Chapter 11 Oscillations and Waves
What is the big rule 2 for waves?
A wave moving from one medium to another will maintain the same frequency AKA is a wave goes from air to water, it will maintain the same frequency (pitch)
What is a mechanical wave?
A wave that requires a medium through which to travel
If the amplitude of an object undergoing simple harmonic motion decreases by a factor of 4, then the frequency of oscillation: A. decreases by a factor of 4. B. stays the same. C. increases by a factor of 2. D. increases by a factor of 4.
B; For simple harmonic motion, frequency (and period) are independent of amplitude (e.g. for a mass/spring system, the frequency depends only on the mass of the object and the spring constant).
Suppose you are tuning a piano and after hitting the tuning fork for A above middle (f = 440 Hz) and playing the same key on a piano, you hear a beat every half second. What is the current frequency of the piano key? A. 438 Hz B. 439.5 Hz C. 438 or 442 Hz D. 539.5 or 440.5 Hz
C; B = |f1-f2| 2 Hz = |440 - f2| Because it is absolute value, f2 can be 438 or 442, and the answer is C
What are wave crests?
Highest point of a wave
For a particular rope, it's found that the second-harmonic frequency is 8 Hz. What's the fifth harmonic frequency?
fn = nf1 f(2) = 2f1 8 = 2f1 f1 = 4 Hz f(5) = nf1 f(5) = 5(4) f(5) = 20 Hz
How do you determine of waves are out of phase?
path difference = (n + 1/2)y
What are nodes of waves?
points of no displacement
How are frequency and period dependent on amplitude?
they aren't
A guitar string is playing notes that are flat (frequency too low). How can the string be made to play the correct notes? A. Increase the tension in the string to increase the wave speed B. Increase the tension in the string to decrease the wavelength C. Decrease the tension in the string to increase the wave speed D. Decrease the tension in the string to decrease the wavelength
A; Intuitively, it makes sense that increasing the tension would cause the string to play higher notes, eliminating choices C and D. Specifically, increasing the tension increases the speed of waves traveling along the string: where T is tension and m / L is the linear mass density of the string. Since the length of the string allowed to vibrate is the same, all harmonic wavelengths will be the same (since λn = 2L / n). This eliminates choice B.
A mass oscillating on a spring travels 5 cm from its equilibrium position to its maximum displacement in 0.2 seconds. What is the frequency of oscillation? A. 1.25 Hz B. 2.5 Hz C. 5 Hz D. 25 Hz
A; The period, T, is the time it takes to complete one full cycle. Traveling from equilibrium to the maximum displacement represents one-quarter of a cycle (a full cycle would occur when the mass travels from equilibrium to the maximum displacement, back to equilibrium, then to the minimum displacement, then back to equilibrium again). Therefore, T = 4(0.2 s) = 0.8 s. The frequency, f, is given by f = 1 / T = 1 / (0.8 s) = 1.25 Hz. Since f is 1 divided by slightly less than 1, the answer must be slightly more than 1, indicating choice A is correct without doing the math. Note that the distance traveled is not needed.
Two transverse waves with identical frequencies interfere, creating a resulting wave with amplitude 8 cm. If one of the original waves has amplitude 12 cm, then the other wave's amplitude could be any of the following EXCEPT: A. 3 cm B. 4 cm C. 12 cm D. 20 cm
A; The principle of superposition for waves states that, when two or more waves interfere, the resulting displacement from equilibrium is the sum of the individual displacements. The two extreme cases are 1) when the crests (and the troughs) line up, or 2) when the crest of one lines up with the trough of the other. In the first case, the amplitudes add up to equal the resultant amplitude, and the in the second case they subtract. If one wave has amplitude 12 cm and the total amplitude is 8 cm, that means that the second wave must have an amplitude somewhere between 12 cm - 8 cm and 12 cm + 8 cm, which is between 4 cm and 20 cm. Choice A falls outside of this range.
Two sound waves are generated in air. Which of the following is true? I. They have the same speed II. They have the same frequency III. The faster wave has a larger wavelength A. I only B. II only C. I and II only D. II and III only
A; With certain exceptions (e.g. the dispersion of light), all waves of the same type in the same medium have the same speed. Therefore Roman numeral I is true and III must be false. This eliminates choices B and D. Roman numeral II is false since waves can be generated with any frequency (e.g. music, where each note or pitch corresponds to a different frequency). This eliminates choice C.
What happens when the wave shown below passes from the thick, heavy rope into the thinner, lighter rope.
According to big rule 2 for waves, when a wave passes into another medium, its speed changes, but its frequency does not. How does the speed change? Because the rope is lighter (lower density), the equation for a wave speed on a string tells us that v will increase. So if v increases but f does not change, then wavelength will also increase.
There are 3 tuning forks: A, B and C. When A and B are played together, 5 beats per second are heard. When tuning forks B and C are played together, 3 beats per second are heard. How many beats per second would be heard if A and C were played together? A. 2 or 4 B. 2 or 8 C. 4 or 8 D. 2, 4 or 8
B; To make the solution concrete, choose a value for one of the frequencies. For example, let fA be 200 Hz. The beat frequency, fbeat, is given by fbeat = |f1- f2| where f1 and f2 are the frequencies of the individual sound waves. Since 5 beats per second are heard when A and B are played together, then fA must be 5 Hz more or less than fB. Given the choice of fA = 200 Hz, this means that fBcould be 205 Hz or 195 Hz. Since 3 beats per second are heard when B and C are played together, fC could be 205 + 3 = 208 Hz, 205 - 3 = 202 Hz, 195 + 3 = 198 Hz or 195 - 3 = 192 Hz. Two of these possibilities (208 Hz and 192 Hz) differ from fA by 8 Hz and the other two possibilities (202 Hz and 198 Hz) differ from fA by 2 Hz. Therefore, when A and C are played together, the beat frequency will be either 2 Hz or 8 Hz.
A 100 kg jumper attached to a bungee cord jumps off a bridge. The bungee cord stretches and the man reaches the lowest spot in his decent before beginning to rise. The force of the stretched bungee cord can be approximated using Hooke's law, where the value of the spring constant is replaced by an elasticity constant, in this case, 100 kg/s^2. If the cord is stretched by 30m beyond its vertical equilibrium length at the lowest spot of the mans descent, then what will his acceleration be at this spot? A. 0 m/s^2 B. 10 m/s^2 C. 20 m/s^2 D. 30 m/s^2
C;
The speed of sound waves depends on the bulk modulus (the resistance to compression) of the medium and its density: v = sqrt (B/p). The general trend is for found waves to travel slowest in gases, faster in liquids, and fastest in solids. Given this fact, which of the following is most likely true? A. When you are under water, and heat people talking on the boat above you, their voices are a higher pitch (frequency) than normal. B. When you are under water, and heat people talking on the boat above you, their voices are louder (more intense, ie, more power per unit area) than normal C. When you are under water, and heat people talking on the boat above you, the wavelength of their voices is longer than it would be in air D. There is no significant difference between the voices heart under water and how they would be heard in the air.
C; A violates rule 2 B violates conservation of energy
A simple pendulum is oscillating alongside a mass on a vertical spring. If both are taken to a planet with a stronger gravitational pull than the earth, how would their respective periods be affected? A. The period of the pendulum would stay the same while the period of the mass/spring would increase. B. The period of the pendulum would decrease while the period of the mass/spring would increase. C. The period of the pendulum would decrease while the period of the mass/spring would stay the same. D. The period of the pendulum would increase while the period of the mass/spring would stay the same.
C; T = 2pi x sqrt (L/g) (this decreases because denominator is getting larger) T = 2pi x sqrt (k/m) (m is constant, mass/spring stays the same)
A block is oscillating horizontally on an ideal spring. If the amplitude is doubled, what would happen to the maximum speed of the block? A. It would stay the same. B. It would increase by a factor of . C. It would increase by a factor of 2. D. It would increase by a factor of 4.
C; Conservation of Mechanical Energy with kinetic energy equal to (1/2)mv2 and elastic potential energy equal to (1/2)kx2. Comparing the energy at x = 0 (where the speed is at its maximum) to the energy at x = A (the amplitude, where speed is zero), we get (1/2)mvmax2 = (1/2)kA2 (based on conservation of energy). Solving for vmax, we see that vmax = A sqrt (k/m)
What is true about sound waves? I. They are longitudinal II. They are electromagnetic in nature III. They have patterns of compressions and rarefactions A. I only B. I and II only C. I and III only D. II and III only
C; Sound waves are mechanical waves (i.e. they require a medium), as opposed to electromagnetic waves which do not require a medium. Roman numeral II is false, which eliminates choices B and D. The particles of the medium in a sound wave oscillate parallel to the direction of propagation, which means sound waves are longitudinal. Roman numeral I is therefore true. The extremes of displacement for longitudinal waves are referred to as compressions and rarefactions, so Roman numeral III is also true.
All of the following are properties of sound waves EXCEPT: A. They are longitudinal. B. They require a medium. C. They can be polarized. D. They can experience refraction.
C; Sound waves are mechanical waves (i.e. they require a medium). This eliminates choice B. The particles of the medium oscillate parallel to the direction of propagation, which means they are longitudinal, eliminating choice A. Any wave that hits the boundary between two different media will bend (i.e. refract) as it enters the new medium. This eliminates choice D. Only transverse waves, such as electromagnetic waves, can be polarized.
For an ideal oscillating mass/spring system, which of the following is true at the object's equilibrium position? A. Both the speed and the acceleration are zero. B. The speed is zero and the acceleration is maximum. C. The speed is maximum and the acceleration is zero. D. Both the speed and the acceleration are maximum.
C; The magnitude of the restoring force exerted by a spring with force constant k on a mass m is given by F = -kx, where x is the displacement from equilibrium. For an ideal mass/spring system (oscillating horizontally), this is the net force acting on the object. Newton's Second Law tells us that Fnet = -kx = ma. At equilibrium, x = 0 and therefore a = 0 (its minimum). This eliminates choices B and D. While conservation of energy can be used to determine where the speed is the maximum, it is obvious that the object is moving as it passes through equilibrium, which eliminates choice A. (Note that answer would still be C even if the mass/spring system were vertical.)
Which of the following is necessarily true about waves? A. Transverse waves require a medium. B. Longitudinal waves can be electromagnetic. C. Electromagnetic waves are always transverse. D. Mechanical waves do not require a medium.
C; This question requires the comparison of mechanical and electromagnetic waves. Beginning with mechanical: Mechanical waves require a medium, since the medium itself oscillates. This eliminates choice D. They can be either transverse (e.g. waves traveling along a string) or longitudinal (e.g. sound). Electromagnetic waves do not require a medium since it is the electric and magnetic fields that oscillate. The fields oscillate perpendicular to the direction of propagation, making them transverse, thus eliminating choices A and B.
A string attached at each end is vibrating in its 4th harmonic mode. If the distance between each node and its closest antinode is 40 cm, then what is the length of the string? A. 1.6 m B. 2.4 m C. 3.2 m D. 6.4 m
C; When a string vibrates in its 4th harmonic mode, it means that there are 4 half-wavelengths (i.e. 2 full wavelengths) along the length of the string. The distance between a node and the closest antinode is one quarter of a wavelength, so λ = 4(40 cm) = 160 cm or 1.6 m. The length of string is therefore 2λ = 3.2 m.
A 2kg mass is attached to a massless, .5m string and is used as a simple pendulum by extending it to an angle theta = 5 degrees and allowing it to oscillate. Which of the following change will increase the prior of the pendulum? A. Replacing the mass with a 1kg mass B. Changing the initial extension of the pendulum to a 10 degree a angle C. Replacing the string with a .25m string D. Moving the pendulum to the surface of the moon
D; T = 2pi x sqrt (l/g) Decreasing g will make the numerator smaller and thus makes the period bigger.
An earthquake generates a longitudinal seismic wave with frequency 15 Hz. The distance from the center of a compression to the center of the closest rarefaction is 0.2 km, how far will the wave travel in 5 seconds? A. 15 m B. 30 m C. 15 km D. 30 km
D; For a longitudinal wave, the distance between a compression and the closest rarefaction is half of a wavelength. Therefore the wavelength, λ, is 2(0.2 km) = 0.4 km or 400 m. The speed, v, of a wave is given by v = fλ, where f is the frequency. So v = (400 m)(15 Hz) = 6000 m/s. The distance traveled by the wave in a time, t, is given by d = vt = (6000 m/s)(5 s) = 30,000 m or 30 km.
A pipe open at both ends supports a standing wave vibrating in its n = 5 harmonic mode. All of the following would decrease the frequency EXCEPT: A. using a longer pipe but keep n the same. B. using the same length pipe and decreasing n. C. closing one end of the pipe and keeping n the same. D. increasing the temperature of the air inside the pipe.
D; For a pipe open at both ends, fn = vn / 2L, where fn is the nth harmonic frequency, v is the speed of sound and L is the length of the pipe. Increasing Lwould decrease the frequency, so choice A can be eliminated. Similarly, decreasing n would also decrease the frequency, so choice B can be eliminated. Increasing the temperature of the air would increase the speed of sound which would increase the frequency. The answer is therefore D. To understand why choice C is incorrect, remember that, for a pipe closed at one end, the nth harmonic frequency is given by fn = vn / 4L. Keeping n and L the same but having a 4 in the denominator instead of a 2 would decrease the frequency.
A block is oscillating horizontally on an ideal spring. When the block is halfway between equilibrium and its maximum displacement, what fraction of total mechanical energy is kinetic? A. 1/4 B. 1/2 C. sqrt 2/2 D. 3/4
D; For an ideal mass/spring system, the total mechanical energy (kinetic plus elastic potential) is conserved. Kinetic energy is (1/2)mv2 and elastic potential energy is (1/2)kx2. Since there is no direct information given about speed, it is easier to focus on potential energy. When x = A (maximum displacement) the speed, and therefore the kinetic energy, is zero and the potential energy is maximized. The total mechanical energy at the maximum displacement can be expressed as E = (1/2)kA2. At x = (1/2)A, the potential energy is (1/2)k(A/2)2, or (1/8)kA2. The fraction of total mechanical energy that is potential = (1/8) / (1/2) = 1/4. Therefore, 1 - 1/4 = 3/4 of the total mechanical energy will be kinetic. PE = 1/2kx^2 KE = 1/2 mv^2 Only PE is dependent on x, so when x is halved, PE decreases by a factor of 4
A person stands at an equal distance between two speakers which produce identical sound waves. If the person moves so that she is 1.5 m closer to one speaker than the other, she is in a "quiet zone" (i.e. the waves experience destructive interference). The wavelength of the sound waves could be all of the following EXCEPT: A. 0.6 m B. 1 m C. 3 m D. 6 m
D; If the path difference from two wave sources to the detector is an integer multiple of wavelengths, then the waves will experience constructive interference (i.e. compressions will line up with compressions, rarefactions will line up with rarefactions). This can be expressed as d2 - d1 = mλ, where d2 and d1 are lengths of the paths traveled by the waves, m is an integer and λis the wavelength. Similarly, if the path difference is equal to an integer-and-a-half times the wavelength, the waves will experience destructive interference: d2 - d1 = (m + 1/2)λ. Since the waves are experiencing destructive interference, 1.5 = (m + 1/2)λ. So 1.5 could equal λ / 2 or 3λ / 2 or 5λ / 2, etc. Solving for λ, we get that λ = 2(1.5) = 3 or (2/3)(1.5) = 1 or (2/5)(1.5) = 0.6, etc. (all answers in meters). Choice D is the only answer not possible.
Which of the following is true concerning the differences between sound waves and light? A. Both can travel through a vacuum B. Both can be polarized C. Both travel faster through water than through air D. Both can experience reflection, refraction and diffraction
D; Sound waves are mechanical, so they need a medium through which to travel. Light waves are electromagnetic and do not need a medium. This eliminates choice A. Sound waves are longitudinal and cannot be polarized, whereas light waves are transverse and can be polarized. This eliminates choice B. Because longitudinal waves travel faster through a more incompressible medium, sound travels faster through water than through air. In contrast, light travels faster through air than water. This eliminates choice C. Both sound and light waves experience wave properties that are not dependent upon the transverse versus longitudinal distinction, such as reflection, refraction and diffraction.
For an ideal oscillating mass/spring system, which of the following is true? A. The speed is maximum at maximum displacement and the acceleration is maximum at equilibrium. B. The speed is maximum at equilibrium and the acceleration is minimum at maximum displacement. C. The speed is minimum at equilibrium and the acceleration is maximum at maximum displacement. D. The speed is minimum at maximum displacement and the acceleration is minimum at equilibrium.
D; The magnitude of the restoring force exerted by a spring with force constant k on a mass m is given by F = -kx, where x is the displacement from equilibrium. For an ideal mass/spring system (oscillating horizontally), this is the net force acting on the object. Newton's Second Law tells us that Fnet = -kx = ma. The maximum acceleration therefore occurs at maximum displacement. Similarly, the minimum acceleration occurs at x = 0 (i.e. equilibrium). This eliminates choices A and B. While conservation of energy can be used to determine the maximum speed, the speed can be more easily determined by recognizing that the speed must be zero at the at maximum displacement, since that is where the object turns around. This eliminates choice C. (Note that answer would still be D even if the mass/spring system were vertical.)
A certain simple pendulum oscillating with a small angle has the same period as an ideal mass/spring system. If the length of the pendulum is doubled, its period will remain the same as the mass/spring system if: A. the spring constant is doubled. B. the pendulum mass is doubled. C. the mass on the spring is halved. D. the mass on the spring is doubled.
D; The period, T, of a simple pendulum (for small angle) is given by , where L is the length of the pendulum and g is the acceleration due to gravity. Period, therefore, does not depend on the mass of the pendulum, so choice B can be eliminated. The period of a mass/spring system is given by , where m is the mass and k is the spring constant. If L is doubled for the pendulum, then the periods will remain the same if either m is doubled or k is halved. This leaves choice D.
All of the following changes would LOWER the frequency played by a plucked string attached at each end EXCEPT: A. decreasing the tension in the string. B. using string with the same length but greater mass. C. using a longer string. D. decreasing the amplitude of oscillation.
D; The speed of waves traveling in a string is given by sqrt (T/mL) where T is tension and m / L is the linear mass density of the string. If tension were to decrease, then speed would decrease. Assuming the length of the string is the same, then all harmonic wavelengths will remain the same (since λn = 2L / n). The equation v = fλ tells us that decreasing the speed would therefore decrease the frequency. This eliminates choice A, since this is a true statement. Similarly, increasing the mass would decrease the speed and the frequency, which eliminates choice B. Increasing the length of the string would increase the wavelength, which would decrease the frequency. This eliminates choice C. The amplitude has no effect on frequency at all, meaning that choice D is correct.
When light passes from Medium #1 (index of refraction n1) to Medium #2 (index of refraction n2), the wavelength becomes smaller. Which of the following is true? A. The frequency in Medium #1 is greater than the frequency in Medium #2 B. The frequency in Medium #1 is less than the frequency in Medium #2 C. n1 > n2 D. n1 < n2
D; When a wave passes into a new medium, its frequency stays the same. This eliminates choices A and B. For waves, v = fλ, where v is the speed, f is the frequency and λ is the wavelength. If λ decreases when entering Medium #2, then v should also decrease. The relationship between v and n is given by n = c / v, where c is the speed of light in a vacuum. Therefore, since v1 > v2, this means than n1 < n2.
A wave travels from Medium 1 to Medium 2 and it is observed that the wavelength decreases. Which of the following is necessarily true? A. The frequency in Medium 2 is greater than the frequency in Medium 1. B. The frequency is Medium 2 is less than the frequency in Medium 1. C. The speed in Medium 2 is greater than the speed in Medium 1. D. The speed in Medium 2 is less than the speed in Medium 1.
D; When any wave passes from one medium to another, its frequency remains the same. This eliminates choices A and B. The relationship between speed, frequency and wavelength is given by v = fλ. If λ decreases and f stays the same, then v must also decrease. This eliminates choice C.
When a sound wave passes from air to water: A. its frequency increases and its wavelength decreases. B. its frequency stays the same and its wavelength decreases. C. its frequency decreases and its wavelength increases. D. its frequency stays the same and its wavelength increases.
D; When any wave passes into a new medium, its frequency remains the same. This eliminates choices A and C. Sound travels faster in water than air since it is a more incompressible medium. The relationship between speed (v), frequency (f) and wavelength (λ) for all waves is given by v = fλ. If v increases and f stays the same, then λ must also increase.
A physics student is doing a wave experiment with a 1m long cord stretched across the lab table. In the middle of the cord, a 2 cm section in painted red. A specially designated machine creates vibrations so that a sine wave will travel on the cord from the east side of the table to the west side. The vibrations of the sine wave are parallel to the table and peak at the north side of the table and the south side of the table. Which of the following best describes the motion of the red spot? A. The spot moves from east to west along the sine wave B. The spot moves from west to east along the sine wave C. The spot remains in a fixed location on the table D. The spot vibrates between the north and south side of the table.
D; sine waves are transverse
A block of mass 200g is oscillating on the end of a horizontal spring of spring constant 100 N/m and a natural length of 12 cm. When the spring is stretched to a length of 14cm, what is the acceleration?
F = -kx F = -(100 Nm)(.02m) F = 2N F = ma a = F/m a = 2/.2 a = 10 m/s^2
Restoring force of a pendulum
F_restoring = m*g*sin(theta)
A parent is pushing a young child on a swing at the playground. When the parent stops pushing, the child's swinging motion continues without assistance. Assume that the chain on the swing has negligible mass and any friction is negligible. Which of the following would need to be true in order for the child's motion on the swing to be considered simple harmonic motion? I. The mass of the child is not too large II. The child is not swinging to high, so the angle between the swing and the vertical is not too big III. The tension in the chain of the swing is negligible
II only
What is the amplitude of a wave?
The maximum displacement of a point on the wave from its undisturbed position
What is the big rule 1 for waves?
The speed of a wave is determined by the type of wave and the characteristic of the medium, NOT the frequency. The exception on the MCAT is dispersion
What is period?
The time it takes to move through one full cycle of motion
What is the principle of superposition of waves?
When two or more waves of the same type arrive at the given point in space at the same time, the resultant displacement of the medium is the vector sum of the individual displacements of the waves
What is constructive interference?
Where two waves arrive in step reinforcing one another (increasing the amplitude)
What is destructive interference?
Where two waves arrive out of step cancelling one another out
The maximum displacement of a hook from equilibrium is called the _________, denoted by A
amplitude
The distance from a trough to crest is 20 cm on a 3 m rope of 1 kg. If the tension in the rope is 3N, what is the period? A. 1/15 s B. 2/15 s C. 15/2 s D. 15 s
B v = (tension / linear density)^.5 v = sqrt (3N / 1/3) v = sqrt 9 v = 3 m/s v = yf (y = 40 cm because wavelength is crest to crest) f = 3 / 4 x 10^-1 f = .75 x 10^1 f = 7.5 s (or 15/2) T = 1/f T = 2/15 s
The speed of a 2kg mass on a spring is 4 m/s as it passes through the equilibrium position., What is the frequency if the amplitude is 2m? A. 1/5 Hz B. 1/3 Hz C. 3 Hz D. 5 Hz
B 1/2kA^2 = 1/2mv^2 k = 8 N/m f = 1/2pi x sqrt (k/m) f = 1/6 x sqrt (8/2) f = (1/6)(2) f = 1/3 Hz
A wave with frequency 50 Hz is generated at one end of a 8 m string, while the other end is attached to a wall. If the wave returns to its original position 2 seconds after it was created, what is its wavelength? A. 8 cm B. 16 cm C. 6.25 m D. 12.5 m
B; The total round-trip distance traveled by the wave is 8 m + 8 m = 16 m. The speed of the wave is given by v = distance / time = (16 m) / (2 s) = 8 m/s. For all waves, v = fλ, where f is the frequency and λ is the wavelength. Therefore, λ = v / f = (8 m/s) / (50 Hz) = 0.16 m or 16 cm.
Two speakers are emitting identical sound waves with frequency 170 Hz (the speed of sound is approximately 340 m/s). If a person stands 3 meters from one speaker and 4 meters from the other speaker, he will experience: A. constructive interference. B. destructive interference. C. beats. D. a Doppler shift.
B; Since neither the speakers nor the person moves, there will be no Doppler shift. This eliminates choice D. Beats occur when sound waves with slightly different frequencies interfere. The problem states that the waves are identical, eliminating choice C. Whether the waves experience constructive or destructive interference depends upon the path difference between the speakers and the person. If the path difference is an integer times the wavelength (e.g. λ, 2λ, 3λ, etc.) then there will be constructive interference. If the path difference is an integer-and-a-half times the wavelength (λ / 2, 3λ / 2, 5λ / 2, etc.) then there will be destructive interference. Since the person is 3 meters from one speaker and 4 meters from the other, the path difference is 4 - 3 = 1 meter. To find the wavelength, use the equation v = fλ, where v is the speed of sound and f is the frequency. So, λ = v / f = (340 m/s) / (170 Hz) = 2 m. Since the path difference is one half the wavelength, the waves (and the person) will experience destructive interference.
A pipe closed at one end supports a standing wave vibrating in its 3rd harmonic mode. If the closed end is then opened, and a standing wave is created with twice the previous frequency, then what is the pipe's new harmonic mode? A. 1.5 B. 3 C. 6 D. 12
B; The harmonic mode, denoted by n, must be an integer. This eliminates choice A. For a pipe closed at one end, the nth harmonic frequency is given by fn = vn / 4L, where v is the speed of sound and L is the length of the pipe. Since the pipe is playing in its 3rd harmonic mode, f = (3v) / (4L). For a pipe open at both ends, fn = vn / 2L. Doubling the original frequency and plugging into this equation yields (2)(3v) / 4L = vn / 2L. Canceling v and L yields 6/4 = n / 2, so n = 3.
An object undergoing simple harmonic motion has an amplitude of 20 cm and a frequency of 10 Hz. What is the shortest time required for the object to travel from its maximum positive displacement to its maximum negative displacement? A. 0.02 seconds B. 0.05 seconds C. 0.1 seconds D. 2 seconds
B; The period, T, is the time takes to complete one full cycle and is given by T= 1 / f, where f is the frequency of oscillation. Therefore, T = 1 / (10 Hz) = 0.1 seconds. Traveling from maximum positive displacement to maximum negative displacement is one half of a period which is 0.05 seconds. Note that the amplitude is not needed.
All of the following are true about waves EXCEPT: A. Electromagnetic waves must be transverse. B. Electromagnetic waves can be either transverse or longitudinal. C. Mechanical waves can be either transverse or longitudinal. D. Mechanical waves require a medium and electromagnetic waves do not.
B; This is a question requiring the comparison of mechanical and electromagnetic waves. Beginning with mechanical: mechanical waves require a medium, since the medium itself oscillates. They can be either transverse (e.g. waves traveling along a string) or longitudinal (e.g. sound). This eliminates choice C, since the statement is true. Electromagnetic waves do not require a medium since it is the electric and magnetic fields that oscillate. This eliminates choice D. The fields oscillate perpendicular to the direction of propagation, making them transverse, thus eliminating choice A.
The speed of a transverse wave along a certain 4 meter long rope is 24 m/s. Which of the following frequencies could cause a standing wave to form on this rope, assuming both ends of the rope are fixed? A. 32 Hz B. 33 Hz C. 34 Hz D. 35 Hz
f1 = nv/2L f1 = (1)(24) / (2)(4) f1 = 3 Hz Has to be a multiple of 3, so answer is B
For a particular rope, it's found that the fundamental frequency is 6 Hz. What is the third harmonic frequency?
fn = nf1 fn = (3)(6Hz) fn = 18 Hz
When x = +A, describe the magnitude of the restoring force and the magnitude of acceleration
max
The bob mass (mass = m) of a simple pendulum is raised to a height h above its lowest point and released. Find an expression for the maximum speed of the pendulum.
mgh = 1/2 mv^2 gh = 1/2v^2 2gh = v^2 v = sqrt 2gh
What is frequency?
number of waves per second
How do you determine if waves are in phase?
path difference = ny (n = 1, 2, 3, etc)
Any motion that repeats is referred to as
periodic or harmonic motion
What are antinodes?
points of maximum amplitude on a standing wave
Frequency and periods are _______
reciprocals
When x = +A, describe the KE and v
0
When x = 0, describe the magnitude of the restoring force and magnitude of acceleration
0 and 0
The distance between any two nodes is ________ the wavelength
1/2
Waves that are directly opposite each other in amplitude are ________ out of phase or ________ radians out of phase
180 pi
Wavelength of the first harmonic of a standing wave
2L
An electrocardiogram responds to changes in the electric potential of the heart from a number of different angles and distances, and represents a different pair of combinations of these signals (voltages) as deflections of several needles under which runs graph paper moving horizontally at constant speed. Suppose a patient has a resting heart rate of 60 beats per minute and the rape runs through the machine at 4 cm/s. What is the wavelength over which the patterns should repeat?
60 beats/min x 1 min/60s = 1 beat/s (frequency) v = fλ λ = v/f λ = 4 cm/s / 1s^-1 λ = 4 cm
The bob on a pendulum moves from point A to point B in .5 seconds. What is the period of oscillation?
A --> B = .5 B --> C = .5 C --> B = .5 B --> A = .5 Period = 2s
A mass is oscillating on a spring with amplitude A and period T. If the motion is simple harmonic, what would the period be if the amplitude were increased to 2A? A. T B. sqrt 2T C. 2T D. 4T
A; For simple harmonic motion, period (and frequency) are independent of amplitude.
A string on a violin, vibrating at an unknown frequency, is played at the same time as another string vibrating at a frequency of 262 Hz. The beat frequency is 4 Hz. If the unknown string is known to be too low in pitch (musically flat), then at what frequency is it vibrating? A. 258 Hz B. 266 Hz C. 258 Hz or 266 Hz D. 258 Hz and 266 Hz
A; The beat frequency, fbeat, is given by fbeat = |f1 - f2| where f1 and f2 are the frequencies of the individual sound waves. Since the beat frequency is 4 Hz, and one of the sound frequencies is 262 Hz, the other frequency must either be 262 + 4 = 266 Hz or 262 - 4 or 258 Hz. Since the string is known to be flat, the frequency must be lower that 262 Hz, so choice A is correct. Note that choice D can be eliminated right away since the string cannot play both frequencies at once.
A wave transmits from a medium of less density to a medium of greater density. Which of the following MUST be the same before and after the wave crossed the boundary (noting that part of the wave reflects backward at the boundary.) I. Wave speed II. Transmitted frequency III. Transmitted energy A. I only B. II only C. II and III only D. III only
B
Suppose two identical springs of constant k are set side by side and attached to a mass m. If the mass is pulled from equilibrium distance A and then released, how do the maximum force and frequency of oscillation compare to those same values with a single spring? A. The values of F and f both increase by a factor of 2 B. The value of F increases by a factor of 2 and f increases by a factor of sqrt 2 C. The value of F stays the same and f increases by a factor of sqrt 2 D. Both values stay the same
B F' = 2F From Hooke's law, F and k are proportional. f is proportional to the square root of k.
How long will it take the wavelength (y) and period (T) to travel a distance of d? A. YTd B. Yd/T C. Td / y D. YT/d
C; Eliminate A and B because units do not work C and D both have s in the numerator C is correct because as d gets bigger, more seconds are needed
A standing wave is created in a 6 m string attached at each end. If the distance between consecutive nodes is 0.5 m, then the string is vibrating in its: A. third harmonic mode. B. sixth harmonic mode. C. twelfth harmonic mode. D. Cannot be determined with the information given.
C; When a string vibrates in its nth harmonic mode, it means there are n half-wavelengths along the length of the string. Since the distance between consecutive nodes is one half of a wavelength, then there must be (6 m) / (0.5 m) = 12 half-wavelengths, and therefore n = 12.
Which one of the following statements is true concerning the amplitude of a wave? A. Amplitude increases with increasing frequency B. Amplitude increases with increasing wavelength C. Amplitude increases with increasing wave speed D. None of the above
D;
Two transverse waves with identical frequencies interfere. If their amplitudes are 10 cm and 15 cm, respectively, what will be resulting amplitude? A. 12.5 cm B. 25 cm C. a value between 0 and 25 cm D. a value between 5 and 25 cm
D; The principle of superposition for waves states that, when two or more waves interfere, the resulting displacement from equilibrium is the sum of the individual displacements. The two extreme cases are 1) when the crests (and the troughs) line up, or 2) when the crest of one lines up with the trough of the other. In the first case, the amplitudes add: 10 cm + 15 cm = 25 cm. In the second case, the amplitudes subtract: 15 cm - 10 cm = 5 cm. Since these represent only the extreme cases, any amplitude between 5 cm and 25 cm is possible.
What is Hooke's Law?
F = -kx F = force k = spring constant (N/m) x = distance (m)
Frequency units
Hertz (Hz)
Elastic potential energy equation
PE = 1/2kx^2 PE = J k = spring constant (N/m) x = distance (m^2)
Period equaiton
T = 1/f
Period of a wave equation
T = 1/f
Period equation for a mass oscillating on a spring
T = 2pi x sqrt m/k
Period of a pendulum equation
T=2π√(L/g)
What is simple harmonic motion?
The back and forth movement of particles when the movement is symmetrical and periodic. Also known as a sine wave
What is the fundamental wavelength?
The first harmony
What are electromagnetic waves?
Transverse waves that transfer energy from the source of the waves to an absorber.
Speed of a wave in a string equation
V = sqrt (tension/linear density)
What are transverse waves?
Waves that move at right angles to the direction of the wave (perpendicular)
As a shortcut, for strings attached at both ends, the number of _________ present tell you which harmonic it is
antinodes
What is the restoring force?
any force that always acts to pull a system back toward equilibrium
Frequency equation for a pendulum
f = 1/2pi x (sqrt g/l) constant is always in the numerator for frequency
Frequency equation for a mass oscillating on a spring
f = 1/2pi x sqrt (k/m) m = mass on spring k = spring constant
Standing wave frequency equation
f = nv/2L
Frequency equation
f=1/T
Frequency of a wave equation
f=1/T
Fundamental frequency equation
fn = nf1
In pendulums, whites negligible?
friction and mass of the string
When two or more waves are superimposed on each other, they will combine to form a single resultant wave. This is called _______
interference
What are wave troughs?
lowest points of a wave
When x = 0, describe the KE and v
max
Examples of mechanical waves
sound waves, water waves, seismic waves
What is the wavelength of a wave?
the distance between two of the same points on a wave
What is wave speed?
the speed at which a wave travels
What is equilibrium position?
the undisturbed position of particles or fields when they are not vibrating
What are standing waves?
two waves of the same amplitude and frequency traveling in opposite directions
A wave of frequency 12 Hz has a wavelength of 3m. What is the speed of this wave?
v = fλ v = 12 (1/s)(3m) v = 36 m/s
Wave speed equation
v = fλ v = speed (m/s) f = frequency (1/s) λ = wavelength (m)
A certain rope transmits a 2 Hz transverse wave of amplitude 10 cm with a speed of 1 m/s. What would be the wavelength of a 5 Hz transverse wave of amplitude 8 cm on the same rope?
v = fλ λ = v/f λ = 1 m/s / 5 s^-1 λ = .2m
vmax equation when given amplitude, spring constant and mass
vmax = A sqrt (k/m)
If the block in example 11-3 above were replaced with a block of mass 800g, how would its maximum speed change?
vmax = A sqrt (k/m) m increases by factor of 4, vmax down by a factor of 2
Length of a standing wave in a second harmonic
λ = 2L/2 λ = L
Wavelength of the second harmonic of a standing wave
λ = 2L/2 = L
Standing wavelength fixed at two ends equation
λ = 2L/n L = length n = harmonic number
If a rope of length 6m supports a standing wave with exactly four nodes (which includes the ends of the rope), what is the wavelength of the standing wave?
λ = 2L/n λ = 2(6) / 3 λ = 4m
Wavelength of a standing wave in the third harmonic
λ = 2L/n λ = 2L/3
Length of a standing wave in the third harmonic
λ = 2L/n λ =2L/3 L = 3λ/2
Fundamental wavelength equation
λ(n) = λ1/n
The second harmonic wavelength for a rope fixed at both ends is .5m. How fast do transverse waves travel along this rope if the fundamental frequency is 4 Hz?
λ(n) = λ1/n λ(2) = λ1/2 2λ2 = λ1 λ 1 = 2λ2 λ 1 = (2)(.5) λ 1 = 1m v = λ1f1 v = 1m x 4 Hz v = 4m/s
Angular velocity equation
ω = θ/t = 2πf = 2π/T
Immediately before a performance, a musician breaks a guitar string. The only string available to repair the guitar is twice the linear density of the string normally used. Hoe can the musician adjust the new string so that it will still have the correct frequency? v = (tension/linear density)^.5 A. The tension in the new string should be twice the tension of the old string B. The tension of the new string should be half the tension of the old string C. The amplitude of the new string should be twice the amplitude of the old string D. The amplitude of the new string should be half the amplitude of the old string
A
When a wave passes from a medium in which it travels slowly to one in which it travels faster, all of the following are true EXCEPT: A. The amplitude in the faster medium is less than or equal to the amplitude in the slower medium. B. The frequency increases when passing into the faster medium. C. The path of the wave, unless it is perpendicular to the interface between the media, bends further from the normal in the faster medium than in the slower. D. The wavelength is longer in the faster medium.
B; When any wave passes from one medium to another, its frequency remains the same, which makes choice B false and the correct answer. The relationship between speed, frequency and wavelength is given by v = fλ. If v increases and f stays the same, then λ must also increase. This eliminates choice D. As a wave enters a faster medium it will bend away from the normal (as an example, think of light travelling into a medium of lower refractive index). This eliminates choice C. Finally, when any wave hits the boundary between two media, some of the wave will reflect back into the original medium. The wave that is transmitted into the faster medium will have less energy, and therefore a smaller amplitude, than the original wave. This eliminates choice A.
A sound wave passes from Medium #1 to Medium #2. If the wavelength in Medium #1 is greater than the wavelength in Medium #2, then which of the following is possible? A. Medium #1 is a gas and Medium #2 is a liquid B. Medium #1 is a liquid and Medium #2 is a gas Correct Answer C. The frequency in Medium #1 is smaller than the frequency in Medium #2 D. The speed in Medium #1 is smaller than the speed in Medium #2
B; When any wave passes into a new medium, its frequency remains the same. This eliminates choice C. The relationship between speed (v), frequency (f) and wavelength (λ) for all waves is given by v = fλ. If λ increases and fstays the same, then v must also increase. This eliminates choice D. Sound travels faster in a liquid than it does in a gas, due to the liquid's comparative incompressibility. Since the speed is greater in Medium #1 than it is in Medium #2, Medium #1 could be a liquid while Medium #2 could be a gas.
A wave of frequency f and speed v is generated in a rope. If a wave of frequency 3f were generated in the same rope, what would its speed be? A. v/3 B. v C. 3v D. Cannot be determined from the information given.
B; With certain exceptions (e.g. the dispersion of light), all waves of the same type in the same medium have the same speed. A wave with three times the frequency of another will travel at the same speed in the same medium, but will have one third its wavelength.
A transverse wave of frequency 4 Hz travels at a speed of 6 m/s along a rope. What would be the speed of a 12 Hz wave along this same rope?
Big rule 1 for waves says that the speed of a wave is determined by the type of wave and the characteristics of the medium. If all we do is change the medium, the wave speed will not change.
A block of mass m attached to a spring with constant k oscillates horizontally on a frictionless surface with amplitude A. In which case does the spring do more work, moving the mass from x =A to x = A/2 or from A/2 to x = 0
In both cases the spring does positive work, since the restoring force is in the same direction as the motion of the block. Since the force is non constant, we cannot use the formula W = Fd cos theta. Instead, the work done by the spring is given by W = -Delta PE(elastic) = -(PE final - PE initial) W = -PE(final - PE initial) (x = A to x = a/2) W = -(1/2k (a/2)^2 - 1/2kA^2 = -(1/8 - 4/8) = 3/8kA x a/2 to x = 0 W = -(1/2k(0)^2 - 1/2k(a/2)^2) = -(0-1/8) = 1/8kA Greater from x =a to x =a/2 than x = a/2 to x = 0
Length of the first harmonic of a standing wave
L = 1/2λ
