In a situation in which data are known to three significant digits, we write $6.379 \mathrm{m}=6.38 \mathrm{m}$ and $6.374 \mathrm{m}=6.37 \mathrm{m} .$ When a number ends in $5,$ we arbitrarily choose to write $6.375 \mathrm{m}=6.38 \mathrm{m} .$ We could equally well write $6.375 \mathrm{m}=$ $6.37 \mathrm{m},$ "rounding down" instead of "rounding up," because we would change the number 6.375 by equal increments in both cases. Now consider an order-of- magnitude estimate, in which factors of change rather than increments are important. We write 500 $\mathrm{m} \sim 10^{3} \mathrm{m}$ because 500 differs from 100 by a factor of $5,$ whereas it differs from 1000 by only a factor of $2 .$ We write 437 $\mathrm{m} \sim$ $10^{3} \mathrm{m}$ and $305 \mathrm{m} \sim 10^{2} \mathrm{m} .$ What distance differs from 100 $\mathrm{m}$ and from 1000 $\mathrm{m}$ by equal factors so that we could equally well choose to represent its order of magnitude either as $\sim 10^{2} \mathrm{m}$ or as $\sim 10^{3} \mathrm{m} ?$

In a particular set of experimental trials, students examine a system described by the equation Qt=kd2(ThTc)4LQt=kd2(ThTc)4L We will see this equation and the various quantities in it in Chapter 20. For experimental control, in these trials all quantities except $d$ and $\Delta t$ are constant. (a) If $d$ is made three times larger, does the equation predict that $\Delta t$ will get larger or smaller? By what factor? (b) What pattern of proportionality of $\Delta t$ to $d$ does the equation predict? (c) To display this proportionality as a straight line on a graph, what quantities should you plot on the horizontal and vertical axes? (d) What expression represents the theoretical slope of this graph?

Three metal rods are located relative to each other as shown in Figure P17.63, where L1+L2=L3L1+L2=L3 . The speed of sound in a rod is given by v=Y/,v=Y/, where YY is Young's modulus for the rod and is the density. Values of density and Young's modulus for the three materials are 1=2.70103kg/m3,Y1=7.001010N/m21=2.70103kg/m3,Y1=7.001010N/m2 2=11.3103kg/m3,Y2=1.601010N/m2,3=2=11.3103kg/m3,Y2=1.601010N/m2,3= 8.80103kg/m3,8.80103kg/m3, and Y3=11.01010N/m2.Y3=11.01010N/m2. (a) If L3=1.50m,L3=1.50m, what must the ratio L1/L2L1/L2 be if a sound wave is to travel the length of rods 1 and 2 in the same time interval required for the wave to travel the length of rod 3?? (b) The frequency of the source is 4.00 kHzkHz . Determine the phase difference between the wave traveling along rods 1 and 2 and the one traveling along rod 3.3.

A spherical shell has an outside radius of 2.60 $\mathrm{cm}$ and an inside radius of $a$ . The shell wall has uniform thickness and is made of a material with density $4.70 \mathrm{g} / \mathrm{cm}^{3} .$ The space inside the shell is filled with a liquid having a density of $1.23 \mathrm{g} / \mathrm{cm}^{3} .$ (a) Find the mass $m$ of the sphere, including its contents, as a function of $a .$ (b) In the answer to part (a), if $a$ is regarded as a variable, for what value of $a$ does $m$ have its maximum possible value? (c) What is this maximum mass? (d) Does the value from part (b) agree with the result of a direct calculation of the mass of a sphere of uniform density? (e) For what value of $a$ does the answer to part (a) have its minimum possible value? (f) What is this minimum mass? (g) Does the value from part (f) agree with the result of a direct calculation of the mass of a uniform sphere? (h) What value of $m$ is halfway between the maximum and minimum possible values? (i) Does this mass agree with the result of part (a) evaluated for $a=2.60 \mathrm{cm} / 2=1.30 \mathrm{cm} ?$ (j) Explain whether you should expect agreement in each of parts $(\mathrm{d}),$ (g), and (i). (k) What If? In part (a), would the answer change if the inner wall of the shell were not concentric with the outer wall?

A spherical shell has an outside radius of 2.60 $\mathrm{cm}$ and an inside radius of $a$ . The shell wall has uniform thickness and is made of a material with density $4.70 \mathrm{g} / \mathrm{cm}^{3} .$ The space inside the shell is filled with a liquid having a density of $1.23 \mathrm{g} / \mathrm{cm}^{3} .$ (a) Find the mass $m$ of the sphere, including its contents, as a function of $a .$ (b) In the answer to part (a), if $a$ is regarded as a variable, for what value of $a$ does $m$ have its maximum possible value? (c) What is this maximum mass? (d) Does the value from part (b) agree with the result of a direct calculation of the mass of a sphere of uniform density? (e) For what value of $a$ does the answer to part (a) have its minimum possible value? (f) What is this minimum mass? (g) Does the value from part (f) agree with the result of a direct calculation of the mass of a uniform sphere? (h) What value of $m$ is halfway between the maximum and minimum possible values? (i) Does this mass agree with the result of part (a) evaluated for $a=2.60 \mathrm{cm} / 2=1.30 \mathrm{cm} ?$ (j) Explain whether you should expect agreement in each of parts $(\mathrm{d}),$ (g), and (i). (k) What If? In part (a), would the answer change if the inner wall of the shell were not concentric with the outer wall?

In physics it is important to use mathematical approximations. Demonstrate that for small angles $\left(<20^{\circ} ight),$ tansin=180tansin=180 where $\alpha$ is in radians and $\alpha^{\prime}$ is in degrees. Use a calculator to find the largest angle for which tan $\alpha$ may be approximated by $\alpha$ with an error less than 10.0$\%$.

Ultrasound is used in medicine both for diagnostic imaging and for therapy. For diagnosis, short pulses of ultra-sound are passed through the patient's body. An echo reflected from a structure of interest is recorded, and the distance to the structure can be determined from the time delay for the echo's return. A single transducer emits and detects the ultrasound. An image of the structure is obtained by reducing the data with a computer. With sound of low intensity, this technique is noninvasive and harmless. It is used to examine fetuses, tumors, aneurysms, gallstones, and many other structures. To reveal detail, the wavelength of the reflected ultrasound must be small compared to the size of the object reflecting the wave. (a) What is the wavelength of ultrasound with a frequency of 2.40 MHzMHz , used in echocardiography to map the beating human heart? (b) In the whole set of imaging techniques, frequencies in the range 1.00 to 20.0 MHzMHz are used. What is the range of wavelengths corresponding to this range of frequencies? The speed of ultrasound in human tissue is about 1500 m/sm/s (nearly the same as the speed of sound in water).

The most soaring vocal melody is in Johann Sebastian Bach's Mass in B Minor. A portion of the score for the Credo section, number 9, bars 25 to 33, appears in Figure Pl7.21. The repeating syllable O in the phrase "resurrectionem mortuorum" (the resurrection of the dead) is seamlessly passed from basses to tenors to altos to first sopranos, like a baton in a relay. Each voice carries the foreground melody up through a rising passage encompassing an octave or more. Together the voices carry it from D below middle CC to A above a tenor's high C. In concert pitch, these notes are now assigned frequencies of 146.8 HzHz and 880.0 HzHz . (a) Find the wavelengths of the initial and final notes. (b) Assume the chorus sings the melody with a uniform sound level of 75.0 dB. Find the pressure amplitudes of the initial and final notes. (c) Find the displacement amplitudes of the initial and final notes. (d) What If? In Bach's time, before the invention of the tuning fork, frequencies were assigned to notes as a matter of immediate local convenience. Assume the rising melody was sung starting from 134.3 HzHz and ending at 804.9 HzHz . How would the answers to parts (a) through (c) change?

Expectant parents are thrilled to hear their unborn baby's heartbeat, revealed by an ultrasonic motion detector. Suppose the fetus's ventricular wall moves in simple harmonic motion with an amplitude of 1.80 mmmm and a frequency of 115 per minute. (a) Find the maximum linear speed of the heart wall. Suppose the motion detector in contact with the mother's abdomen produces sound at 2000000.0 HzHz which travels through tissue at 1.50 km/skm/s . (b) Find the maximum frequency at which sound arrives at the wall of the baby's heart. (c) Find the maximum frequency at which reflected sound is received by the motion detector. By electronically "Istening" for echoes at a frequency different from the broadcast frequency, the motion detector can produce beeps of audible sound in synchronization with the fetal heartbeat.

This problem represents a possible (but not recommended) way to code instantaneous pressures in a sound wave into 16 -bit digital words. Example 17.2 mentions that the pressure amplitude of a 120dB120dB sound is 28.7 N/m2N/m2 . Let this pressure variation be represented by the digital code 65536 . Let the digital word 0 on the recording represent zero pressure variation. Let other intermediate pressures be represented by digital words of intermediate size, in direct proportion to the pressure. (a) What digital word would represent the maximum pressure in a 40dB40dB sound? (b) Explain why this scheme works poorly for soft sounds. ( c)c) Explain how this coding scheme would clip off half of the waveform of any sound, ignoring the actual shape of the wave and turning it into a string of zeros. By introducing sharp corners into every recorded waveform, this coding scheme would make everything sound like a buzzer or a kazoo.

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A large set of unoccupied football bleachers has solid seats and risers. You stand on the field in front of the bleachers and sharply clap two wooden boards together once. The sound pulse you produce has no definite frequency and no wavelength. The sound you hear reflected from the bleachers has an identifiable frequency and may remind you of a short toot on a trumpet or of a buzzer or kazoo. Account for this sound. (a) Compute order-of- magnitude estimates for the frequency, wavelength, and duration of the sound, on the basis of data you specify. (b) Each face of a great Mayan pyramid is like a steep stairway with very narrow steps. Can it produce an echo of a handclap that sounds like the call of a bird? Explain your answer.

Explain how the wave function P(r,t)=(25.0Pamr)sin(1.36rrad/m2030trad/s)P(r,t)=(25.0Pamr)sin(1.36rrad/m2030trad/s) can apply to a wave radiating from a small source, with rr being the radial distance from the center of the source to any point outside the source. Give the most detailed description of the wave that you can. Include answers to such questions as the following. Does the wave move more toward the right or the left? As it moves away from the source, what happens to its amplitude? Its speed? Its frequency? Its wavelength? Its power? Its intensity? What are representative values for each of these quantities? What can you say about the source of the wave? About the medium through which it travels?

For a certain type of steel, stress is always proportional to strain with Young's modulus as shown in Table 12.1.12.1. The steel has the density listed for iron in Table 14.1.14.1. It will fail by bending permanently if subjected to compressive stress greater than its yield strength y=y= 400 MPa. A rod 80.0 cmcm long, made of this steel, is fired at 12.0 m/sm/s straight at a very hard wall or at another identical rod moving in the opposite direction. (a) The speed of a one-dimensional compressional wave moving along the rod is given by v=Y/,v=Y/, where YY is Young's modulus for the rod and is the density. Calculate this speed. (b) After the front end of the rod hits the wall and stops, the back end of the rod keeps moving as described by of a one-dimensional compressional wave moving along the rod is given by v=Y/,v=Y/, where YY is Young's modulus for the rod and is the density. Calculate this speed. (b) After the front end of the rod hits the wall and stops, the back end of the rod keeps moving as described by

Prove that sound waves propagate with a speed given by Equation 17.1.17.1. Proceed as follows. In Active Figure 17.2,17.2, consider a thin, cylindrical layer of air in the cylinder, with face area AA and thickness xx . Draw a free-body diagram of this thin layer. Show that Fx=maxFx=max implies that

(P)xAx=Ax2st2(P)xAx=Ax2st2 By substituting P=B(s/x),P=B(s/x), derive the following wave equation for sound: B2sx2=2st2B2sx2=2st2 To a mathematical physicist, this equation demonstrates the existence of sound waves and determines their speed. As a physics student, you must take another step or two. Substitute into the wave equation the trial solution s(x,t)=s(x,t)= smaxcos(kxt)smaxcos(kxt) . Show that this function satisfies the wave equation provided that /k=B//k=B/ . This result reveals that sound waves exist provided they move with the speed v=f=(2f)(/2)=/k=B/v=f=(2f)(/2)=/k=B/

Equation 17.7 states that at distance rr away from a point source with power Pavg,Pavg, the wave intensity is

I=Pavg4r2I=Pavg4r2 Study Active Figure 17.9 and prove that at distance rr straight in front of a point source with power PargParg moving with constant speed vsvs the wave intensity is I=Pavg4r2(vvsv)I=Pavg4r2(vvsv)

The Doppler equation presented in the text is valid when the motion between the observer and the source occurs on a straight line so that the source and observer are moving either directly toward or directly away from each other. If this restriction is relaxed, one must use the more general Doppler equation f=(v+vOcosOvvScosS)ff=(v+vOcosOvvScosS)f where oo and ss are defined in Figure P17.62aP17.62a (a) Show that if the observer and source are moving directly away from each other, the preceding equation reduces to Equation 17.13 with negative values for both vovo and vSvS . (b) Use the preceding equation to solve the following problem. A train moves at a constant speed of 25.0 m/sm/s toward the intersection shown in Figure P17.62bP17.62b . A car is stopped near the crossing, 30.0 mm from the tracks. If the train's horn emits a frequency of 500 HzHz , what is the frequency heard by the passengers in the car when the train is 40.0 mm from the intersection? Take the speed of sound to be 343 m/sm/s .

A compact disc was modeled as a rigid object under constant angular acceleration to find the total angular displacement during the playing time of the disc. In reality, the angular acceleration of a disc is not constant. In this problem, let us explore the actual time dependence of the angular acceleration. (a) Assume the track on the disc is a spiral such that adjacent loops of the track are separated by a small distance $h$ . Show that the radius $r$ of a given portion of the track is given by r=ri+h2r=ri+h2 where $r_{i}$ is the radius of the innermost portion of the track and $\theta$ is the angle through which the disc turns to arrive at the location of the track of radius $r .$ (b) Show that the rate of change of the angle $\theta$ is given by ddt=vri+(h/2)ddt=vri+(h/2) where $v$ is the constant speed with which the disc surface passes the laser. (c) From the result in part (b), use integration to find an expression for the angle $\theta$ as a function of time. (d) From the result in part (c), use differentiation to find the angular acceleration of the disc as a function of time.

A cylindrical rod 24.0 $\mathrm{cm}$ long with mass 1.20 $\mathrm{kg}$ and radius 1.50 $\mathrm{cm}$ has a ball of diameter 8.00 $\mathrm{cm}$ and mass 2.00 $\mathrm{kg}$ attached to one end. The arrangement is originally vertical and stationary, with the ball at the top. The system is free to pivot about the bottom end of the rod after being given a slight nudge. (a) After the rod rotates through $90^{\circ},$ what is its rotational kinetic energy? (b) What is the angular speed of the rod and ball? (c) What is the linear speed of the ball? (d) How does this speed compare with the speed if the ball had fallen freely through the same distance of 28 $\mathrm{cm}$ ?