AP Physics 2 Assignment – Waves & Interference

 

Reading   Sections:  16.9 – 11; 17.1 – 7; 24.1 – 4; 27.1 – 5, 8 Open Stax College Physics
                Sections:  11.1 – 10; 22.1 – 7; 24.1, 3, 5; 25.1 – 5 Etkina et. al.

 

Objectives/HW

 

 

The student will be able to:

HW:

1

Define, apply, and give examples of the following concepts:  wave, pulse vs. continuous wave, source, medium, longitudinal wave, transverse wave, surface wave, crest, trough, compression, rarefaction, polarization.

1 – 5

2

Define, apply and give examples of the following wave parameters:  speed, wavelength, frequency, period, and amplitude and state the influence of source and medium on each wave parameter.  Solve problems using the relation between speed, wavelength, and frequency.  Solve problems using the relation between wave speed, tension, and mass per length.

6 – 11

3

Identify the wave type, medium, and speed of: mechanical waves, sound, light, and electromagnetic radiation and solve related problems.      

12 – 16

4

Explain, illustrate, and apply the concept of the Doppler effect in qualitative terms relative frequency and/or wavelength to relative motion.

17 – 19

5

Solve problems analyzing graphs and/or sinusoidal functions to determine a wave’s parameters.

20 – 22

6

Define and apply the following concepts:  superposition, constructive and destructive interference, phase, beat frequency and solve related problems.

23 – 26

7

Explain the requirements for the creation of a standing wave.  Define and identify nodes and antinodes in standing wave patterns.  Solve problems involving harmonics for strings or pipes.  Define resonance and identify and give examples of this phenomenon.

27 – 38

8

Model diffraction and interference of light involving slits or gratings by Huygen’s principle and analyze and solve problems relating geometry of openings and/or path difference to patterns of interference.

39 – 47

 

 

Homework Problems

 

1.      Suppose an astronaut standing on the Moon holds and rings a metal bell.  Would that same astronaut hear the bell?  How about any other astronauts in the vicinity? Explain.

2.      Suppose an earthquake occurs in California.  The resulting seismic waves are recorded by special equipment (seismographs) in laboratories all over the world (not just in California).  (a) What is the source of seismic waves?  (b) What is the medium of seismic waves?  (c) There are both P-type and S-type seismic waves – what are the differences?  (do a little research online!)  (d) The Earth’s molten core can only transmit one of these two types – which one and why?  (This has helped scientists determine the size of the core!)

3.      A physics teacher creates waves that travel along a spring that is stretched across the room.  (a) Without changing the length of the spring, can the teacher change the speed of the waves in the spring?  Explain.  (b) Can the teacher change the frequency of the waves in the spring?  Explain.  (c) Can the teacher change the wavelength of the waves in the spring?  Explain.

4.      Sports fans “doing the wave” at a stadium might be viewed as creating a “wave pulse”.  The pulse travels around the stadium, a distance of 400 m, once every 40 s.  It takes 2.0 s for a fan to stand up and sit back down.  (a)  Explain why a wave like this does not have a defined period or wavelength.  (b) Determine the speed of the wave.  (c) Determine the width of the pulse.  (d) Explain why this is not actually a real wave, scientifically speaking.

5.      A certain type of radar gun for measuring speed emits microwaves that are moving horizontally when pointed at a baseball pitch from behind home plate.  It is possible to block the signal of the radar gun using a metal grate consisting of parallel metal rods separated by a few millimeters.  When the rods are vertical the radar beam passes through but when the rods are horizontal the beam is blocked and does not pass through the grate.  (a) Explain how this is evidence that the microwaves are polarized.  (b) If one assumes that the microwave radiation is absorbed by the metal rods, is the electric field of the wave vertical or horizontal and what becomes of the “lost” energy of the wave?   

6.      The speed of waves in the ocean depends on the depth of the water.  The cresting of waves at the shoreline can be attributed to this dependency.  (a) Decreasing depth of water causes wave speed to increase or decrease?  Explain your reasoning.  (b) As the wave comes in, does the wavelength increase, decrease, or stay the same?  Explain.

7.      A physics nerd makes longitudinal waves in a “slinky” spring that is lying atop a table.  The waves have frequency 2.50 Hz and wavelength 18.0 cm.  (a) Calculate the speed of the rarefactions moving through the slinky.  (b) Determine the time for a single coil in the spring to oscillate back and forth once, as the wave goes by.  (c) Determine the new wavelength if the nerd decreases the frequency to 1.50 Hz. 

8.      Two kids are resting on floaties in a lake, separated by 3.0 m.  When a wave goes by they notice that when one kid is at the bottom of a trough, the other kid is 0.30 m higher, at the top of an adjacent crest.  They also count 14 crests pass by 20.0 s time (wow, these must be physics students!).    (a) Determine the frequency of the wave.  (b) Determine the wavelength.  (c) Determine the speed of the waves.

9.      A transverse wave in a wire has speed v, wavelength λ, and amplitude A.  A particle on the wire has a certain average speed vavg as it moves from the bottom of a trough to the top of a crest.  Derive an expression for vavg in terms of v, λ, and A and any appropriate constants.   

10.  A particular piece of elastic cord has mass 5.50 g and length 1.80 m when the tension is 2.00 N.  (a) Determine the value of λ, the mass per unit length.  (b) What is the speed of transverse waves in this cord?  (c) A wave of what frequency in this cord would produce three complete cycles across its length?

11.  The B-string on a guitar has diameter 0.28 mm and is made of steel (density 7.9 g/cm3).  When plucked, the waves in the string have frequency 247 Hz and wavelength 1.3 m.  (a) Determine the tension in the string.  (b) Increasing the tension by 4.5 N results in what frequency waves, given the wavelength remains at 1.3 m (constrained by the length of the guitar).

12.  When a sound wave goes from air to water, its speed and wavelength change, but its frequency does not.  (a) Explain why the frequency is constant.  (b) If the speed increases what must happen to the wavelength going from air to water?  Explain.

13.  Suppose you hear the sound made by a certain tuning fork.  And then you hear the sound made by the same tuning fork but the room’s temperature is higher than before. How do the speed, wavelength, and frequency of this sound at higher temperature compare to the speed, wavelength, and frequency at lower temperature?

14.  A sound wave of wavelength 70.0 cm and speed 330 m/s is produced by a tuning fork that vibrates for 0.500 s.  (a) What is the frequency of the tuning fork?  (b) How many complete cycles are emitted from the tuning fork in this time interval?  (c) For this traveling group of waves (a “wave train”), how far ahead is the compression of the first wave from the compression of the last wave?

15.  The speed of sound in water is 1498 m/s.  A sonar signal of frequency 225 kHz is sent from a ship at a point just below the water surface and 1.80 s later the reflected signal is detected.  (a) How deep is the ocean beneath the ship? (b) How many complete cycles will “fit” between the ship and the ocean floor?  (c) The sound waves transition into the air above the ship – determine the frequency and wavelength, given the speed of sound in air is 343 m/s.

16.  A certain laser emits a light wave consisting of oscillating electric fields of maximum strength 1.70 kV/m and oscillating magnetic fields of maximum strength 5.67 mT.  Both fields oscillate with frequency 462 THz.  (a) Show that the calculation v = E/B gives the speed of light, including correct units.  (b) Determine the wavelength of the laser light.
(c) To graph this wave with an equation y = A sin(ωx) what value of ω should be used?

17.  Neutral hydrogen atoms emit electromagnetic radiation with wavelength 21 cm.  Because hydrogen is the most common element in the universe, this is of great interest to astronomers.  (a) Determine the frequency of this type of radiation.  What type of telescope is needed to detect this – radio, infrared, visible, ultraviolet, X-ray, or gamma?  (b) If scientists determine this type of hydrogen emission has wavelength 23 cm when studying a certain galaxy, is it redshift or blueshift?  And is the galaxy moving away from us or toward us? (c) Determine the frequency of this doppler-shifted radiation.

18.  A radar gun for measuring the speed of a baseball uses microwaves of frequency 10.525 GHz and incorporates the doppler effect.  Microwaves reflected by the baseball will have a wavelength changed by a percentage approximately equal to the ratio v/c.  (a) Determine the wavelength of the waves emitted by the device.  (b) Find the change in wavelength and change in frequency of the reflected waves if the baseball has velocity 45 m/s directly toward the radar gun.

19.  The pitch of a certain train whistle is 800 Hz.  If the train is moving toward you at 40.0 m/s, then you will hear the whistle as a frequency of 906 Hz.  If the train is stationary and you are moving toward it at 40.0 m/s, then you will hear the whistle as a frequency of 893 Hz. (a) When the train is moving, what happens to the wavelength of the whistle’s sound ahead of the train (where you are)?  (b) When the train is not moving, why would you hear an increased frequency due to your own motion?  (No calculations required either answer!)

20.  Turn in the 4 pairs of graphs that were done (or started) in class.  (Can be found at www.milliganphysics.com/Physics2/HW_Wave_Graphs.pdf)  Show work:  for A, λ, and T, simply label the graphs; for f and v, show the calculations done.

21.  A certain sound wave is described by the following two graphs:  y(x) = 2.00 sin(2π x/5) + sin(2π x/3)  y(t) = 2.00 sin(150π t) + sin(250π t)  x = distance in meters, t = time in seconds, y = disturbance level in Pascals.  Use a graphing calculator in Radian mode or other graphing technology such as Desmos to graph the above equations one at a time.  You will need to adjust your viewing window for each graph – try 0 < x < 25 m and 0 < t < 0.10 s.  (a) Make a sketch of the distance graph showing the wave’s shape.  (b) Make a sketch of the time graph showing the wave’s shape.  (These do not have to be graphed on graph paper!)

22.  Use the tracing features of your calculator and any necessary calculations to determine the following parameters for the sound wave of the previous problem:  (a) Wavelength, (b) Period, (c) Frequency, (d) Amplitude, (e) Speed.

23.  The sound wave graphed by the functions given in the problem above could be produced by the superposition of two sounds of what frequencies and amplitudes?

24.  Suppose two sine waves are produced – each with wavelength 10.0 m.  One wave has amplitude 7.00 cm and the other has amplitude 2.00 cm.  Determine the wavelength and amplitude of the superposition of these two waves: (a) if the two waves are exactly in phase, and (b) if the two waves are exactly out of phase.

25.  A guitarist tunes his A-string by comparing it to a sound known to have the correct frequency of 220.0 Hz.  (a) If he hears beats at a frequency of 2 Hz and his string is sharp (pitch is too high) what is the frequency of his string?  (b) If it is desired to get within 219.9 to 220.1 Hz simply by listening, it will be necessary to hear beats with what period?

26.  When waves of any type, mechanical or otherwise, overlap or meet in the same medium the result is called a “superposition” and is equivalent to the summation of the disturbance of the medium caused by each wave.  When electromagnetic waves meet, the same principle applies – explain how “wave superposition” is consistent with a different use of the word “superposition” – the principle for determining a net electric or magnetic field.

27.  Explain what happens at a node in a standing wave produced in a guitar string – why does this part of the string remain at rest even though there are transverse waves in the string?

28.  A standing wave is produced in a thin strip of metal at a frequency of 125 Hz.  The nodes in the pattern are 5.00 cm apart. For the waves in the metal strip, find the following:  (a) wavelength, and (b) speed.

29.  (a) How could a standing wave be created with the light coming from a red laser?  (b) Would it be possible to see the nodes and antinodes of such a pattern?

30.  A string 45.0 cm long is fixed at both ends to a musical instrument.  Waves in the string travel at speed 520 m/s.  Sketch the standing wave, find the frequency and wavelength for each of the following:  (a) the fundamental, (b) the 2nd harmonic, (c) the 3rd harmonic.

31.  A violin string is 25.4 cm long and produces a fundamental frequency of 440.0 Hz (an A).  The violinist frets the string to change the length of the string.  This changes the fundamental.  (a) What is the speed of waves in the string?  (b) What change in length is required to produce a frequency of 523 Hz (a C)?  (c) At its new length what wavelength of sound is produced by the violin?

32.  A kid blows across one end of a hollow tube that is open at each end.  This causes the column of air to resonate at the fundamental frequency so that a certain pitch is heard by the kid.  Describe at least two things the kid can do to create a higher pitch with the same tube.  Describe two ways to get a lower pitch.  

33.  A 12.0 cm pipe is open at both ends.  Sketch the standing wave, find the frequency and wavelength for each of the following:  (a) the 2nd harmonic, (b) the 3rd harmonic.

34.  A 0.560 m pipe is closed at one end.  Sketch the standing wave, find the frequency and wavelength for each of the following:  (a) the fundamental, (b) the next possible harmonic.

35.  (a) A flute sounds a note with a pitch 370 Hz (F#).  Given that it is basically a pipe with openings at each end, what are the frequencies of the four lowest pitched harmonics when the flute makes this sound.  (b) A clarinet sounds the same note as the flute, however, the reed in the mouthpiece essentially closes off that end.  What are the frequencies of the four lowest pitched harmonics that the clarinet makes playing the same note?

36.  An opera singer can cause a crystal glass to shatter just by the sound of her voice.  However this only works if the singer hits a certain note and not other notes.  (a) Explain why the glass shatters at a certain note. (b) Explain why the glass does not shatter at other notes that are just as loud.

37.  A kid plays around with a 76 cm tube of wrapping paper pretending that it is a horn or a megaphone.  However, in reality, the tube can actually serve to amplify the sound of the kids voice.  Determine two frequencies at which the tube will resonate.

38.  An open vertical tube is filled to the top with water and a tuning fork vibrates over its mouth.  The water level is lowered in the tube, and resonance is heard when the water level is 17 cm below the top and again when the water level is 51 cm below the top.  What is the frequency of the tuning fork? 

39.  Diffraction is typically more prominent for waves with longer wavelengths.  How does this help explain the fact that AM radio stations can often be received behind a hill or mountain while FM radio stations cannot?

40.  Light falls on a pair of slits 0.00190 cm apart.  The slits are 80.0 cm from the screen.  The first-order bright line is 1.90 cm from the central bright line.  (a) What is the wavelength of the light?  (b) How far away from the central bright line are the 3rd order bright lines?  (c) Explain why the 3rd order lines are dimmer than the center, 1st, or 2nd lines in the pattern.

41.  Light of wavelength 542 nm falls on a double slit.  First order bright bands appear 4.00 cm from the central bright line on a screen that is 1.20 m from the slits.  (a) How far apart are the slits?  (b) Describe three changes in this type of experiment that would result in doubling the distance from the central line to the first order lines.

42.  An illuminated grating with 4850 lines/cm gives a second-order image at an angle of 37.4°.  Calculate the wavelength of the light.

43.  A certain replica grating has 400 lines/mm.  “Rainbow images” appear on either side of a white light bulb when viewed through the grating held next to the eye.  (a) What is the distance between two adjacent lines in the grating?  (b) Determine the angular width of the second-order rainbow given the range of visible light 400 nm to 750 nm.  (c) Determine how many complete rainbow images this grating forms – support the result mathematically.

44.  A radio station uses two antennas and broadcasts at 600 kHz.  The antennas broadcast the same signal in phase.  Determine the type of interference that would occur at the following three homes where radio receivers are tuned to that station:  (a) Home A is 17.75 km from one tower and 19.25 km from the other.  (b) Home B is 5.00 km from one tower and 5.75 km from the other.  (c) Home C is 8.70 km from one tower and 8.80 km from the other.

45.  Suppose you are listening to a stereo system.  You are located 2.50 m away from the left speaker and 1.75 m away from the right speaker.  Assume the speakers are in phase with one another.  (a) At what frequencies would you experience constructive interference?  (b) At what frequencies would you experience destructive interference?

46.  Laser light of wavelength 532 nm illuminates a single slit of width 0.10 mm.  (a) Determine the angle between the central bright line and the first dark line on either side.  (b) If the interference pattern is projected onto a piece of paper 2.00 m away from the slit how far apart are the dark lines where the brightness is minimized?

47.  Light passing through a tiny rectangular opening of height 0.400 mm and width 0.0800 mm falls upon a screen 95.0 cm away, forming a central bright rectangle of height 0.36 cm and width 1.80 cm.  (a) Determine the wavelength of the light.  (b) Changing the wavelength to half its original value would cause the central rectangle to take on what dimensions?

 

 

Selected Answers

 

6

236

203000

2.90 × 1015 radians/s

9.68 × 106 radians/m

0.30°

8.3°

0.0400 s

0.400 s

10 s

– 1600 Hz

0.70 Hz

25.0 Hz

42.6 Hz

75 Hz, 2.0 Pa and 125 Hz, 1.0 Pa

153 Hz, 2.24 m

222 Hz

any two multiples of 226 Hz

229 Hz, 686 Hz, 1140 Hz, . . .

260 Hz

370 Hz, 740 Hz, 1110 Hz, 1480 Hz

370 Hz, 1110 Hz, 1850 Hz, 2590 Hz

457 Hz, 915 Hz, 1370 Hz, . . .

459 Hz, 0.747 m

471 Hz

500 Hz

578 Hz, 0.900 m

225 kHz

1160 Hz, 0.450 m

1730 Hz, 0.300 m

2860 Hz, 0.120 m

4290 Hz, 0.080 m

1300 MHz

1400 MHz

– 4.0 cm

– 4.3 nm

451 nm

626 nm

649 nm

7.58 × 10–7 m

2.5 μm

16.3 µm

1.50 mm, 1.80 s, 0.556 Hz, 36.0 cm, 0.200 m/s

1.52 mm

10.0 mm, 0.440 s, 2.27 Hz, 927 m, 2100 m/s

0.18 cm × 0.900 cm

2.1 cm

2.85 cm

5.71 cm

10.0 cm

0.300 m

65.6 cm

4.00 m, 0.600 s, 1.67 Hz, 12.0 m, 20.0 m/s

6.0 m

λ = 10.0 m, A = 5.00 cm

λ = 10.0 m, A = 9.00 cm

15.0 m

20 m

165 m

1350 m

0.450 m/s

4.2 m/s

10 m/s

12.5 m/s

25.6 m/s

224 m/s

375 m/s

3.06 g/m

50 N

3.4 mPa, 46.0 ms, 21.7 Hz, 16.0 m, 348 m/s

2.77 Pa