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Chapter 2: Basic Physics SE-SHM Wave

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“I often think about music. I daydream about music. I see my life in the form of music.” - Albert Einstein

The Big Idea

The development of devices to measure time like a pendulum led to the analysis of periodic motion. Motion that repeats itself in equal intervals of time is called harmonic motion. When an object moves back and forth over the same path in harmonic motion it is said to be oscillating. If the amount of motion of an oscillating object (the distance the object travels) stays the same during the period of motion, it is called simple harmonic motion(SHM). A grandfather clock’s pendulum and the quartz crystal in a modern watch are examples of SHM.

Objects in motion that return to the same position after a fixed period of time are said to be in harmonic motion. Objects in harmonic motion have the ability to transfer some of their energy over large distances. They do so by creating waves in a medium. Imagine pushing up and down on the surface of a bathtub filled with water. Water acts as the medium that carries energy from your hand to the edges of the bathtub. Waves transfer energy over a distance without direct contact of the initial source. In this sense waves are phenomena not objects.

Key Concepts

  • The oscillating object does not lose any energy in SHM. Friction is assumed to be zero.
  • In harmonic motion there is always a restorative force, which acts in the opposite direction of the velocity. The restorative force changes during oscillation and depends on the position of the object. In a spring the force is the spring force; in a pendulum it is the component of gravity along the path. In both cases, the force on the oscillating object is directly opposite that of the direction of velocity.
  • Objects in simple harmonic motion do not obey the “Big Three” equations of motion because the acceleration is not constant. As a spring compresses, the force (and hence acceleration) increases. As a pendulum swings, the tangential component of the force of gravity changes, so the acceleration changes.
  • The period, T, is the amount of time for the harmonic motion to repeat itself, or for the object to go one full cycle. In SHM, T is the time it takes the object to return to its exact starting point and starting direction.
  • The frequency, f, is the number of cycles an object goes through in 1 second. Frequency is measured in Hertz (Hz). 1 Hz = 1 cycle per sec.
  • The amplitude, A, is the distance from the equilibrium (or center) point of motion to either its lowest or highest point (end points). The amplitude, therefore, is half of the total distance covered by the oscillating object. The amplitude can vary in harmonic motion but is constant in SHM.
  • A medium is the substance through which the wave travels. For example, water acts as the medium for ocean waves, while air molecules act as the medium for sound waves.
  • When a wave passes through a medium, the medium is only temporarily disturbed. When an ocean wave travels from one side of the Mediterranean Sea to the other, no actual water molecules move this great distance. Only the disturbance propagates (moves) through the medium.
  • An object oscillating with frequency f will create waves which oscillate with the same frequency f.
  • The speed v and wavelength \lambda of a wave depend on the nature of the medium through which the wave travels.
  • There are two main types of waves we will consider: longitudinal and transverse waves.
  • In longitudinal waves, the vibrations of the medium are in the same direction as the wave motion. A classic example is a wave traveling down a line of standing dominoes: each domino will fall in the same direction as the motion of the wave. A more physical example is a sound wave. For sound waves, high and low pressure zones move both forward and backward as the wave moves through them.
  • In transverse waves, the vibrations of the medium are perpendicular to the direction of motion. A classic example is a wave created in a long rope: the wave travels from one end of the rope to the other, but the actual rope moves up and down, and not from left to right as the wave does.
  • Water waves act as a mix of longitudinal and transverse waves. A typical water molecule pretty much moves in a circle when a wave passes through it.
  • Most wave media act like a series of connected oscillators. For instance, a rope can be thought of as a large number of masses (molecules) connected by springs (intermolecular forces). The speed of a wave through connected harmonic oscillators depends on the distance between them, the spring constant, and the mass. In this way, we can model wave media using the principles of simple harmonic motion.
  • The speed of a wave on a string depends on the material the string is made of, as well as the tension in the string. This fact is why tightening a string on your violin or guitar will increase the frequency, or pitch, of the sound it produces.
  • The speed of a sound wave in air depends subtly on pressure, density, and temperature, but is about 343 m/s at room temperature.

Key Equations

  • T = \frac{1}{f} ; period and frequency are inversely related
  • v = f \lambda ; wave speed equals wavelength times oscillation frequency
  • f_{\text{beat}} = |f_1 -f_2| ; two interfering waves create a beat wave with frequency equal to the difference in their frequencies.
  • T_p =2\pi \sqrt{\frac{L}{g}} ; the period of oscillation for a pendulum (i.e. a mass swinging string) swinging at small angles (say, \theta < 15^\circ) withdepends on the length of the pendulum and the acceleration due to gravity.

Key Applications

  • Constructive interference occurs when two waves combine to create a larger wave. This occurs when the peaks of two waves line up.
  • Destructive interference occurs when two waves combine and cancel each other out. This occurs when a peak in one wave lines up with a trough in the other wave.
  • When waves of two different frequencies interfere, a phenomenon known as beating occurs. The frequency of a beat is the difference of the two frequencies.
  • When a wave meets a barrier, it reflects and travels back the way it came. The reflected wave may interfere with the original wave. If this occurs in precisely the right way, a standing wave can be created. The types of standing waves that can form depend strongly on the speed of the wave and the size of the region in which it is traveling.
  • A couple typical standing waves are shown below. The first is the motion of a simple jump-rope. Nodes are the places where the rope doesn’t move at all; antinodes occur where the motion is greatest. The second is a string in its 5^{th} harmonic

  • Longitudinal waves, like sound waves, have compression and rarefaction zones. The compression zones are where, for example in a sound wave traveling through air, the air molecules are densely packed. The rarefaction zones are areas where the air molecules are loosely packed, like a vacuum zone. Us teachers will often draw longitudinal waves as transverse waves, but remember the difference.

  • The DopplerEffect occurs when either the source of a wave or the observer of a wave (or both) are moving. When a source of a wave is moving towards you, the apparent frequency of the wave you detect is higher than that emitted. For instance, if a car approaches you while playing a note at 500 Hz, the sound you hear will be slightly higher. The opposite occurs (the frequency observed is lower than emitted) for a receding wave or if the observer moves away from the source. It’s important to note that the speed of the wave does not change –it’s traveling through the same medium so the speed is the same. Due to the relative motion between the source and the observer the frequency changes, but not the speed of the wave. Note that while the effect is similar for light and electromagnetic waves the formulas are not exactly the same as for sound.

Solved Examples

Example 1: A bee flaps its wings at a rate of approximately 190 Hz. How long does it take for a bee to flap its wings once (down and up)?

Question: T = ? [sec]

Given: f = 190 \ Hz

Equation: T = \frac{1}{f}

Plug n’ Chug: T = \frac{1}{f} = \frac{1}{190 \ Hz} = 0.00526 \ s = 5.26 \ ms

Answer: \boxed{\mathbf{5.26 \ ms}}

Example 2: Wild 94.9 in San Francisco operates at a frequency of 94.9 MHz. What is the wavelength of these waves?

Question: \lambda = ? [m]

Given: f = 94.9 \ MHz = 94.9 \times 10^6 \ Hz

{\;} \qquad \quad v = c = 3.00 \times 10^8 \ m/s

Equation: v = f \cdot \lambda therefore \lambda = \frac{v}{f}

Plug n’ Chug: \lambda = \frac{v}{f} = \frac{3.00 \times 10^8 \ m/s}{94.9 \times 10^6 \ Hz} = 3.16 \ m

Answer: \boxed{\mathbf{3.16 \ m}}

Example 3: While on vacation in Hawaii you observe waves at the Banzai Pipeline approaching the shore at 6.0 m/s. You also note that the distance between waves is 28 m. Calculate (a) the frequency of the waves and (b) the period.

Question a: f= ? [Hz]

Given: v = 6.0 \ m/s

{\;} \qquad \quad \lambda = 28 \ m

Equation: v = f \cdot \lambda therefore f = \frac{v}{\lambda}

Plug n’ Chug: f = \frac{v} {\lambda} = \frac{6.0 \ m/s}{28 \ m} = 0.21 \ Hz

Answer: \boxed{\mathbf{0.21 \ Hz}}

Question b: T = ? [s]

Given: f = 0.21 \ Hz

Equation: T = \frac{1}{f}

Plug n’ Chug: T = \frac{1}{f} = \frac{1}{0.21 \ Hz} = 4.76 \ s

Answer: \boxed{\mathbf{4.76 \ s}}

SHM, Wave Motion and Sound Problem Set

Be sure to show all your work in the space provided!

  1. Take two classic examples of simple harmonic motion and answer the following questions.
    1. Does the period of a pendulum depend upon the pendulum’s mass or length? If the pendulum is lengthened, what happens to its frequency of oscillation? What happens to its period?
    2. Does the period of a mass on a spring depend upon the mass or length of the spring? If the spring is replaced with a ‘stiffer’ spring (and thus a larger spring constant), what happens to its frequency of oscillation? What happens to its period?
  2. If you tap your toe into a calm swimming pool, circular waves result.
    1. What does the fact that those waves are circular tell you about the speed of the wave in different directions?
    2. If you tap your toe more frequently, what happens to the wavelength of those waves? What happens to the period of the waves?
  3. Blue light has a shorter wavelength than red light. Which color has the higher frequency? Which moves faster in a vacuum?
  4. What changes about a wave in the Doppler effect: Frequency? Wavelength? Speed? Is it correct to say that if an ambulance is moving towards you the speed of the sound from its siren is faster than normal? Discuss.
  5. Astronomers can tell from looking at the spectrum of some galaxies that they’re moving – the light is “red-shifted”, meaning the wavelengths look longer than if the galaxies were at rest relative to us.
    1. Are those galaxies moving towards us or away from us?
    2. If an astronomer looked at the sun and saw light “red-shifted” on one side and “blue-shifted” on the other, what would that information tell them about the sun?
  6. A boat moving through the water does not always produce a bow wave, or wake. What has to be true about the boat’s speed to produce a bow wave?
  7. Describe the pressure changes in the air as a sound wave passes a given point, then explain why a very loud sound can damage your tympanic membrane (ear drum).
  8. If the frequency of a sound wave is tripled, what happens to its speed and wavelength? Explain briefly
  9. The Indian instrument called a “sitar” uses two sets of strings, one above the other. Only one set of strings is played but both make sound. Research the sitar and explain briefly how this works. For more information about the Sitar, see http://www.britannica.com/EBchecked/topic/546792/sitar .
  10. “Noise-cancelling” headphones are useful for listening to music on noisy airplane flights – they cancel out the background sound on the plane. They’re also used by jackhammer operators to protect their ears. To be used properly the headphones briefly sample the background sound, then the “noise cancelling” comes on. Explain why this sampling is necessary and how the headphones work. Do a little research online if necessary to answer this question.
  11. While treading water, you notice a buoy way out towards the horizon. The buoy is bobbing up and down in simple harmonic motion. You only see the buoy at the most upward part of its cycle. You see the buoy appear 10 times over the course of one minute.
    1. What is the restoring force that is leading to simple harmonic motion?
    2. What are the period (T) and frequency (f) of its cycle? Use the proper units.
  12. Bored in class, you start tapping your finger on the table. Your friend, sitting right next to you also starts tapping away. But while you are tapping once every second, you’re friend taps twice for every one tap of yours.
    1. What is the Period and frequency of your tapping?
    2. What is the Period and frequency of your friend’s tapping?
    3. Your tapping starts small waves going down the desk. Sort of like hitting a bell with a hammer. The frequency of the sound you hear is 1200 Hz. You know the wave speed in wood is about 3600 m/s. Find the wavelengths generated by your tapping.
  13. Give some everyday examples of simple harmonic motion.
  14. The pitch of a Middle C note on a piano is 263 Hz. This means when you hear this note, the hairs (cilia) in your inner ears wiggle back and forth at this frequency.
    1. What is the period of oscillation for your ear hairs?
    2. What is the period of oscillation of the struck wire within the piano?
  15. You’re sitting on Ocean Beach in San Francisco one fine afternoon and you notice that the waves are crashing on the beach about 6 times every minute.
    1. Calculate the frequency and period of the waves.
    2. You estimate that it takes 1 wave about 4 seconds to travel from a surfer 30 m off shore to the beach. Calculate the velocity and average wavelengths of the wave.
  16. A mass is oscillating up and down on a spring. Below is a graph of its vertical position as a function of time.
    1. Determine the
      1. amplitude,
      2. period and
      3. frequency.
    2. What is the amplitude at t = 32 seconds?
    3. At what times is the mass momentarily at rest? How do you know?
    4. Velocity is defined as change in position over time. Can you see that would be the slope of this graph? (slope = rise over run and in this case the ‘rise’ is position and the ‘run’ is time). Find the instantaneous speed at t = 20 sec.
  17. The Sun tends to have dark, Earth-sized spots on its surface due to kinks in its magnetic field. The number of visible spots varies over the course of years. Use the graph of the sunspot cycle below to answer the following questions. (Note that this is real data from our sun, so it doesn’t look like a perfect sine wave. What you need to do is estimate the best sine wave that fits this data.)
    1. Estimate the period T in years.
    2. When do we expect the next “solar maximum?”
  18. Human beings can hear sound waves in the frequency range 20 Hz – 20 kHz. Assuming a speed of sound of 343 m/s, answer the following questions.
    1. What is the shortest wavelength the human ear can hear?
    2. What is the longest wavelength the human ear can hear?
  19. The speed of light cis 300,000 km/sec.
    1. What is the frequency in Hz of a wave of red light (\lambda = 0.7 \times 10^{-6} \ m)?
    2. What is the period T of oscillation (in seconds) of an electron that is bouncing up and down in response to the passage of a packet of red light? Is the electron moving rapidly or slowly?
  20. Radio signals are carried by electromagnetic waves (i.e. light waves). The radio waves from San Francisco radio station KMEL (106.1 FM) have a frequency of 106.1 MHz. When these waves reach your antenna, your radio converts the motions of the electrons in the antenna back into sound.
    1. What is the wavelength of the signal from KMEL?
    2. What is the wavelength of a signal from KPOO (89.5 FM)?
    3. If your antenna were broken off so that it was only 2 cm long, how would this affect your reception?
  21. Below you will find actual measurements of acceleration as observed by a seismometer during a relatively small earthquake. An earthquake can be thought of as a whole bunch of different waves all piled up on top of each other.
    1. Estimate (using a ruler) the approximate period of oscillation T of the minor aftershock which occurs around t = 40 sec.
    2. Convert your estimated period from part (a) into a frequency f in Hz.
    3. Suppose a wave with frequency f from part (b) is traveling through concrete as a result of the earthquake. What is the wavelength \lambda of that wave in meters? (The speed of sound in concrete is approximately v = 3200 \ m/s.)
  22. The speed of sound v in air is approximately 331.4 \ m/s + 0.6 \ T, where T is the temperature of the air in Celsius. The speed of light c is 300,000 km/sec, which means it travels from one place to another on Earth more or less instantaneously. Let’s say on a cool night (air temperature 10^\circ Celsius) you see lightning flash and then hear the thunder rumble five seconds later. How far away (in km) did the lightning strike?
  23. Reread the difference between transverse and longitudinalwaves. For each of the following types of waves, tell what type it is and why. (Include a sketch for each.)
    • sound waves
    • water waves in the wake of a boat
    • a vibrating string on a guitar
    • a swinging jump rope
    • the vibrating surface of a drum
    • the “wave” done by spectators at a sports event
    • slowly moving traffic jams
  24. For every vertical dashed line, add the two waves together and sketch the resultant wave on the third graph. Be as exact as possible. The two waves have different frequencies, but the same amplitude of 1 m. What is the frequency of the resultant wave? How will the resultant wave sound different?
  25. The simple bridge shown here oscillated up and down pretty violently four times every second as a result of an earthquake.
    1. What was the frequency of the shaking in Hz?
    2. Why was the bridge oscillating so violently?
    3. Calculate two other frequencies that would be considered “dangerous” for the bridge.
    4. What could you do to make the bridge safer?
  26. The length of the western section of the Bay Bridge is 2.7 km.
    1. Draw a side-view of the western section of the Bay Bridge and identify the seven nodes in this section of the bridge.
    2. Assume that the bridge is concrete (the speed of sound in concrete is 3200 m/s). What is the lowest frequency of vibration for the bridge? (You can assume that the towers are equally spaced, and that the central support is equidistant from both middle towers. The best way to approach this problem is by drawing in a wave that “works.”)
    3. What might happen if an earthquake occurs that shakes the bridge at precisely this frequency?

  27. Aborigines, the native people of Australia, play an instrument called the Didgeridoo like the one shown above. The Didgeridoo produces a low pitch sound and is possibly the world’s oldest instrument. The one shown above is about 1.3 m long and open at both ends.
    1. Knowing that when a tube is open at both ends there must be an antinode at both ends, draw the first 3 harmonics for this instrument.
    2. Calculate the frequency of the first 3 harmonics assuming room temperature and thus a velocity of sound of 340 m/s. Then take a shot at deriving a generic formula for the frequency of the nth standing wave mode for the Didgeridoo, as was done for the string tied at both ends and for the tube open at one end.
  28. At the Sunday drum circle in Golden Gate Park, an Indian princess is striking her drum at a frequency of 2 Hz. You would like to hit your drum at another frequency, so that the sound of your drum and the sound of her drum “beat” together at a frequency of 0.1 Hz. What frequencies could you choose?
  29. Students are doing an experiment to determine the speed of sound in air. They hold a tuning fork above a large empty graduated cylinder and try to create resonance. The air column in the graduated cylinder can be adjusted by putting water in it. At a certain point for each tuning fork a clear resonance point is heard. The students adjust the water finely to get the peak resonance then carefully measure the air column from water to top of air column. (The assumption is that the tuning fork itself creates an anti-node and the water creates a node.) The following data were collected:
Frequency of tuning fork (Hz) Length of air column (cm) Wavelength (m) Speed of sound (m/s)
184 46
328 26
384 22
512 16
1024 24
    1. Fill out the last two columns in the data table.
    2. Explain major inconsistencies in the data or results.
    3. The graduated cylinder is 50 cm high. Were there other resonance points that could have been heard? If so what would be the length of the wavelength?
    4. What are the inherent errors in this experiment?
  1. A train, moving at some speed lower than the speed of sound, is equipped with a gun. The gun shoots a bullet forward at precisely the speed of sound, relative to the train. An observer watches some distance down the tracks, with the bullet headed towards him. Will the observer hear the sound of the bullet being fired before being struck by the bullet? Explain.
  2. Peter is playing tones by blowing across the top of a glass bottle partially filled with water. He notices that if he blows softly he hears a lower note, but if he blows harder he hears higher frequencies. 
    1. In the 120 cm long tubes below draw three diagrams showing the first three harmonics produced in the tube. Please draw the waves as transverse even though we know sound waves are longitudinal (reason for this, obviously, is that it is much easier to draw transverse waves rather than longitudinal). Note that the tube is CLOSED at one end and OPEN at the other.  
    2. Calculate the frequencies of the first three harmonics played in this tube, if the speed of sound in the tube is 340 m/s. 
    3. The speed of sound in carbon dioxide is lower than in air. If the bottle contained CO2 instead of air, would the frequencies found above be higher or lower? Knowing that the pitch of your voice gets higher when you inhale helium, what can we say about the speed of sound in He.

Answers:

  1. 6s, 0.167 Hz

    1. 1.0s, 1 Hz
    2. 0.5s, 2 Hz
    3. 3 m

    1. 0.0038 s
    2. 0.0038 s

    1. 0.1 Hz, 10.0 s
    2. 7.5 m/s, 75 m


      1. 8 m;
      2. 20 s ;
      3. 0.05 Hz;
    1. – 4 m
    2. \sim 2.3 m/s

    1. About 11 years
    2. About 2014

(see http://solarscience.msfc.nasa.gov/SunspotCycle.shtml for more info)


    1. 1.7 cm
    2. 17 m
    1. 4.3 \times 10^{14} \ Hz
    2. 2.3 \times 10^{-15} \ s

    1. 2.83 m
    2. 3.35 m
    3. rule of thumb, antenna should be \frac{1}{4} \lambda, so quality of reception will suffer
    1. about 1.5 s
    2. about 0.67 Hz
    3. 4.8 km (for above estimates)
  1. 1.7 km

    1. 4 Hz
    2. It was being driven near its resonance frequency
    3. 8 Hz, 12 Hz

    1. 3 wavelengths, 6^{th} harmonic
    2. 3.56 Hz
    3. Resonance will occur, the amplitude will go out of control, bridge will collapse
  1. b. 131 Hz, 262 Hz, 393 Hz; formula is same as closed at both ends
  2. 1.9 Hz or 2.1 Hz
  3. Discuss in class
  4. Struck by bullet first
  5. b. 70.8 Hz, 213 Hz, 354 Hz  c. voice gets lower pitch. Speed of sound in He must be faster by same logic.

OPTIONAL PROBLEMS

  1. A violin string vibrates, when struck, as a standing wave with a frequency of 260 Hz. When you place your finger on the same string so that its length is reduced to 2/3 of its original length, what is its new vibration frequency?
  2. On the moon, how long must a pendulum be if the period of one cycle is one second? The acceleration of gravity on the moon is 1/6 th that of Earth.
  3. You have been chosen by your physics teacher and NASA to be the first Menlo student on Mars – congratulations! Your mission, should you choose to accept it (you do), is to measure the acceleration due to gravity on Mars using only a stopwatch and a rock tied to a 12-cm string. You must use both pieces of equipment, nothing more or less. i. Describe your experiment. ii. If the pendulum completes 10 full swings in 11.5 sec, calculate g on Mars. iii. Look up the accepted value for g on Mars – are you close? iv. Suppose you want to use a new string that will give a period twice as long as your original one. How long should the string be? See if you can reason this out without actually calculating the length using the pendulum formula.

Answers to Optional problems:

  1. 390 Hz
  2. 4.1 cm
  3. ii. 3.5 \ m/s^2

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    Oct 09, 2013

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