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physics
oscillations mechanical waves
Questions and Answers of
Oscillations Mechanical Waves
Two sources have a phase difference δ0 that is proportional to time: δ0 = Ct, where C is a constant. The amplitude of the wave from each source at some point P is A0.(a) Write the wave functions
The wave functions of two standing waves on a string of length L arey1(x, t) = A1 cos ώ1 t sin k1xandy2(x, t) = A2 cos ώ2 t sin k2x, where kn = nπ/L, and ώn =
A 2-m wire fixed at both ends is vibrating in its fundamental mode. The tension in the wire is 40 N and the mass of the wire is 0.1 kg. At the midpoint of the wire, the amplitude is 2 cm.(a) Find the
A string 3.2 m long and with a linear mass density of 0.008 kg/m is kept under tension so that traveling waves propagate at 48 m/s along the string. The ends of the string are clamped and the string
(Multiple choice)(1) True / False(a) The waves from two coherent sources that are radiating in phase interfere constructively everywhere in space. (b) Two wave sources that are out of phase by
Show that the periods of a pendulum bob are the same for linear oscillation and circular motion (for small displacements).
Two bodies of masses m and 3m are attached to each other and to two fixed points by three identical light springs as shown in figure The whole arrangement rests on a smooth horizontal table. The
Two bodies of mass M1 and M2 are connected by a spring but are otherwise free to move along a horizontal line. A periodic force F cos wt is exerted on the body of mass M1 along the ling. Show that
Derive the equation of motion of a particle of mass m subject to restoring and frictional forces of magnitude kx and b dx/dt respectively, where x is its displacement and k and b are positive
A steady force of 40N is required to lift a mass of 2kg vertically through water at a constant speed of ms?1. Assuming that the effect of viscosity can be described by a force proportional to
The equation of motion for a damped simple harmonic oscillator is ?where k and ? are constants, m is the mass and x is the displacement of the system. Describe the conditions for lightly damped,
A particle of mass m moves in one dimension and is subject to a restoring force which is proportional to its displacement x and a damping force which is proportional to its velocity. Derive the
Show that the steady state complex amplitude of a damped oscillator driven by an external force F exp (iwt) is given by the expression Explain the meanings of the symbols used and the condition for
Estimate the quality factor Q of the bell ‘Big Ben’.
Explain why the displacement A sin (k.r – wt) at position r and time t is called a plane wave. In which direction does it travel? A wave displacement is y = 0.5 sin (0.1 x – 0.4t), where all
A wave is represented by Ψ1 = 10cos (5x + 25t), where x is measured in metres and t in seconds. Show that this represents a travelling wave and deduce its wavelength, frequency, speed and direction
A uniform inextensible string of length I and total mass M is suspended vertically and tapped at the top end so that a transverse impulse runs down it. At the same moment a body is released from rest
A long string of mass per unit length 0.2kg m–1 is stretched to a tension of 500 M. Find the speed of transverse waves on the string and the mean power required to maintain a travelling wave of
The phase velocity of a surface wave on a liquid of surface tension T and density ρ is where g is the acceleration due to gravity and ? is the wavelength of the wave. Find the group velocity vg of
The phase velocity v of gravity waves in a liquid of depth h is given by the formulav2 = g/k tanh kh,where g is the acceleration of free fall and k = 2π/λ is the wave number, λ being the wave
Prove that the group velocity vg of electromagnetic waves in dispersive medium with refractive index n is given by where c is the free-space velocity of light and w is the angular frequency of the
1. Deezo the Clown slept in again. As he roller-skates out the door at breakneck speed on his way to a lunchtime birthday party, his superelastic suspenders catch on a fence post, and he flies back
1. If the amplitude of a simple harmonic oscillator is tripled, by what factor is the energy changed?2. A 2.4-kg object is attached to a horizontal spring of force constant k = 4.5 kN/m. The spring
1. The length of the string or wire supporting a pendulum increases slightly when its temperature is raised. How would this affect a clock operated by a simple pendulum?2. An object oscillates on a
1. Show that the total energy of a simple pendulum undergoing oscillations of small amplitude ф0 is approximately E ≈ 1/2mgLф02.2. What would be the period of the pendulum in Problem 50
1. The effect of the mass of a spring on the motion of an object attached to it is usually neglected. Describe qualitatively its effect when it is not neglected.2. Show that the dampening constant,
1. A rope hangs vertically from the ceiling. Do waves on the rope move faster, slower, or at the same speed as they move from bottom to top? Explain.2. A rope hangs vertically. You shake the bottom
1. The sound level of a dog's bark is 50 dB. The intensity of a rock concert is 10,000 times that of the dog's bark. What is the sound level of the rock concert?2. Show that if the intensity is
1. Everyone at a party is talking equally loudly. If only one person were talking, the sound level would be 72 dB. Find the sound level when all 38 people are talking.2. If the source and receiver
A police radar unit transmits microwaves of frequency 3 × 1010 Hz. The speed of these waves in air is 3.0 × 108 m/s. Suppose a car is receding from the stationary police car at a speed of 140 km/h.
In a common lecture demonstration of wave pulses, a piece of rubber tubing is tied at one end to a fixed post and is passed over a pulley to a weight hanging at the other end. Suppose that the
1. Two violinists standing a few feet apart are playing the same notes. Are there places in the room where certain notes are not heard because of destructive interference? Explain.2. What is the
1. Two waves having the same frequency, wavelength, and amplitude are traveling in the same direction. If they differ in phase by π/2 and each has an amplitude of 0.05 m, what is the amplitude of
1. Two tuning forks have frequencies of 256 and 260 Hz. If the forks are vibrating at the same time, what is the beat frequency?2. How do the resonance frequencies of an organ pipe change when the
1. 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?2. When two waves interfere constructively or destructively,
What is the maximum acceleration of a platform that oscillates at amplitude 2.20 cm and frequency 6.60 Hz?
A loudspeaker produces a musical sound by means of the oscillation of a diaphragm whose amplitude is limited to 1.00 µm.(a) At what frequency is the magnitude a of the diaphragm's acceleration equal
In Figure, the pendulum consists of a uniform disk with radius r = 10.0 cm and mass 500 g attached to a uniform rod with length L = 500 mm and mass 270 g.(a) Calculate the rotational inertia of the
(a) If the physical pendulum of Figure and the associated sample problem is inverted and suspended at point P, what is its period of oscillation?(b) Is the period now greater than, less than, or
A rectangular block, with face lengths a = 35 cm and b = 45 cm, is to be suspended on a thin horizontal rod running through a narrow hole in the block. The block is then to be set swinging about the
In a damped oscillator with m = 250 g, k = 85 N/m, and b = 70 g/s, what is the ratio of the amplitude of the damped oscillations to the initial amplitude at the end of 20 cycles?
For Eq.15-45, suppose the amplitude xm is given bywhere Fm is the (constant) amplitude of the external oscillating force exerted on the spring by the rigid support in Figure.At resonance, what are
Hanging from a horizontal beam are nine simple pendulums of the following lengths:(a) 0.10,(b) 0.30,(c) 0.40,(d) 0.80,(e) 1.2,(f) 2.8,(g) 3.5,(h) 5.0, and(i) 6.2 m.Suppose the beam undergoes
In Figure, three 10,000 kg ore cars are held at rest on a mine railway using a cable that is parallel to the rails, which are inclined at angle θ = 30°. The cable stretches 15 cm just before the
A massless spring with spring constant 19 N/m hangs vertically. A body of mass 0.20 kg is attached to its free end and then released. Assume that the spring was unstretched before the body was
A 55.0 g block oscillates in SHM on the end of a spring with k = 1500 N/m according to x = xm cos(ωt + Ф). How long does the block take to move from position +0.800xm to(a) Position +0.600xm and(b)
A flat uniform circular disk has a mass of 3.00 kg and a radius of 70.0 cm. It is suspended in a horizontal plane by a vertical wire attached to its center. If the disk is rotated 2.50 rad about the
A wave has an angular frequency of 110 rad/s and a wavelength of 1.80 m. Calculate(a) The angular wave number and(b) The speed of the wave.
At time t = 0 and at position x = 0 m along a string, a traveling sinusoidal wave with an angular frequency of 440 rad/s has displacement y = +4.5 mm and transverse velocity u = – 0.75 m/s. If the
A traveling wave on a string is described bywhere x and y are in centimeters and t is in seconds.(a) For t = 0, plot y as a function of x for 0 ¤ x ¤ 160 cm.(b) Repeat (a)
In Figure sound waves A and B, both of wavelength λ, are initially in phase and traveling rightward, as indicated by the two rays. Wave A is reflected from four surfaces but ends up traveling in its
Figure shows an air-filled, acoustic interferometer, used to demonstrate the interference of sound waves. Sound source S is an oscillating diaphragm; D is a sound detector, such as the ear or a
A 1.50 µF capacitor is connected as in Figure to an ac generator with ξm = 30.0 V. What is the amplitude of the resulting alternating current if the frequency of the emf is(a) 1.00 kHz and(b) 8.00
Figure shows an ac generator connected to a "black box" through a pair of terminals. The box contains an RLC circuit, possibly even a multiloop circuit, whose elements and connections we do not
In Figure, let the rectangular box on the left represent the (high-impedance) output of an audio amplifier, with r = 1000 Ω. Let R = 10 Ω represent the (low-impedance) coil of a
Consider the circuit shown in Figure. With switch S1 closed and the other two switches open, the circuit has a time constant τC. With switch S2 closed and the other two switches open, the
Light is an electromagnetic wave and travels at a speed of 3.00 × 108 m/s. The human eye is most sensitive to yellow-green light, which has a wavelength of 5.45 × 10-7 m. What is the frequency of
In Figure (c) the hand moves the end of the Slinky up and down through two complete cycles in one second. The wave moves along the Slinky at a speed of 0.50 m /s. find the distance between two
The middle C string on a piano is under a tension of 944 N. The period and wavelength of a wave on this string are 3.82 ms and 1.26 m, respectively. Find the linear density of the string.
The drawing shows two transverse waves traveling on strings. The linear density of each string is 0.065 kg/m. The tension is provided by a 26-N block that is hanging from the string. Find the speed
A wave traveling in the +x direction has an amplitude of 0.35 m, a speed of 5.2 m/s, and a frequency of 14 Hz. Write the equation of the wave in the form given by either Equation.
The tension in a string is 15 N, and its linear density is 0.85 kg/m. A wave on the string travels toward the -x direction; it has amplitude of 3.6 cm and a frequency of 12 Hz. What are they?(a)
For research purposes a sonic buoy is tethered to the ocean floor and emits an infrasonic pulse of sound (speed = 1522 m/s). The period of this sound is 71ms. Determine the wavelength of the sound.
The speed of a sound in a container of hydrogen at 201 K is 1220 m/s. What would be the speed of sound if the temperature were raised to 405 K? Assume that hydrogen behaves like an ideal gas.
A hunter is standing on flat ground between two vertical cliffs that are directly opposite one another. He is closer to one cliff than to the other. He fires a gun and, after a while, hears three
When an earthquake occurs, two types of sound waves are generated and travel through the earth. The primary, or P, wave has a speed of about 8.0 km/s and the secondary, or S, wave has a speed of
As a prank, someone drops a water-filled balloon out of a window. The balloon is released from rest at a height of 10.0 m above the ears of a man who is the target. Then, because of a guilty
At a distance of 3.8 m from a siren, the sound intensity is 3.6 × 10-2 W/m2. Assuming that the siren radiates sound uniformly in all directions, find the total power radiated.
A loudspeaker has a circular opening with a radius of 0.0950 m. The electrical power needed to operate the speaker is 25.0 W. The average sound intensity at the opening is 17.5 W/m2. What percentage
A dish of lasagna is being heated in a microwave oven. The effective area of the lasagna that is exposed to the microwaves is 2.2 × 10-2 m2. The mass of the lasagna is 0.35 kg, and its specific heat
A middle-aged man typically has poorer hearing than a middle-aged woman. In one case, a woman can just begin to hear a musical tone, while a man can just begin to hear the tone only when its
When one person shouts at a football game, the sound intensity level at the center of the field is 60.0 dB. When all the people shout together, the intensity level increases to 109 dB. Assuming that
Suppose that when a certain sound intensity level (in dB) triples, the sound intensity (in W/m2) also triples. Determine this sound intensity level.
From a vantage point very close to the track at a stock car race, you hear the sound emitted by a moving car. You detect a frequency that is 0.86 times as small as the frequency emitted by the car
Two trucks travel at the same speed. They are far apart on adjacent lanes and approach each other essentially head-on. One driver hears the horn of the other truck at a frequency that is 1.14 times
Two submarines are under water and approaching each other head-on. Sub A has a speed of 12 m/s and sub B has a speed of 8 m/s. Sub A sends out a 1550-Hz sonar wave that travels at a speed of 1522
At 20° C the densities of fresh water and ethyl alcohol are, respectively, 998 and 789 kg/m3. Find the ratio of the adiabatic bulk modulus of fresh water to the adiabatic bulk modulus of ethyl
Suppose that the linear density of the A string on a violin is 7.8 × 10-4 kg/m. A wave on the string has a frequency of 440 Hz and a wavelength of 65 cm. What is the tension in the string?
When Gloria wears her hearing aid, the sound intensity level increases by 30.0 dB. By what factor does the sound intensity increase?
Suppose that the amplitude and frequency of the transverse wave in Figure (c) are, respectively, 1.3 cm and 5.0 Hz. Find the total vertical distance (in cm) through which the colored dot moves in
You are riding your bicycle directly away from a stationary source of sound and hear a frequency that is 1.0% lower than the emitted frequency. The speed of sound is 343 m/s. What is your speed?
The sound intensity level at a rock concert is 115 dB, while that at a jazz fest is 95 dB. Determine the ratio of the sound intensity at the rock concert to the sound intensity at the jazz fest.
Two blocks are connected by a wire that has a mass per unit length of 8.50 × 10-4 kg/m. One block has a mass of 19.0 kg, and the other has a mass of 42.0 kg. These blocks are being pulled across a
The drawing graphs a string on which two rectangular pulses are traveling at a constant speed of 1 cm/s at time t = 0 s. Using the principle of linear superposition, draw the shape of the string at t
Two waves are traveling in opposite directions on the same string. The displacements caused by the individual waves are given by y1 = (24.0 mm) sin (9.00πt – 1.25πx) and y2 = (35.0 mm) sin
The drawing shows a loudspeaker A and point C, where a listener is positioned. A second loudspeaker B is located somewhere to the right of loudspeaker A. Both speakers vibrate in phase and are
Sound exits a diffraction horn loudspeaker through a rectangular opening like a small doorway. Such a loudspeaker is mounted outside on a pole. In winter, when the temperature is 273 K, the
Two pure tones are sounded together. The drawing shows the pressure variations of the two sound waves, measured with respect to atmospheric pressure. What is the beatfrequency?
Two out-of-tune flutes play the same note. One produces a tone that has a frequency of 262 Hz, while the other produces 266 Hz. When a tuning fork is sounded together with the 262-Hz tone, a beat
The approach to solving this problem is similar to that taken in Multiple-Concept Example. On a cello, the string with the largest linear density (1.56 × 10-2 kg/m) is the C string. This string
Suppose that the strings on a violin are stretched with the same tension and each has the same length between its two fixed ends. The musical notes and corresponding fundamental frequencies of two of
The E string on an electric bass guitar has a length of 0.628 m and, when producing the note E, vibrates at a fundamental frequency of 41.2 Hz. Players sometimes add to their instruments a device
The note that is three octaves above middle C is supposed to have a fundamental frequency of 2093 Hz. On a certain piano, the steel wire that produces this note has a cross-sectional area 7.85 ×
Sound enters the ear, travels through the auditory canal, and reaches the eardrum. The auditory canal is approximately a tube open at only one end. The eardrum closes the other end. A typical length
A tube is open only at one end. A certain harmonic produced by the tube has a frequency of 450 Hz. The next higher harmonic has a frequency of 750 Hz. The speed of sound in air is 343 m/s.(a) What is
A person hums into the top of a well and finds that standing waves are established at frequencies of 42, 70.0, and 98 Hz. The frequency of 42 Hz is not necessarily the fundamental frequency. The
Review Example in the text. Speaker A is moved further to the left, while ABC remains a right triangle. What is the separation between the speakers when constructive interference occurs again at
The A string on a string bass vibrates at a fundamental frequency of 55.0 Hz. If the string’s tension were increased by a factor of four, what would be the new fundamental frequency?
The arrangement in the drawing shows a block (mass = 15.0 kg) that is held in position on a frictionless incline by a cord (length = 0.600 m). The mass per unit length of the cord is 1.20 × 10-2
The pendulum on a grandfather clock is calibrated so its period equals 1.00 s. (a) What is the length of the pendulum cable? (b) If the grandfather clock is moved to the moon where the acceleration
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