this problem is to refine the proportionality FWHM fres/Q into the equation FWHM=fres/Q, i.e. to prove that the constant of proportionality equals 1.

(a) Show that the work done by a damping force F=-bv over one cycle of steady-state motion equals Wdamp=-2π2bfA2. Hint: It is less confusing to calculate the work done over half a cycle, from x=-A to x=+A, and then double it.

(b) Show that the fraction of the undriven oscillator's energy lost to damping over one cycle is |Wdamp| / E = 4π2bf / k.

(c) Use the previous result, combined with the result of problem 4, to prove that Q equals k/2πbf .

(d) Combine the preceding result for Q with the equation FWHM=b/2πm from section 2.4 to prove the equation FWHM=fres/Q.

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Last Update: 2010-11-11