Chapter 11 Oscillations and Waves

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

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

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 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 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 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.

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.

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.

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.

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.

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.

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

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

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

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)

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;

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.

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.

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.

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

What is Hooke's Law?

F = -kx F = force k = spring constant (N/m) x = distance (m)

Restoring force of a pendulum

F_restoring = m*g*sin(theta)

Frequency units

Hertz (Hz)

What are wave crests?

Highest point of a wave

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

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 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 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)

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

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

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

Frequency equation

f=1/T

Frequency of a wave equation

f=1/T

Fundamental frequency equation

fn = nf1

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

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

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 = +A, describe the magnitude of the restoring force and the magnitude of acceleration

max

When x = 0, describe the KE and v

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 of waves are out of phase?

path difference = (n + 1/2)y

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

What are nodes of waves?

points of no displacement

Frequency and periods are _______

reciprocals

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

How are frequency and period dependent on amplitude?

they aren't

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