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physics
oscillations mechanical waves

Questions and Answers of Oscillations Mechanical Waves

Figure shows a solid half-cylinder of mass M and radius R resting on a horizontal surface. If one side of this cylinder is pushed down slightly and then released, the object will oscillate about its
Repeat Problem 123 replacing the half-cylinder with a half-sphere.
A straight tunnel is dug through the earth as shown in Figure. Assume that the walls of the tunnel are frictionless.(a) The gravitational force exerted by the earth on a particle of mass m at a
A damped oscillator has a frequency ώ’ that is 10% less than its undamped frequency.(a) By what factor is the amplitude of the oscillator decreased during each oscillation?(b) By what factor is
Show by direct substitution that Equation 14-48 is a solution of Equation 14-47. Equation 14-47 reads
A block of mass m on a horizontal table is attached to a spring of force constant k as shown in Figure. The coefficient of kinetic friction between the block and the table is mk. The spring is
In this problem, you will derive the expression for the average power delivered by a driving force to a driven oscillator (Figure).(a) Show that the instantaneous power input of the driving force is
In this problem, you are to use the result of Problem 129 to derive Equation 14-45, which relates the width of the resonance curve to the Q value when the resonance is sharp. At resonance, the
(Multiple choice)(1) True/false(a) In simple harmonic motion, the period is proportional to the square of the amplitude. (b) In simple harmonic motion, the frequency does not depend on the
(Multiple choice)(1) True or false:The motion of a simple pendulum is periodic for any initial angular displacement.(2)Two clocks have simple pendulums of identical lengths L. The pendulum of clock A
(Multiple choice)(1)A crystal wineglass shattered by an intense sound is an example of (a) Resonance. (b) Critical damping. (c) An exponential decrease in energy. (d)
(a) The bulk modulus for water is 2.0 x 109 N/m2. Use it to find the speed of sound in water.(b) The speed of sound in mercury is 1410 m/s. What is the bulk modulus for mercury (r = 13.6 x 103 kg/m3)?
Calculate the speed of sound waves in hydrogen gas at T = 300 K. (Take M = 2 g/mol and g = 1.4.)
A steel wire 7 m long has a mass of 100 g. It is under a tension of 900 N. What is the speed of a transverse wave pulse on this wire?
Transverse waves travel at 150 m/s on a wire of length 80 cm that is under a tension of 550 N. What is the mass of the wire?
A steel piano wire is 0.7 m long and has a mass of 5 g. It is stretched with a tension of 500 N.(a) What is the speed of transverse waves on the wire?(b) To reduce the wave speed by a factor of 2
The cable of a ski lift runs 400 m up a mountain and has a mass of 80 kg. When the cable is struck with a transverse blow at one end, the return pulse is detected 12 s later.(a) What is the speed of
A common method for estimating the distance to a lightning flash is to begin counting when the flash is observed and continue until the thunder clap is heard. The number of seconds counted is then
A method for measuring the speed of sound using an ordinary watch with a second hand is to stand some distance from a large flat wall and clap your hands rhythmically in such a way that the echo from
A man drops a stone from a high bridge and hears it strike the water below exactly 4 s later.(a) Estimate the distance to the water based on the assumption that the travel time for the sound to reach
(a) Compute the derivative of the speed of a wave on a string with respect to the tension dv/dF, and show that the differentials dv and dF obey dv/v = ½ dF/F.(b) A wave moves with a speed of 300 m/s
(a) Compute the derivative of the velocity of sound with respect to the absolute temperature, and show that the differentials dv and dT obey dv/v = ½ dT/T.(b) Use this result to compute the
In this problem, you will derive a convenient formula for the speed of sound in air at temperature t in Celsius degrees. Begin by writing the temperature as T = T0 + ∆T, where T0 = 273 K
While studying physics in her dorm room, a student is listening to a live radio broadcast of a baseball game. She is 1.6 km due south of the baseball field. Over her radio, the student hears a noise
A coiled spring, such as a Slinky, is stretched to a length L. It has a force constant k and a mass m.(a) Show that the velocity of longitudinal compression waves along the spring is given by v = L
Show explicitly that the following functions satisfy the wave equation:(a) y(x, t) = k(x + vt )3;(b) y(x ,t) = Aeik(x - vt) , where A and k are constants and i = √– 1 ; and(c) y(x, t) = ln k(x
Show that the function y = A sin kx cos wt satisfies the wave equation.
Consider the following equation: where a is a constant. Show that y(x, t) = A sin(kx - ?t) is not a solution of this equation but that the functions y(x, t) = Aei(kx - ?t) and y(x, t) = Aei(kx -
One end of a string 6 m long is moved up and down with simple harmonic motion at a frequency of 60 Hz. The waves reach the other end of the string in 0.5 s. Find the wavelength of the waves on the
Equation 15-13 expresses the displacement of a harmonic wave as a function of x and t in terms of the wave parameters k and ?. Write the equivalent expressions that contain the following pairs of
Equation 15-10 applies to all types of periodic waves, including electromagnetic waves such as light waves and microwaves, which travel at 3 x 108 m/s in a vacuum. (a) The range of wavelengths of
A harmonic wave on a string with a mass per unit length of 0.05 kg/m and a tension of 80 N has an amplitude of 5 cm. Each section of the string moves with simple harmonic motion at a frequency of 10
A rope 2 m long has a mass of 0.1 kg. The tension is 60 N. A power source at one end sends a harmonic wave frequency of the power source if the power transmitted is 100 W?
The wave function for a harmonic wave on a string is y(x, t) = (0.001 m) sin (62.8 m-1 x + 314 s-1t).(a) In what direction does this wave travel, and what is its speed?(b) Find the wavelength,
A harmonic wave with a frequency of 80 Hz and an amplitude of 0.025 m travels along a string to the right with a speed of 12 m/s.(a) Write a suitable wave function for this wave.(b) Find the maximum
Waves of frequency 200 Hz and amplitude 1.2 cm move along a 20-m string that has a mass of 0.06 kg and a tension of 50 N.(a) What is the average total energy of the waves on the string?(b) Find the
In a real string, a wave loses some energy as it travels down the string. Such a situation can be described by a wave function whose amplitude A(x) depends on x:y = A(x) (kx-
Power is to be transmitted along a stretched wire by means of transverse harmonic waves. The wave speed is 10 m/s, and the linear mass density of the wire is 0.01 kg/m. The power source oscillates
A sound wave in air produces a pressure variation given bywhere p is in pascals, x is in meters, and t is in seconds. Find(a) The pressure amplitude of the sound wave,(b) The
(a) Middle C on the musical scale has a frequency of 262 Hz. What is the wavelength of this note in air?(b) The frequency of the C an octave above middle C is twice that of middle C. What is the
(a) What is the displacement amplitude for a sound wave having a frequency of 100 Hz and a pressure amplitude of 104 atm?(b) The displacement amplitude of a sound wave of frequency 300 Hz is 107 m.
(a) Find the displacement amplitude of a sound wave of frequency 500 Hz at the pain-threshold pressure amplitude of 29 Pa.(b) Find the displacement amplitude of a sound wave with the same pressure
A typical loud sound wave with a frequency of 1 kHz has a pressure amplitude of about 10-4 atm.(a) At t = 0, the pressure is a maximum at some point x1. What is the displacement at that point at t =
(a) Find the displacement amplitude of a sound wave of frequency 500 Hz at the threshold-of-hearing pressure amplitude of 2.9 x 105 Pa.(b) Find the displacement amplitude of a wave of the same
A piston at one end of a long tube filled with air at room temperature and normal pressure oscillates with a frequency of 500 Hz and an amplitude of 0.1 mm. The area of the piston is 100 cm2.(a) What
A spherical source radiates sound uniformly in all directions. At a distance of 10 m, the sound intensity level is 104 W/m2.(a) At what distance from the source is the intensity 106 W/m2?(b) What
A loudspeaker at a rock concert generates 102 W/m2 at 20 m at a frequency of 1 kHz. Assume that the speaker spreads its energy uniformly in three dimensions.(a) What is the total acoustic power
When a pin of mass 0.1 g is dropped from a height of 1 m, 0.05% of its energy is converted into a sound pulse with a duration of 0.1 s.(a) Estimate the range at which the dropped pin can be heard if
Find the intensity of a sound wave if(a) b = 10 dB, and(b) b = 3 dB.(c) Find the pressure amplitudes of sound waves in air for each of these intensities.
Normal human speech has a sound intensity level of about 65 dB at 1 m. Estimate the power of human speech.
A spherical source radiates sound uniformly in all directions. At a distance of 10 m, the sound intensity level is 80 dB.(a) At what distance from the source is the intensity level 60 dB?(b) What
A spherical source of intensity I0 radiates sound uniformly in all directions. Its intensity level is b1 at a distance r1, and b2 at a distance r2. Find b2 /b1.
A loudspeaker at a rock concert generates 102 W/m2 at 20 m at a frequency of 1 kHz. Assume that the speaker spreads its energy uniformly in all directions.(a) What is the intensity level at 20 m?(b)
An article on noise pollution claims that sound intensity levels in large cities have been increasing by about 1 dB annually.(a) To what percentage increase in intensity does this correspond? Does
Three noise sources produce intensity levels of 70, 73, and 80 dB when acting separately. When the sources act together, their intensities add.(a) Find the sound intensity level in decibels when the
The equation I = Pav/4πr2 is predicated on the assumption that the transmitting medium does not absorb any energy. It is known that absorption of sound by dry air results in a decrease of intensity
When a violinist pulls the bow across a string, the force with which the bow is pulled is fairly small, about 0.6 N. Suppose the bow travels across the A string, which vibrates at 440 Hz, at 0.5 m/s.
The noise level in an empty examination hall is 40 dB. When 100 students are writing an exam, the sounds of heavy breathing and pens traveling rapidly over paper cause the noise level to rise to 60
Stars often occur in pairs revolving around their common center of mass. If one of the stars is a black hole, it is invisible. Explain how the existence of such a black hole might be inferred from
A conveyor belt moves to the right with a speed v = 300 m/min. A very fast piemaker puts pies on the belt at a rate of 20 per minute, and they are received at the other end by a pie eater.(a) If the
For the situation described in Problem 63, derive general expressions for the spacing of the pies l and the frequency f with which they are received by the pie eater in terms of the speed of the belt
The sound source described above moves with a speed of 80 m/s relative to still air toward a stationary listener.(a) Find the wavelength of the sound between the source and the listener.(b) Find the
Consider the situation in Problem 65 from the reference frame in which the source is at rest. In this frame, the listener moves toward the source with a speed of 80 m/s, and there is a wind blowing
The source moves away from the stationary listener at 80 m/s.(a) Find the wavelength of the sound waves between the source and the listener.(b) Find the frequency heard by the listener.
The listener moves at 80 m/s relative to still air toward the stationary source.(a) What is the wavelength of the sound between the source and the listener?(b) What is the frequency heard by the
Consider the situation in Problem 68 in a reference frame in which the listener is at rest.(a) What is the wind velocity in this frame?(b) What is the speed of the sound from the source to the
The listener moves at 80 m/s relative to the still air away from the stationary source. Find the frequency heard by the listener.
A jet is traveling at Mach 2.5 at an altitude of 5000 m.(a) What is the angle that the shock wave makes with the track of the jet? (Assume that the speed of sound at this altitude is still 340
If you are running at top speed toward a source of sound at 1000 Hz, estimate the frequency of the sound that you hear. Suppose that you can recognize a change in frequency of 3%. Can you use your
A radar device emits microwaves with a frequency of 2.00 GHz. When the waves are reflected from a car moving directly away from the emitter, a frequency difference of 293 Hz is detected. Find the
A stationary destroyer is equipped with sonar that sends out pulses of sound at 40 MHz. Reflected pulses are received from a submarine directly below with a time delay of 80 ms at a frequency of
Two airplanes, one flying due east and the other due west, are on a near collision course separated by 15 km when the pilot of one plane, traveling at 900 km/h, observes the other on his Doppler
Suppose the police car of Problem 76 is moving in the same direction as the other vehicle at a speed of 60 km/h. What then is the difference in frequency between the emitted and the reflected signals?
At time t = 0, a supersonic plane is directly over point P flying due west at an altitude of 12 km and a speed of Mach 1.6. Where is the plane when the sonic boom is heard?
A small radio of 0.10 kg mass is attached to one end of an air track by a spring. The radio emits a sound of 800 Hz. A listener at the other end of the air track hears a sound whose frequency varies
A sound source of frequency f0 moves with speed us relative to still air toward a receiver who is moving with speed ur relative to still air away from the source.(a) Write an expression for the
Two students with vibrating 440-Hz tuning forks walk away from each other with equal speeds. How fast must they walk so that they each hear a frequency of 438 Hz from the other fork?
A physics student walks down a long hall carrying a vibrating 512-Hz tuning fork. The end of the hall is closed so that sound reflects from it. The student hears a sound of 516 Hz from the wall. How
A small speaker radiating sound at 1000 Hz is tied to one end of an 0.8-m-long rod that is free to rotate about its other end. The rod rotates in the horizontal plane at 4.0 rad/s. Derive an
You have won a free trip on the Queen Elizabeth II and are in mid-Atlantic steaming due east at 45 km/h as the Concorde passes directly overhead flying due west at Mach 1.6 at an altitude of 12,500
A balloon driven by a 36-km/h wind emits a sound of 800 Hz as it approaches a tall building.(a) What is the frequency of the sound heard by an observer at the window of this building?(b) What is the
A car is approaching a reflecting wall. A stationary observer behind the car hears a sound of frequency 745 Hz from the car horn and a sound of frequency 863 Hz from the wall.(a) How fast is the car
The driver of a car traveling at 100 km/h toward a vertical cliff briefly sounds the horn. Exactly one second later she hears the echo and notes that its frequency is 840 Hz. How far from the cliff
You are on a transatlantic flight traveling due west at 800 km/h. A Concorde flying at Mach 1.6 and 3 km to the north of your plane is also on an east-to-west course. What is the distance between the
Astronomers can deduce the existence of a binary star system even if the two stars cannot be visually resolved by noting an alternating Doppler shift of a spectral line. Suppose that an astronomical
A physics student drops a vibrating 440-Hz tuning fork down the elevator shaft of a tall building. When the student hears a frequency of 400 Hz, how far has the tuning fork fallen?
When a guitar string is plucked, is the wavelength of the wave it produces in air the same as the wavelength of the wave on the string?
Figure shows a wave pulse at time t = 0 moving to the right. At this particular time, which segments of the string are moving up? Which are moving down? Is there any segment of the string at the
Make a sketch of the velocity of each string segment versus position for the pulse shown inFigure.
Consider a long line of cars equally spaced by one car length and moving slowly with the same speed. One car suddenly slows to avoid a dog and then speeds up until it is again one car length behind
At time t = 0, the shape of a wave pulse on a string is given by the functionwhere x is in meters.(a) Sketch y(x, 0) versus x. Give the wave function y(x, t) at a general time t if(b)
A wave with frequency of 1200 Hz propagates along a wire that is under a tension of 800 N. The wavelength of the wave is 24 cm. What will be the wavelength if the tension is decreased to 600 N and
A boat traveling at 10 m/s on a still lake makes a bow wave at an angle of 20o with its direction of motion. What is the speed of the bow wave?
A whistle of frequency 500 Hz moves in a circle of radius 1 m at 3rev/s. What are the maximum and minimum frequencies heard by a stationary listener in the plane of the circle and 5 m away from its
Ocean waves move toward the beach with a speed of 8.9 m/s and a crest-to-crest separation of 15.0 m. You are in a small boat anchored off shore.(a) What is the frequency of the ocean waves?(b) You
Two connected wires with linear mass densities that are related by μ1 = 3μ2 are under the same tension. When the wires oscillate at a frequency of 120 Hz waves of wavelength 10 cm travel down the
A 12.0-m wire of mass 85 g is stretched under a tension of 180 N. A pulse is generated at the left end of the wire, and 25 ms later a second pulse is generated at the right end of the wire. Where do
A harmonic wave moves down a string with speed 12.4 m/s. A particle on the string has a maximum displacement of 4.5 cm and a maximum speed of 9.4 m/s. Find(a) The wavelength of the wave, and(b) The
Find the speed of a car for which the tone of its horn will drop by 10% as it passes you. Let a = us/v.
A loudspeaker diaphragm 20 cm in diameter is vibrating at 800 Hz with an amplitude of 0.025 mm. Assuming that the air molecules in the vicinity have this same amplitude of vibration, find(a) The

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