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Principles of Physics: A Calculus-Based Text

5th Edition

ISBN:9781133104261

Author:Raymond A. Serway, John W. Jewett

Publisher:Raymond A. Serway, John W. Jewett

Chapter14: Superposition And Standing Waves

Section: Chapter Questions

Problem 31P

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Similar questions

Table 17.1 shows the speed of sound is typically an order of magnitude larger in solids than in gases. To what can this higher value be most directly attributed? (a) the difference in density between solids and gases (b) the difference in compressibility between solids and gases (c) the limited size of a solid object compared to a free gas (d) the impossibility of holding a gas under significant tension
As shown in Figure P14.37, water is pumped into a tall, vertical cylinder at a volume flow rate R. The radius of the cylinder is r, and at the open top of the cylinder a tuning fork is vibrating with a frequency f. As the water rises, what time interval elapses between successive resonances? Figure P14.37 Problems 37 and 38.
Review. As a sound wave passes through a gas, the compressions are either so rapid or so far apart that thermal conduction is prevented by a negligible time interval or by effective thickness of insulation. The compressions and rarefactions are adiabatic. (a) Show that the speed of sound in an ideal gas is v=RTM where M is the molar mass. The speed of sound in a gas is given by Equation 16.35; use that equation and the definition of the bulk modulus from Section 12.4. (b) Compute the theoretical speed of sound in air at 20.0C and state how it compares with the value in Table 16.1. Take M = 28.9 g/mol. (c) Show that the speed of sound in an ideal gas is v=kBTm0 where m0 is the mass of one molecule. (d) Slate how the result in part (c) compares with the most probable, average, and rms molecular speeds.
Explain why you expect an object made of a stiff material to vibrate at a higher frequency than a similar object made of a spongy material.
Review. A steel guitar string with a diameter of 1.00 mm is stretched between supports 80.0 cm apart. The temperature is 0.0C. (a) Find the mass per unit length of this siring. (Use the value 7.86 103 kg/m4 for the density.) (b) The fundamental frequency of transverse oscillations of the string is 200 Hz. What is the tension in the string? Next, the temperature is raised to 30.0C. Find the resulting values of (c) the tension and (d) the fundamental frequency. Assume both the Youngs modulus of 20.0 1010 N/m2 and the average coefficient of expansion = 11.0 10-6 (C)-1 have constant values between 0.0C and 30.0C.
In Section 16.7, we derived the speed of sound in a gas using the impulsemomentum theorem applied to the cylinder of gas in Figure 16.20. Let us find the speed of sound in a gas using a different approach based on the element of gas in Figure 16.18. Proceed as follows. (a) Draw a force diagram for this element showing the forces exerted on the left and right surfaces due to the pressure of the gas on either side of the element. (b) By applying Newtons second law to the element, show that (P)xAx=Ax2st2 (c) By substituting P = (B s/x) (Eq. 16.30), derive the following wave equation for sound: B2sx2=2st2 (d) 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) = smax cos (kx t). Show that this function satisfies the wave equation, provided /k=v=B/.
A dolphin (Fig. P17.7) in seawater at a temperature of 25C emits a sound wave directed toward the ocean floor 150 m below. How much time passes before it hears an echo?
Parcels of air (small volumes of air) in a stable atmosphere (where the temperature increases with height) can oscillate up and down, due to the restoring force provided by the buoyancy of the air parcel. The frequency of the oscillations are a measure of the stability of the atmosphere. Assuming that the acceleration of an air parcel can be modeled as 2zt2=g(z)0zz , prove that z=z0etN2 is a solution, where N is known as the Brunt-Väisälä frequency. Note that in a stable atmosphere, the density decreases with height and parcel oscillates up and down.
By what fraction will the frequencies produced by a wind instrument change when air temperature goes from 10.0°C to 30.0°C? That is, find the ratio of the frequencies at those temperatures.
A cable with a linear density of =0.2 kg/m is hung from telephone poles. The tension in the cable is 500.00 N. The distance between poles is 20 meters. The wind blows across the line, causing the cable resonate. A standing waves pattern is produced that has 4.5 wavelengths between the two poles. The air temperature is T=20C . What are the frequency and wavelength of the hum?
Consider the simplified single-piston engine in Figure CQ15.13. Assuming the wheel rotates with constant angular speed, explain why the piston rod oscillates in simple harmonic motion.
In about 1657, Otto von Guericke, inventor of the air pump, evacuated a sphere made of two brass hemispheres (Fig. P15.62). Two teams of eight horses each could pull the hemispheres apart only on some trials and then with greatest difficulty, with the resulting sound likened to a cannon firing. Find the force F required to pull the thin-walled evacuated hemispheres apart in terms of R, the radius of the hemispheres; P, the pressure inside the hemispheres; and atmospheric pressure P0. Figure P15.62

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Do wind instruments warm up by becoming more sharp or flat? What about string instruments? Provide justification.