Assignment 10: Vibrations and Waves. Sound

How many decibels correspond to the lowest-intensity sound we can hear?

0 decibels

If the frequency of a particular wave is 20 Hz, its period is _________.

1/20 second

Recall that in the equilibrium position, the upward force of the spring balances the force of gravity on the weight. Use this concept to estimate the spring constant k of spring #1. Express your answer in N/m to one significant figure.

10 N/m

How many vibrations per second are represented in a radio wave of 101.7 MHz?

101,700,000 vibrations

Now, for parts F-I, you will investigate how the period of oscillation depends on the properties of the pendulum. The period of oscillation is the amount of time it takes for the pendulum to take a full swing, going from the original angle to the other side, and returning to the original angle. You can determine the period by selecting other tools, which gives you a stopwatch. With the pendulum swinging, you can start the stopwatch when the pendulum is at its original angle and time how long it takes to complete 10 swings. The period will be this time interval divided by 10 (this method is more accurate than trying to time one swing). Set the length of the pendulum to 1.0 m and the mass to 1.0 kg. Click Reset, and then drag the pendulum to an angle (with respect to the vertical) of 30∘ and release it. What is the period of oscillation?

2.0 s A 1.0-m/1.0-kg pendulum completes one oscillation in 2.0 s.

What beat frequency will occur when a 370-Hz and a 374-Hz sound source are sounded together?

4 Hz

If the fundamental frequency of a note is 200 Hz, what is the frequency of the third harmonic?

600 hz

Now, keep the mass fixed to any value you choose and measure the period for several different pendulum lengths. How does the period of the pendulum depend on the length?

A longer pendulum has a longer period. A longer length causes the period to increase. In fact, as you can verify with the stopwatch, the period of oscillation is proportional to the square root of the length; if you increase the length from 0.5 m to 2.0 m, the period doubles.

What is meant by the fundamental frequency of a musical note?

A partial tone at the lowest frequency

What is a reverberation?

A sound that persists after many reflections

The speed of sound in air varies with _________.

Air temperature

What kind of waves can exhibit interference?

All kinds of waves

What is meant by a blue shift and a red shift for light?

An approaching source has increased light frequency - a blue shift. A receding source has a decreasing frequency - a red shift.

Distinguish between analog and digital for how the sound signal is captured on phonograph records versus a CD?

Analog recordings on a record store a continuous trace of motion, whereas digital recordings on a CD record digital values sampled many times a second.

The graphs shown in (Figure 1) represent pressure variation versus time recorded by a microphone. Which could correspond to a sound wave? Enter the letters of all the correct answers in alphabetical order. Do not use commas. For example, if you think all three graphs could represent sound waves, enter ABC.

BC

How does the cooler air above the lake affect the movement of sound at night?

Because it is colder just above the surface and warmer higher above, the sound wave is refracted toward the ground.

Now, for parts D-F, you'll investigate the energetics of the spring. Select 1 in the Show Energy of box, which shows an energy bar diagram. Select the g = 0 option (under the planet names), which simulates what happens without any gravitational forces (and consequently removes gravitational potential energy from the energetics). Adjust the friction slider to none (this prevents any thermal energy from being generated). Place a weight on spring #1, stretch it, and release it. Watch how the kinetic energy and elastic potential energy vary with time. (You can slow down or stop time using the buttons next to the list of planets.) When is the elastic potential energy of the spring a maximum?

Both when the spring is most compressed and when the spring is most stretched The elastic potential energy depends on the magnitude of the change in the length of the spring. Mathematically, the elastic potential energy is given by Uelas=(1/2)kx2, where k is the spring constant and x is the difference between the length of the spring and its unweighted length. A compression of x therefore results in the same elastic potential energy as a stretch of x.

Distinguish between constructive interference and destructive interference.

Constructive interference occurs when the crests of two waves add together. Destructive interference occurs when a crest of one wave is reduced by the trough of another.

True or false: A sonic boom occurs only when an aircraft is breaking through the sound barrier. Defend your answer.

False. An aircraft creates a shock wave continuously as it is supersonic. This is heard as a sonic boom by an observer.

What ultimately happens to the energy of sound in the air?

Faster in warm air, faster in humid air

A high-fidelity sound system may have a frequency range that extends up to or beyond 20,000 Hz. What is the purpose of this extended range?

High frequencies are needed to accurately reproduce the quality of sounds.

How does a high-pitch musical note relate to frequency?

High pitch is how the ear perceives high frequencies.

Distinguish between a compression and a rarefaction.

In a compression, the molecules are closer together than average; in a rarefaction, they are further apart.

Distinguish between infrasonic and ultrasonic sound waves.

Infrasonic is too low in frequency (below 20 Hz) for humans to hear, whereas ultrasonic is too high (above 20,000 Hz).

What feature of sound is measured in decibels?

Intensity

What is the difference between sound intensity and loudness?

Intensity is power per meter squared, whereas loudness is the physiological perception of intensity.

How does the highest pitch one can hear vary with age?

It decreases.

What factors does the speed of sound depend upon? What are some factors that it does not depend upon?

It depends on temperature and humidity, not on loudness and frequency.

What is the difference in the speed of sound on a warm day versus on a cold day?

It is faster on a warm day and slower on a cold day.

A certain sound is recorded by a microphone. The same microphone then detects a second sound, which is identical to the first one except that it has twice the frequency. In addition to the higher frequency, what distinguishes the second sound from the first one?

It is perceived as higher in pitch.

A certain sound is recorded by a microphone. The same microphone then detects a second sound, which is identical to the first one except that the amplitude of the pressure fluctuations is larger. In addition to the larger amplitude, what distinguishes the second sound from the first one?

It is perceived as louder.

In the Doppler effect, does frequency change? Does wave speed change?

Only frequency changes

In what direction are the vibrations relative to the direction of wave travel in a longitudinal wave?

Parallel to the direction of wave travel

Distinguish between these different aspects of a wave: period, amplitude, wavelength, and frequency.

Period is the time from one crest to the next, amplitude is the distance from the midpoint to a crest, wavelength is the distance from one crest to the next, and frequency is the number of crests that pass per second.

What are the three principal characteristics of musical tones?

Pitch, loudness, quality

Select to show the energy of pendulum 1. Be sure that friction is set to none. Drag the pendulum to an angle (with respect to the vertical) of 30∘, and then release it. When the pendulum is at−30∘, what form(s) of energy does it have?

Potential energy The pendulum starts off with no kinetic energy since it is released from rest, so it initially only has potential energy. When the pendulum is at−30∘, it is just as high above the ground as when it started, so it must have the same amount of potential energy as it initially had. Since the total energy is conserved, it can't have other forms of energy at−30∘ (if it did, it would have more energy there than it initially had) so it again has only potential energy.

Having established that a sound wave corresponds to pressure fluctuations in the medium, what can you conclude about the direction in which such pressure fluctuations travel?

Pressure fluctuations travel along the direction of propagation of the sound wave.

Based on the information presented in the introduction of this problem, what is a sound wave?

Propagation of pressure fluctuations in a medium

What is an echo?

Reflection of sound is an echo.

Relative to solids and liquids, how does air rank as a conductor of sound?

Relative to solids and liquids, air is a poor conductor of sound.

What is ultrasound?

Sound above the frequency of human hearing

Sound waves transport energy from one place to another.

Sound waves transport energy from one place to another.

Drag the pendulum to an angle (with respect to the vertical) of 30∘, and then release it. Select to show the acceleration vector. With the pendulum swinging back and forth, at which locations is the acceleration equal to zero?

The acceleration is never equal to zero as it swings back and forth. The pendulum is moving in a circular path so its velocity is never constant. In fact, for most locations, the acceleration has both a radial component (the centripetal acceleration, which is directed along the rope) and a tangential component (due to the speed changing, directed along the path; the only place the tangential acceleration is zero is when the angle is 0∘).

Given a wave of a particular wavelength and amplitude, what must be the amplitude, wavelength, and phase change of a wave you add to this wave to create a wave of twice the amplitude?

The added wave must have the same amplitude, the same wavelength, and a phase difference of 0 degrees with respect to the original wave.

Given a wave of a particular wavelength and amplitude, what must be the amplitude, wavelength, and phase change of a wave you add to the existing wave to completely destroy it?

The added wave must have the same amplitude, the same wavelength, and a phase difference of 180 degrees with respect to the original wave.

How does the V shape of a bow wave depend on the speed of the source?

The angle of the V gets smaller as the speed of the source gets faster.

Why do the same notes plucked on a banjo and on a guitar have distinctly different sounds?

The banjo and guitar have different amplitude partials when playing the same note, so they have a different quality of sound, too.

For maximum constructive interference between two waves, how must their crests and troughs align with each other?

The crests and troughs of one wave should align with the crests and troughs of the other wave respectively.

For complete destructive interference between two waves, how must their crests and troughs align with each other?

The crests and troughs of one wave should align with the troughs and crests of the other wave respectively.

Now, for parts G-I, you'll investigate what determines the frequency of oscillation. For these parts, turn off the friction using the slider bar. Select the stopwatch, and time how long it takes for a weight to oscillate back and forth 10 times. The period of oscillation is this time divided by 10. The frequency of oscillation is one divided by the period. How does the frequency of oscillation depend on the mass of the weight?

The frequency decreases as the mass increases. A greater mass results in a lower frequency and a longer period of oscillation.

What happens to the frequency of the wave if you increase the wavelength, keeping the velocity of the wave constant?

The frequency decreases.

The spring constant of spring #3 can be adjusted with the softness spring 3 slider bar (harder means a greater spring constant, or stiffer spring). How does the frequency of oscillation depend on the spring constant?

The frequency increases as the spring constant increases. It turns out that the frequency of oscillation depends on the square root of the ratio of the spring constant to mass: f=(k/m)−−−−−√/2π, where f is the frequency. A stiffer spring constant causes the frequency to increase. Sports cars use stiff springs, whereas large plush Cadillacs use soft springs for their suspension.

What happens to the frequency of the wave if you increase the wave velocity, keeping the wavelength of the wave constant?

The frequency increases.

The amplitude of oscillation is the maximum distance between the oscillating weight and the equilibrium position. Determine the frequency of oscillation for several different amplitudes by pulling the weight down different amounts. How does the frequency depend on the amplitude of oscillation?

The frequency is independent of the amplitude. Even though the weight has to travel farther each oscillation if the amplitude is greater, the spring on average exerts a stronger force, causing a greater acceleration and a greater average speed. The effects of the longer distance and faster speed cancel out so that the period of oscillation doesn't change!

Which has the longer period, a short or a long pendulum?

The long pendulum has the longer period.

Keeping the length of the pendulum fixed, determine the period for a few different masses. (Alternatively, you can set up two pendulums by selecting Show 2nd pendulum. Adjust the lengths to be the same, and have one pendulum with a higher mass. You can release one and then release the other, with the same angle, when the first one is back at that angle.) How does the period of the pendulum depend on mass?

The period is independent of the pendulum's mass. The period of a pendulum does not depend on mass. The reason for this result is very similar to the reason that, without air resistance, all objects fall to the ground at the same rate (changing the mass changes both the inertia and the force of gravity by the same amount).

How does the period of oscillation depend on the initial angle of the pendulum when released? (Be sure to measure the period for initial angles much greater than

The period is longer when the initial angle is greater. Unlike a harmonic oscillator such as a mass on a spring, the period actually depends on the initial angle. For small angles (e.g.,<30∘), it is a pretty good approximation that the period doesn't change, but for larger angles the period does in fact increase.

Now, change the planet where the experiment takes place to see how the period of oscillation depends on the acceleration due to gravity, g (on Earth, g≈10 m/s/s; g is larger than this value on Jupiter and smaller than this value on the Moon). How does the period of oscillation depend on the value of g?

The period of oscillation is shorter on planets with a higher value of g. A higher value of g gives rise to greater accelerations, more energy, and thus greater average speeds. In fact, for small angles of oscillation, the period is inversely proportional to the square root of g; T∝L/g−−−√, where L is the length and T is the period.

How did wind-generated resonance affect the Tacoma Narrows Bridge in the state of Washington in 1940?

The resonant amplification of bridge motion destroyed the bridge.

How does the direction of sound travel compare to the shapes of the sound waves?

The sound is at right angles to the waves.

What is the source of all waves?

The source is something that is vibrating.

Place a 50 g weight on spring #1, and release it. Eventually, the weight will come to rest at an equilibrium position, with the spring somewhat stretched compared to its original (unweighted) length. At this point, the upward force of the spring balances the force of gravity on the weight. With the weight in its equilibrium position, how does the amount the spring is stretched depend on the mass of the weight

The spring stretches more for a heavier weight. Since the force of gravity on the weight increases as the mass increases, the upward force of the spring must increase for the two forces to balance (and the weight to therefore be in equilibrium). The force the spring exerts on the weight increases the more the spring is stretched from its unweighted length.

Why will a struck tuning fork sound louder when it is held against a table?

The table is forced to vibrate, and its large surface vibrates more air than the tuning fork.

Why did the campers in Dr. Hewitt's story hear the sound of the campers across the lake more clearly at night than during the day?

The temperature change from cold at the surface to warm at the top changed the shape of the sound waves.

With the pendulum swinging back and forth, how does the tension of the rope compare to the force of gravity when the angle is?

The tension is greater than the force of gravity. Since the acceleration of the pendulum is directed up when the angle is 0∘, the net force must be directed up (Newton's 2nd law). This means that the upward force of tension must be stronger than the downward force of gravity.

Drag the pendulum to an angle (with respect to the vertical) of 90∘, and then release it. With the pendulum swinging back and forth, where is the tension equal to zero?

The tension is zero at the angles+90∘ and−90∘. At these locations, the acceleration is solely due to gravity and directed downward. Thus, the net force acting on the pendulum is also directed downward, meaning there are no horizontal forces. This requires the tension to be zero.

How do the waves of sound travel on the night that produces the temperature inversion, and why?

The waves bend toward the ground because it is warmer above and colder lower below, and waves travel more slowly through the colder air.

What are the three principal classes of musical instruments?

Vibrating strings, vibrating air, and percussion

What is the relationship among frequency, wavelength, and wave speed?

Wave speed is frequency multiplied by wavelength.

How do forced vibrations relate to resonance?

When forced vibrations match an object's natural frequency, resonant amplification occurs.

Select Earth in the menu box so that there is now a force of gravity. Now the total energy of the mass/spring system is the sum of the kinetic energy, the elastic potential energy, and the gravitational potential energy. When is the kinetic energy a maximum? (It may help to watch the simulation in slow motion - 1/16 time.)

When the mass is at the equilibrium position The total potential energy (the gravitational potential energy plus elastic potential energy) is a minimum at the equilibrium position, even though there is some elastic potential energy when the mass is at this location. The kinetic energy is always a maximum when the total potential energy is a minimum (since the total energy is conserved).

When is the kinetic energy of the mass a maximum?

When the spring is at its unweighted length (when it isn't stretched or compressed) The total energy of the system is equal to the kinetic energy of the mass (since the spring has negligible mass) plus the elastic potential energy. Since this total energy is conserved, the kinetic energy is a maximum when the elastic potential energy is a minimum, which occurs when the spring is at its unweighted length.

How can you double the frequency of a wave if you have control over both the wavelength and the wave velocity?

You can either halve the wavelength, keeping the wave speed constant; or double the wave speed, keeping the wavelength constant.

Interference is a phenomenon that occurs primarily in _________.

all kinds of waves

What kinds of waves can show interference?

all waves show interference

The vibrations along a longitudinal wave move in a direction _________.

along and parallel to the wave

Digital signals for music are a direct measure of sound _________.

amplitude

Drag the pendulum to an angle (with respect to the vertical) of 30∘, and then release it. Where is the pendulum swinging the fastest?

at 0∘ The pendulum has the least potential energy at this location since it is at the lowest point in the arc (in fact, for this simulation, the potential energy reference location is here, so it has no potential energy). This means that the kinetic energy is greatest here, so the pendulum is moving the fastest

Can standing waves be formed with transverse waves or longitudinal waves?

both

Does sound tend to bend upward or downward when its speed is less near the ground?

downward

The next graph in (Figure 2) shows a sound wave consisting of a sinusoidal displacement of air particles versus time, as recorded at a fixed location. For sinusoidal waves, it is possible to identify a specific frequency (rate of oscillation) and wavelength (distance in space corresponding to one complete cycle). Taking the speed of sound in air to be 344 m/s, what are the frequency f and the wavelength λ of the sound wave shown in the graph? Express your answers in, respectively, hertz and meters to three significant figures. Separate the two answers with a comma.

f, λ = 1000,0.344 Hz,m

A shock wave occurs when an aircraft travels _________.

faster than the speed of sound

When a sound source approaches a stationary listener, the listener experiences an increase in sound _________.

frequency

The phenomenon of beats is most closely related to _________.

interference

Resonance occurs when forced vibration _________.

matches natural frequency

Does the medium in which a wave travels move with the wave?

no

What most distinguishes noise from music is the presence of _________.

regularity

Forced vibrations mainly occur in _________.

sounding boards

Refraction of sound occurs with changes in wave _________.

speed

What varies between two tones that are different in timbre, that is, two tones that have the same fundamental frequency but are produced, say, by different musical instruments? Note that the graphs B and C in (Figure 1) could represent tones with different timbre

the harmonic content

The pitch of a note on a guitar string can be increased by increasing the _________.

tightness of the string

Compared with the first harmonic of a musical note, the second harmonic has _________.

twice the frequency

Does air play a role in the propagation of the human voice from one end of a lecture hall to the other?

yes