Suppose that f is a positive function such that f’ is continuous.
(a) How is the graph of y = f(x) sin nx related to the graph of y = f(x) What happens as n → ∞?
(b) Make a guess as to the value of the limit
based on graphs of the integrand.
(c) Using integration by parts, confirm the guess that you made in part (b). [Use the fact that, since f’ is continuous, there is a constant M such that | f'(x) | ≤ M for 0 ≤ x ≤ 1.]
Jimmy has caught two fish in Yellow Creek. He has tied the line holding the 3.0 kg steelhead trout to the tail of the 1.5 kg carp. To show the fish to a friend, he lifts upward on the carp with a force of 60 N.
a. Draw separate free-body diagrams for the trout and the carp. Label all forces, then use dashed lines to connect action/reaction pairs or forces that act as if they are a pair.
b. Rank in order, from largest to smallest, the magnitudes of all the forces shown on your free-body diagrams. Explain your reasoning.
a. A large nuclear power plant has a power output of 1000 MW. In other words, it generates electric energy at the rate 1000 MJ/s. How much energy does this power plant supply in one day?
b. The oceans are vast. How much energy could be extracted from 1 km3 of water if its temperature were decreased by 1°C? For simplicity, assume fresh water.
c. A friend of yours who is an inventor comes to you with an idea. He has done the calculations that you just did in parts a and b, and he’s concluded that a few cubic kilometers of ocean water could meet most of the energy needs of the United States. This is an insignificant fraction of the U.S. coastal waters. In addition, the oceans are constantly being reheated by the sun, so energy obtained from the ocean is essentially solar energy. He has sketched out some design plans—highly secret, of course, because they’re not patented—and now he needs some investors to provide money for a prototype. A working prototype will lead to a patent. As an initial investor, you’ll receive a fraction of all future royalties. Time is of the essence because a rival inventor is working on the same idea. He needs $10,000 from you right away. You could make millions if it works out. Will you invest? If so, explain why. If not, why not? Either way, your explanation should be based on scientific principles.
An inventor wants you to invest money with his company, offering you 10% of all future profits. He reminds you that the brakes on cars get extremely hot when they stop and that there is a large quantity of thermal energy in the brakes. He has invented a device, he tells you, that converts that thermal energy into the forward motion of the car. This device will take over from the engine after a stop and accelerate the car back up to its original speed, thereby saving a tremendous amount of gasoline. Now, you’re a smart person, so he admits up front that this device is not 100% efficient, that there is some unavoidable heat loss to the air and to friction within the device, but the upcoming research for which he needs your investment will make those losses extremely small. You do also have to start the car with cold brakes after it has been parked awhile, so you’ll still need a gasoline engine for that. Nonetheless, he tells you, his prototype car gets 500 miles to the gallon and he expects to be at well over 1000 miles to the gallon after the next phase of research. Should you invest? Base your answer on an analysis of the physics of the situation.
Ships measure the distance to the ocean bottom with sonar. A pulse of sound waves is aimed at the ocean bottom, then sensitive microphones listen for the echo. Figure P20.45 shows the delay time as a function of the ship’s position as it crosses 60 km of ocean. Draw a graph of the ocean bottom. Let the ocean surface define y = 0 and ocean bottom have negative values of y. This way your graph will be a picture of the ocean bottom. The speed of sound in ocean water varies slightly with temperature, but you can use 1500 m/s as an average value.
The broadcast antenna of an AM radio station is located at the edge of town. The station owners would like to beam all of the energy into town and none into the countryside, but a single antenna radiates energy equally in all directions. Figure CP21.80 shows two parallel antennas separated by distance L. Both antennas broadcast a signal at wavelength l, but antenna 2 can delay its broadcast relative to antenna 1 by a time interval Δt in order to create a phase difference ΔΦ0 between the sources. Your task is to find values of L and Δt such that the waves interfere constructively on the town side and destructively on the country side.
Let antenna 1 be at x = 0. The wave that travels to the right is asin[2π(x/λ – t/T)]. The left wave is a sin[2π(–x/λ – t/T)]. (It must be this, rather than a sin[2π(x/λ + t/T)], so that the two waves match at x = 0. ) Antenna 2 is at x = L. It broadcasts wave a sin[2π((x – L)/λ – t/T) + Φ20] to the right and wave a sin[2π(–(x – L)/λ – t/T) + Φ20] to the left.
a. What is the smallest value of L for which you can create perfect constructive interference on the town side and perfect destructive interference on the country side? Your answer will be a multiple or fraction of the wavelength λ .
b. What phase constant Φ20 of antenna 2 is needed?
c. What fraction of the oscillation period T must Δt be to produce the proper value of Φ20?
d. Evaluate both L and Δt for the realistic AM radio frequency of 1000 KHz.
Two radio antennas are 100 m apart along a north-south line. They broadcast identical radio waves at a frequency of 3.0 MHz. Your job is to monitor the signal strength with a handheld receiver. To get to your first measuring point, you walk 800 m east from the midpoint between the antennas, then 600 m north.
a. What is the phase difference between the waves at this point?
b. Is the interference at this point maximum constructive, perfect destructive, or somewhere in between? Explain.
c. If you now begin to walk farther north, does the signal strength increase, decrease, or stay the same? Explain.
Two loudspeakers face each other from opposite walls of a room. Both are playing exactly the same frequency, thus setting up a standing wave with distance λ/2 between antinodes. Assume that λ is much less than the room width, so there are many antinodes.
a. Yvette starts at one speaker and runs toward the other at speed vY. As the does so, she hears a loud-soft-loud modulation of the sound intensity. From your perspective, as you sit at rest in the room, Yvette is running through the nodes and antinodes of the standing wave. Find an expression for the number of sound maxima she hears per second.
b. From Yvette’s perspective, the two sound waves are Doppler shifted. They’re not the same frequency, so they don’t create a standing wave. Instead, she hears a loud-soft-loud modulation of the sound intensity because of beats. Find an expression for the beat frequency that Yvette hears.
c. Are your answers to parts a and b the same or different? Should they be the same or different?
An important characteristic of the heart, one used to diagnose heart disease, is the pressure difference between the blood pressure inside the heart and the blood pressure in the aorta, the large artery leading away from the heart. The blood inside the heart is essentially at rest, but it speeds up significantly as it enters the aorta—and its speed can be measured by using the Doppler shift of reflected ultrasound.
a. The Doppler effect enters twice in calculating the frequency of the reflection from a moving object. Suppose the object’s speed v0 is very small compared to the wave speed v. Show that a good approximation for the Doppler shift—the difference between the reflected frequency and the incident frequency—is
b. A doctor using 2.5 MHz ultrasound measures a 6000 Hz Doppler shift as the ultrasound reflects from blood ejected from the heart into the aorta. What is the blood pressure difference, in mm of Hg, between the inside of the heart and the aorta? Assume the patient is lying down so that there is no height difference between the heart and the aorta. The density of blood is 1060 kg/m3.
A string with linear density 2.0 g/m is stretched along the positive x-axis with tension 20 N. One end of the string, at x = 0 m, is tied to a hook that oscillates up and down at a frequency of 100 Hz with a maximum displacement of 1.0 mm. At t = 0 s, the hook is at its lowest point.
a. What are the wave speed on the string and the wavelength?
b. What are the amplitude and phase constant of the wave?
c. Write the equation for the displacement D(x, t) of the traveling wave.
d. What is the string’s displacement at x = 0.50 m and t = 15 ms?