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physics
oscillations mechanical waves

Questions and Answers of Oscillations Mechanical Waves

One end of a horizontal rope is attached to a prong of an electrically driven tuning fork that vibrates the rope transversely at 120 Hz. The other end passes over a pulley and supports a 1.5D-kg
A 1.5D-m string of weight 1.25 N is tied to the ceiling at its upper end, and the lower end supports a weight W. When you pluck the string slightly, the waves traveling up the string obey the
A thin, 75.0-cm wire has a mass of 16.5 g. One end is tied to a nail, and the other end is attached to a screw that can be adjusted to vary the tension in the wire. (a) To what tension (in
Weighty Rope. If in Example 15.3 (Section 15.4) we do not neglect the weight of the rope, what is the wave speed?(a) At the bottom of the rope; (b) At the middle of the rope; (c) At the top of the
A simple harmonic oscillator at the point x = 0 generates a wave on a rope. The oscillator operates at a frequency of 40.0 Hz and with an amplitude of 3.00 cm. The rope has a linear mass density of
A piano wire with mass 3.00 g and length 80.0 cm is stretched with a tension of 25.0 N. A wave with frequency 120.0 Hz and amplitude 1.6 mm travels along the wire. (a) Calculate the average power
A jet plane at take-off can produce sound of intensity 10.0 W 1m2 at 30.0 m away. But you prefer the tranquil sound of normal conversation, which is 1.0µW/m2. Assume that the plane behaves like a
You are investigating the report of a UFO landing in an isolated portion of New Mexico, and you encounter a strange object that is radiating sound waves uniformly in all directions. Assume that the
By measurement you determine that sound waves are spreading out equally in all directions from a point source and that the intensity is 0.026 W 1m2 at a distance of 4.3 m from the source. (a) What is
A fellow student with a mathematical bent tells you that the wave function of a traveling wave on a thin rope is y(x, t) = 2.30 mm cos [(6.98 rad/m) x + (742 rad/s) t]. Being more practical, you
What is the total power output of the siren in Example 15.5?
A wave pulse on a string has the dimensions showninFig.15.31 at t = 0. The wave speed is 40 cm/s. (a) If point o is a fixed end, draw the total wave on the string at t = 15 ms, 20ms, 25 ms, 30 ms, 35
A wave pulse on a string has the dimensions showninFJg.15.32att = 0. The wave speed is 5.0 m/s.(a) If point o is a fixed end, draw the total wave on the string at t = 1.0 ms, 2.0 ms, 3.0 ms. 4.0 ms,
Two triangular wave pulses are traveling toward each other on a stretched string as shown in Fig.1S.33. Each pulse is identical to the other and travels at 2.00 cm/s. The leading edges of the pulses
Suppose that the left-traveling pulse in Exercise I5.28 is below the level of the unstretched string instead of above it. Make the same sketches that you did in that exercise. 15.30. Two pulses are
Two pulses are moving in opposite directions at 1.0 cm/s on a taut string, as shown in Fig. IS.34. Each square is 1.0 cm. Sketch the shape of the string at the end of(a) 6.0 s;(b) 7.0 s;(c) 8.0 s.
Interference of Rectangular Pulses. Figure I5.3S shows two rectangular wave pulses on a stretched string traveling toward each other. Each pulse is traveling with a speed of 1.00 mm/s and has the
Two traveling waves moving on a string are identical except for opposite velocities. They obey the equation y(x, t) = Asin (kx + wt), where the plus-or-minus sign in the argument Depends on the
Standing waves on a wire are described by Eq. (IS.28), with As w = 2.S0mm, cu = 942rad/s, and k = 0.7S01rrad/m. The left end of the wire is at x = 0. At what distances from the left end are? (a) The
Adjacent antinodes of a standing wave on a string are I5.0 cm apart. A particle at an antinode oscillates in simple harmonic motion with amplitude 0.850 cm and period 0.0750 s. The string lies along
Wave Equation and Standing Waves. (a) Prove by direct substitution that y(x, t) = (Aswsinkx) sin wt is a solution of the wave equation, Eq. (I5.12), for u = w/k.(b) Explain why the relationship u =
Give the details of the derivation of Eq. (I5.28) from y, (x, t) + Y2(x, t) = A [-cos (kx +wt) + cos (kx - wt)].
Let y1(x, t) = Acos (k1x – w1t) and Y2(x,t) =Acos(k, x - w2t) be two solutions to the wave equation, Eq. (I5.12), for the same u. Show that y(x, t) =y,(x, t) + Y2(x, t) is also a solution to the
A 1.50-rn-long rope is stretched between two supports with a tension that makes the speed of transverse waves 48.0 m/s. What are the wavelength and frequency of? (a) The fundamental; (b) The second
A wire with mass 40.0 g is stretched so that its ends are tied down at points 80.0 cm apart. The wire vibrates in its fundamental mode with frequency 60.0 Hz and with amplitude at the antinodes of
A piano tuner stretches a steel piano wire with a tension of 800 N. The steel wire is 0.400 rn long and has a mass of 3.00 g.(a) What is the frequency of its fundamental mode of vibration?(b) What is
A thin, taut string tied at both ends and oscillating in its third harmonic has its shape described by the equation y(x, t) = (S.60 cm) sin [(0.0340 rad/cm) x] sin [(50.0rad/s) t], where the origin
The wave function of a standing wave is y(x, t) = 4.44 mm sin [(32.5 rad/m) x] sin [(754rad/s) t]. For the two traveling waves that make up this standing wave, find the (a) Amplitude;(b) Wavelength;
Consider again the rope and traveling wave of Exercise 15.24. Assume that the ends of the rope are held fixed and that this traveling wave and the reflected wave are traveling in the opposite
One suing of a certain musical instrument is 75.0 cm long and has a mass of 8.75 g. It is being played in a room where the speed of sound is 344 m/s. (a) To what tension must you adjust the string so
The portion of the string of a certain musical instrument between the bridge and upper end of the finger board (that part of the string that is free to vibrate) is 60.0 cm long, and this length of
(a) A horizontal string tied at both ends is vibrating in its fundamental mode. The traveling waves have speed v, frequency f, amplitude A, and wavelength A. Calculate the maximum transverse velocity
Guitar String one of the 63.5-cm-Iong strings of an ordinary guitar is tuned to produce the note B3 (frequency 245 Hz) when vibrating in its fundamental mode. (a) Find the speed of transverse waves
Waves on a Stick a flexible stick 2.0 m long is not fixed in any way and is free to vibrate. Make clear drawings of this stick vibrating in its first three harmonics, and then use your drawings to
A transverse sine wave with an amplitude of 2.50 mm and a wavelength of 1.80 m travels from left to right along a long, horizontal, stretched string with a speed of 36.0 m/s. Take the origin at the
A transverse wave on a rope is given by y (x, t} = (0.750cm} cos1t [(0.400cm-2} x + (250s-2} t] (a) Find the amplitude, period, frequency, wavelength, and speed of propagation. (b) Sketch the shape
Three pieces of string, each of length L, are joined together end to end, to make a combined string of length 3L. The first piece of string has mass per unit length µ1, the second piece has mass per
A 1750-N irregular beam is hanging horizontally by its ends from the ceiling by two vertical wires (A and B), each 1.25 m long and weighing 2.50 N. The center of gravity of this beam is one third of
Ant Joy Ride you place your pet ant Klyde (mass m) on top of a horizontal, stretched rope, where he holds on tightly. The rope has mass M and length L and is under tension F. You start a sinusoidal
Weightiess Ant. An ant with mass m is standing peacefully on top of a horizontal. stretched rope. The rope has mass per unit length µ and is under tension F. Without warning, Cousin Throckmorton
When a transverse sinusoidal wave is present on a string, the particles of the string undergo SHM. This is the same motion as that of a mass m attached to an ideal spring of force constant k', for
A 5.00-m, 0.732-kg wire is used to suppose two uniform 235-N posts of equal length (Fig. 15.37). Assume that the wire is essentially horizontal and that the speed of sound is 344 m/s. A strong wind
A Nonsinusoidal Wave. The shape of a wave on a string at a specific instant is shown in Fig. 15.38. The wave is propagating to the right, in the + x-direction.(a) Determine the direction of the
A continuous succession of sinusoidal wave pulses is produced at one end of a very long string and travel along the length of the string. The wave has frequency 40.0 Hz, amplitude 5.00 mm, and
Two-Dimensional Waves. A stretched string lies along the x-axis. The string is displaced along both the y- and z-directions, so that the transverse displacement of the string is given byY(x, t) =
A vertical, 1.20-m length of 18-gauge (diameter of 1.024 mm) copper wire has a l00.0-N ball hanging from it (a) What is the wavelength of the third harmonic for this wire? (b) A 500.0-N ball now
Waves of Arbitrary Shape.(a) Explain why any wave described by a function of the form y(x, t) = fix - ut) moves in the +x-direction with speed u.(b) Show that y(x, t) = f(x - ut) satisfies the wave
Equation (15.7) for a sinusoidal wave can be made more general by including a phase angle Ф, where 0≤Ф≤2 π (in radians). Then the wave function y(x, t) becomes y(x, t) =
(a) Show that Eq. (15.25) can also be written as Pay = 1/2FkwA2, where k is the wave number of the wave. (b) If the tension F in the string is quadrupled while the amplitude A is kept the same, how
Energy in a Triangular Pulse a triangular wave pulse on a taut string travels in the positive x-direction with speed u. The tension in the string is F, and the linear mass density of the string is
A sinusoidal transverse wave travels on a string. The string has length 8.00 m and mass 6.00 g. The wave speed is 30.0 m/s, and the wavelength is 0.200 m. (a) IT the wave is to have an average power
Instantaneous Power in a Wave. (a) Graph y(x, t) as given by Eq. (15.7) as a function of x for a given time r (say, t = 0). On the same axes, make a graph of the instantaneous power p(x, t) as given
A metal wire, with density p and Young's modulus Y, is stretched between rigid supports. At temperature T, the speed of a transverse wave is found to be v1 When the temperature is increased to T +
A vibrating string 50.0 cm long is under a tension of 1.00 N. The results from five successive stroboscopic pictures are shown in Fig. 15.39. The strobe rate is set at 5000 flashes per minute, and
Cousin Throckmorton is once again playing with the clothesline in Example 15.2 (Section 15.3). One end of the clothesline is attached to a vertical post. Throcky holds the other end loosely in his
A guitar string is vibrating in its fundamental mode, with nodes at each end. The length of the segment of the string that is free to vibrate is 0.386 m. The maximum transverse acceleration of a
As shown in Exercise 15.35, a standing wave given by Eq. (15.28) satisfies the wave equation Eq. (15.12).(a) Show that a standing wave given by Eq. (15.28) also satisfies the equationInterpret this
(a) The red and blue waves in Fig. 15.20 combine so that the displacement of the string at 0 is always zero. To show this mathematically for a wave of arbitrary shape, consider a wave moving to the
A string that lies along the + x-axis has a free end at x = 0. (a) By using steps similar to those used to derive Eq. (15.28), show that an incident raveling wave y1x, t) = Acos (kx+ wt) gives rise
A string with both ends held fixed is vibrating in its third harmonic. The waves have a speed of 192 m/s and a frequency of 240 Hz. The amplitude of the standing wave at an antinode is 0.400 cm. (a)
A uniform cylindrical steel wire, 55.0 cm long and 1.14 mm in diameter, is fixed at both ends. To what tension must it be adjusted so that, when vibrating in its first overtone, it produces the note
Holding Up Under Stress. A string or rope will break apart if it is placed under too much tensile stress [Eq. (11.8)]. Thicker ropes can withstand more tension without breaking because the thicker
Combining Standing Waves. A guitar string of length L is plucked in such a way that the total wave produced is the sum of the fundamental and the second harmonic. That is, the standing wave is given
When a massive aluminum sculpture is hung from a steel wire, the fundamental frequency for transverse standing waves on the wire is 250.0 Hz. The sculpture (but not the wire) is then completely
Tuning an Instrument. A musician tunes the C-string of her instrument to a fundamental frequency of 65.4 Hz. The vibrating portion of the string is 0.600 m long and has a mass of 14.4 g. (a) With
Longitudinal Waves on a spring. A long spring such as a SlinkyTM is often used to demonstrate longitudinal waves. (a) Show that if a spring that obeys Hooke's law has mass m, length L, and force
(a) Show that for a wave on a string, the kinetic energy per unit length of string is Where /L is the mass per unit length. (b) Calculate Uk(X, t) for a sinusoidal wave given by Eq. (15.7). (c)
A deep-sea diver is suspended beneath the surface of Loch Ness by a l00-m-Iong cable that is attached to a boat on the surface (Fig. 15.40). The diver and his suit have a total mass of 120 kg and a
A uniform rope with length L and mass m is held at one end and whirled in a horizontal circle with angular velocity w. You can ignore the force of gravity on the rope. Find the time required for a
Instantaneous Power in a Standing Wave. From Eq. (15.21), the instantaneous rate at which a wave transmits energy along a string (instantaneous power) isWhere F is the tension.(a) Evaluate f (x, t}
The B-string of a guitar is made of steel (density 7800 kg/m3), is 63.5 cm long, and has diameter 0.406 mm. The fundamental frequency is f= 247.0 Hz.(a) Find the string tension.(b) If the tension F
Example 16.1 (Section 16.1) showed that for sound waves in air with frequency 1000 Hz, a displacement amplitude of 1.2 X 10-8 m produces a pressure amplitude of3.0 X 10-2 Pa. (a) What is the
Example 16.1 (Section 16.1) showed that for sound waves in air with frequency 1000 Hz, a displacement amplitude of 1.2 X 10-8 m produces a pressure amplitude of 3.0 X 10-2 PaWater at 20°C has a bulk
Consider a sound wave in air that has displacement amplitude 0.0200 mm. Calculate the pressure amplitude for frequencies of (a) 150 Hz; (b) 1500 Hz;(c) 15,000 Hz. In each case compare the result to
A loud factory machine produces sound having a displacement amplitude of 1.00 µm. but the frequency of this sound can be adjusted. In order to prevent ear damage to the workers, the maximum pressure
(a) In a liquid with density 1300 kg/m3longitudinal waves with frequency 400 Hz are found to have wavelength 8.00 m. Calculate the bulk modulus of the liquid (b) A metal bar with a length of 1.50 m
A major earthquake centered on Lorna Prieta, California, near San Francisco, occurred at 5:04 P.M. local time on October 17, 1989 (in UTC, Coordinated Universal Tune, 0h 4m 15s on October 18, 1989).
A submerged scuba diver Figure 16.39 Exercise 16.1.Hears the sound of a boat horn directly above her on the surface of the lake. At the same time, a friend on dry land 22.0 m from the boat also hears
At a temperature of 27.0°C, what is the speed of longitudinal waves in (a) Hydrogen molar mass 2.02 g/mol)? (b) Helium (molar mass 4.00 g/mol); (c) Argon (molar mass 39.9 g/mol)? See Table 19.1 for
An oscillator vibrating at 1250 Hz produces a sound wave that travels through an ideal gas at 325 m/s when the gas temperature is 22.0°C. For a certain experiment, you need to have the same
(a) Show that the fractional change in the speed of sound (du/u) due to a very small temperature change d T is given by du/u = 1/2dT/T. (Hint: Start with Eq. 16.10.) (b) The speed of sound in air at
An 80.0-m-long brass rod is struck at one end. A person at b the other end hears two sounds as a result of two longitudinal waves, one traveling in the metal rod and the other traveling in the air.
What is the difference between the speed of longitudinal waves in air at 21.0°C and their speed at -13.0C?
What must be the stress (FIA) in a stretched wire of a material whose Young's modulus is for the speed of longitudinal waves to equal 30 times the speed of transverse waves?
Use information from Table 16.2 to answer the following questions about sound in air. At 20oC the bulk modulus for air is 1.42 X 105 Pa and its density is 1.20 kg/m3. At this temperature, what are
Longitudinal Waves in Different Fluids. (a) A longitudinal wave propagating in a water-filled pipe has intensity 3.00 X 10-6w/m2 and frequency 3400 Hz. Find the amplitude A and wavelength A of the
Derive Eq. (16.14) from the equations that precede it.
A sound wave in air at 20°C has a frequency of 150 Hz and displacement amplitude of 5.00 X 10-3 mm. For this sound wave calculate the (a) Pressure amplitude (in Pa); (b) Intensity fm W/m2); (c)
What is the sound intensity level in a car when the sound intensity is 0.500 µw/m2? What is the sound intensity level in the air near a jackhammer when the pressure amplitude of the sound is 0.150
For a person with normal hearing, the faintest sound that can be heard at a frequency of 400 Hz has a pressure amplitude of about 6.0 X 10-5 Pa. Calculate the (a) Intensity; (b) Sound intensity
The intensity due to a number of independent sound sources is the sum of the individual intensities. (a) When four quadruplets cry simultaneously, how many decibels greater is the sound intensity
A baby's mouth is 30 cm from her father's ear and 1.50 m from her mother's ear. What is the difference between the sound intensity levels heard by the father and by the mother?
The Sacramento City Council recently adopted a law to reduce the allowed sound intensity level of the much despised leaf blowers from their current level of about 95 dB to 70 dB. With the new law,
(a) By what factor must the sound intensity be increased to raise the sound intensity level by 13.0 dB? (b) Explain why you don't need to know the original sound intensity.
The fundamental frequency of a pipe that is open at both ends is 594 Hz. (a) How long is this pipe? If one end is now closed, find (b) The wavelength and (c) The frequency of the new fundamental.
Standing sound waves are produced in a pipe that is 1.20 m long. For the fundamental and first two overtones, determine the locations along the pipe (measured from the left end) of the displace
Find the fundamental frequency and the frequency of the first three overtones of a pipe 45.0 cm long (a) If the pipe is open at both ends and (b) If the pipe is closed at one end. Use v = 344 m/s.
The Human Voice. The human vocal tract is a pipe that extends about 17 cm from the lips to the vocal folds (also called ''vocal cords") near the middle of your throat. The vocal folds behave rather
Human Ear. The auditory canal of the ear (see Fig. 16.4) is filled with air. One end is open, and the other end is closed by the eardrum. A particular person's auditory canal is 2.40 cm long and can
A certain pipe produces a fundamental frequency of 262 Hz in air. (a) If the pipe is filled with helium at the same temperature, what fundamental frequency does it produce? (The molar mass of air is

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