1

MW Assignments: Mechanical Waves

Questions:

1. The figure at right shows three waves that are separately sent along a string that isstretched under a certain tension along the x-axis. Rank the waves according to their (a) wavelengths, (b) speeds, and (c) angular frequencies, greatest first. (HRW7e Q16.1 , i.e. Ch. 16, question #1 on p. 436)

2. Two waves travel on the same string. Is it possible for them to have (a) differentfrequencies; (b) different wavelengths; (c) different speeds; (d) different amplitudes; (e) thesame frequency but different wavelengths? Explain your reasoning. (YF12e Q15.1 on p. 518 in our text)

3. The four strings on a violin have different thicknesses, but all are under approximately the same tension. Do thewaves travel faster on the thick strings or the thin strings? Why? How does the fundamental vibration frequencycompare for the thick versus the thin strings? (YF12e Q15.12)

4. A long rope with mass m is suspended from the ceiling and hangs vertically. A wave pulse is produced at thelower end of the rope, and the pulse travels up the rope. Does the speed of the wave pulse change as it moves upthe rope? If not, why not? If so, does it increase or decrease? (YF12e Q15.15)

Problems:

1. The following wave functions represent traveling waves:

(a) y2 (x , t ) = Acos [k(x + 34t)]

(b) y3 (x , t ) = Ae-k (x - 20 t)

(c) y1 (x , t ) = B/[C + (x - 10t)2]

where x is in meters, t is in seconds, and A, k, B, and C are constants that have the proper units for y to be inmeters. Give the direction of propagation and the speed of the wave for each wave function. (T13.6)

2. A transverse traveling wave has the equation:

y = (6.0 cm) sin (0.02Bx + 4Bt),

where x and y are in centimeters and t is in seconds. Find (a) the amplitude, (b) the wavelength, (c) the frequency,(d) the speed, (e) the direction, and (f) the maximum transverse speed of the wave. (HRW6e17.6; see also YF12e

15.5 & 8)

3. A continuous sinusoidal wave is traveling on a string with velocity 80 cm/s. The displacement of the particlesof the string at x = 10 cm is found to vary with time according to the equation:

y = (5.0 cm) sin(1.0 - 4.0 t).

The linear density of the string is 4.0 g/cm. What is (a) the frequency of the wave, (b) the wavelength of the wave,(c) the general equation of the wave, and (d) the tension in the string? (HR19.10)

2

4. Two connecting wires with linear mass densities that are related by :1 = 3:2 are under the same tension. When the wires oscillate at a frequency of 120 Hz, waves of wavelength 10 cm travel down the first wire with thelinear density of :1 . (a) What is the wave speed in the first wire? (b) What is the wave speed in the second wire? (c) What is the wavelength in the second wire? (T13.58)

5. A wire 10.0 m long and having a mass of 100 g is stretched under a tension of 250 N. If two pulses, separatedin time by 30.0 ms, are generated, one at each end of the wire, where will the pulses first meet? (HRW6e 17.21)

6. The wave function for a certain standing wave on a string fixed at both ends isgiven by:

y(x,t) = 0.5 sin (0.025x) cos (500t)

where y and x are in centimeters and t is in seconds. (a) Find the speed andamplitude of the two traveling waves that result in this standing wave. (b) Whatis the distance between successive nodes on the string? (c) What is the shortestpossible length of the string? (T13.36)

7. In the figure at right, an aluminum wire, of length L1 = 60.0 cm, cross-sectional area 1.00 × 10-2 cm2, anddensity 2.60 g/cm3, is joined to a steel wire of density 7.80 g/cm3 and the same cross-sectional area. Thecompound wire, loaded with a block of mass m = 10.0 kg, is arranged so that the distance L2 from the joint to thesupporting pulley is 86.6 cm. Transverse waves are set up on the wire by an external source of variable frequency;a node is located at the pulley. (a) Find the lowest frequency that generates a standing wave having the joint as oneof the nodes. (b) How many nodes are observed at this frequency? (HRW7e 16.55; see also YF12e 15.15 & 40 & 68)

See Also Problem Summary: Ch. 15 - problems 5, 8, 15, 40, & 68 in the text (YF12e) - pp. 519...524(Brief solutions to the see also problems are posted on the class Blackboard site.)

Brief Answers to Questions:

1. (a) 3 > 2 = 1 for wavelength(b) speed is the same for all: since all waves travel on the same string & tension F is constant

(c) 6 frequency is larger when wavelength is smaller.

So: 1 = 2 > 3 for frequency f and angular frequency T.

2. (a) yes (b) yes (c) no (d) yes (e) no

3. Speed of waves is larger on thinner wires. The fundamental frequency is the lower on the thicker wires.

4. Speed increases as the pulse moves up the rope.

3

SW Assignments: Sound Waves

Questions:

1. In the figure at right, two point sources S1 & S2, which are in phase, emitidentical sound waves of wavelength 2.0 m. In terms of wavelengths, what isthe phase difference between the waves arriving at point P on the far right if(a) L1 = 38 m and L2 = 34 m, and (b) L1 = 39 m and L2 = 36 m? (c) Assumingthat the source separation is much smaller than L1 & L2 , what type ofinterference occurs at P in situations (a) & (b)? (HRW7e Q17.1)

2. The figure at right shows a stretched string of length L andpipes a, b, c, & d of lengths L, 2L, L/2, & L/2 respectively. Thestring’s tension is adjusted until the speed of the waves on thestring equals the speed of sound waves in air. The fundamentalmode of oscillation is then set up on the string. In which pipewill the sound produced by the string cause resonance, andwhat oscillation mode will that sound set up? (HRW7e Q17.7)

3. If you wait at a railroad crossing as a train approaches and passes, you hear a Doppler shift in the sound emittedby the train’s whistle. But if you listen closely, you hear that the change in frequency is continuous; it does notsuddenly go from one high frequency to another low frequency. Instead the frequency smoothly (but ratherquickly) changes from high to low as the train passes. Why is the change smooth rather than sudden? (YF12e

Q16.21 reworded)

Problems:

1. You are at a large outdoor concert, seated 300 m from the speaker system. The concert is also being broadcastlive via satellite (at the speed of light). Consider a listener 5000 km away who receives the broadcast. Who hearsthe music first, you or the listener, and by what time difference? (HRW 6e 18.2)

2. A stone is dropped into a well. The sound of the splash is heard 3.00 s later. What is the depth of the well? (HRW6e 18.7P)

3. Two loudspeakers are driven in phase by an audio amplifier at a frequencyof 600 Hz. The speakers are on the y-axis, one at y = +1.00 m and the other aty = - 1.00 m. A listener begins at y = 0 and walks along a line parallel to they-axis at a very large distance D away. (See diagram at right.) (a) At whatangle 2 will the person first hear a minimum in the sound intensity? (b) Atwhat angle will a maximum first be heard (after 2 = 0)? (c) How many maximawill be heard if the person keeps walking in the same direction? (T14-80; see

also YF12e 16.33; HINT: D is very large, so you may assume that the two lines from

the two loudspeakers are parallel. In that case the path difference between them is (2.00

m)sin 2.)

4

4. The water level in a vertical glass tube 1.0 m long can be adjusted to any position in the tube. A tuning forkvibrating at 686 Hz is held just over the open top end of the tube. At what positions of the water level will there beresonance? (HRW 6e 18.32; see also YF12e 16.27 & 28)

5. Two identical piano wires have a fundamental frequency of f1 = 600 Hz when kept under the same tension. What fractional increase in the tension of one wire will lead to 6 beats per second when both wires vibrate? (HRW6e 18.45)

6. Trooper B is chasing speeder A along a straight stretch of road. Both are moving at a speed of 100 mi/h. Trooper B, failing to catch up, sounds his siren again. Take the speed of sound in air to be 1100 ft/s and thefrequency of the source to be 500 Hz. What is the Doppler shift in the frequency heard by speeder A? (HRW6e

18.46E - modified)

7. Two students with vibrating 440-Hz tuning forks walk away from each other with equal speeds. How fast mustthey walk to hear a beat frequency of 2-Hz? (T14.67; see also YF12e 40)

8. A bat is flitting about in a cave, navigating via ultrasonic bleeps. Assume that the sound emission frequency ofthe bat is 39,000 Hz. During one fast swoop directly toward a flat wall surface, the bat is moving at 0.025 timesthe speed of sound in air. What frequency does the bat hear reflected off the wall? (HRW6e 18.54P; see also YF12e

16.43)

9. A girl is sitting near the open window of a train that is moving at a velocity of 10.00 m/s to the east. The girl’suncle stands near the tracks and watches the train move away. The locomotive whistle emits sound at frequency500.0 Hz. The air is still. (a) What frequency does the uncle hear? (b) What frequency does the girl hear? Awind begins to blow from the east at 10.00 m/s. (c) What frequency does the uncle now hear? (d) What frequencydoes the girl now hear? (HRW6e 18.55; see also YF12e 50)

See Also Problem Summary: Ch. 16 (YF12e) - problems 27, 28, 33, 40, 43, 50(Brief solutions to the see also problems are posted on the class Blackboard site.)

Brief Answers to Questions:

1. (a) zero wavelength phase difference (peak to peak) 6 constructive interference(b) half a wavelength phase difference (peak to trough) 6 destructive interference

2. Pipe “d” - fundamental mode

3. The Doppler frequency shift for a stationary listener & moving source depends on vs - where vs is theline of sight velocity of the source (along a straight line from the listener to the source of the sound).

5

EM Assignments: Electromagnetic Waves

Questions:

1. The figure at right shows the electric and magnetic fields of an electromagnetic wave at acertain instant of time. In which direction is the wave traveling? (HRW7e 33.1Q)

2. If the magnetic field of a light wave oscillates parallel to the y-axis and is given by

,

a. in what direction does the wave travel, andb. parallel to which axis does the associated electric field oscillate? (HRW7e 33.2Q)

3. Give several examples of electromagnetic waves that are encountered in everyday life. How are all alike? Howdo they differ? (YF12e 32.3Q)

4. If a light beam carried momentum, should a person who turns on a flashlight feel a recoil analogous to the recoilof a rifle when it is fired? Why is this recoil not actually observed? (YF12e 32.11)

Problems:

1. An electric heater emits 1.0 kW uniformly in all directions. How close would you have to stand to feel anintensity of 100 W/m2 ? (HRW5e 34.21E; see also YF12e 32.15 on p. 1116)

2. An electromagnetic wave has a frequency of 100 MHz and is traveling in a vacuum. The magnetic field isgiven by :

B (z, t) = (10-8 T) cos (kz - Tt) i .

(a) Find the direction of propagation of the wave. (b) Find the electric vector E(z,t). (c) Give the Poynting vector and find the intensity of the wave. (T 29.28; see also YF12e 32.6 & 11)

3. What is the radiation pressure 1.5 m away from a 500 W light bulb? Assume that the surface on which thepressure is exerted faces the bulb and is perfectly absorbing and that the bulb radiates uniformly in all directions. (HRW6e 34.23; see also YF12e 32.26)

4. The earth (radius R = 6.4 x 106 m) orbits the sun at an average distance D = 1.5 x 1011 m.

(a) What fraction of the electromagnetic radiation emitted by the sun is intercepted by the earth? (b) The intensity of sunlight striking the earth's upper atmosphere is known as the solar constant. What is the solarconstant? (NOTE: The sun has power 3.83 x 1026 W.) (c) Calculate the force on the earth due to this radiation if all of it is absorbed and none of it is reflected.(d) What is the ratio of the force of radiation on the earth to the gravitational force on the earth due to the Sun? (D. Brown)

6

5. Small particles might be blown out of planetary systems around other stars by the radiation pressure of starlight. Assume spherical particles of radius r and density 1.0 g/cm3 that absorb all the radiation in a cross-sectional areaBr2 . Assume they are a distance D from a solar-type star which has power 3.83 × 1026 W and mass 1.99 × 1030 kg. How small must a particle be (i.e. what is r ) for the radiation force of repulsion to exactly balance the gravitationalattraction of the star? (T 29.47; see also YF12e 32.51 & 54)

See Also Problem Summary: Ch. 32 (YF12e) - 6, 11, 15, 26, 51, & 53 on pp. 1117-1120(Brief solutions to the see also problems are posted on the class Blackboard site.)

Brief Answers to Questions:

1. into the page 2. (a) +z direction (b) x-axis3. optical radiation from the Sun - radio waves that transmit the signal from the radio station to your home -microwaves heating your frozen foods - all travel at the same speed in a vacuum - but all have differentwavelengths & frequencies.4. yes - it happens - but is too small to be noticed compared to the other forces that act on a person

7

OP Assignments: Optics

Questions:

1. For what range of object positions does a concave spherical mirror form a real image? What about aconvex spherical mirror? (YF 12e Q34.6)

2. When a T. rex pursues a jeep in the movie Jurassic Park, we see a reflected image of the T. rex via aside-view mirror, on which is printed the (then darkly humorous) warning: “Objects in mirror are closer thanthey appear.” Is the mirror flat, convex, or concave? (HRW7e Q34.5)

3. Can an image formed by one reflecting or refracting surface serve as an object for a second reflection orrefraction? Does it matter whether the first image is real or virtual? Explain. (YF12e Q34.19)

Problems:

1. A bottom-weighted vertical pole extends 2.0 m above the bottom of a swimming pool and 0.5 m abovethe water. Sunlight is incident at 45o. What is the length of the shadow ofthe pole on the bottom of the pool? (HR 43.7)

2. A ray of light is incident normally on the face ab of a glass prism (n =1.52) as shown in the figure at right. What is N so that the ray is totallyreflected at face ac if the prism is immersed in:

(a) air? (b) water? (HRW6e 34.55)

3. A point source of light is located 5 m below the surface of a large pool of water. Find the area of thelargest circle on the pool's surface through which light coming directly from the source can emerge. (T30.25)

4. You are standing 5.0 m in front of a plane mirror. You see an object in the mirror. It is 5.0 m to yourright and 1.0 m closer to the mirror. For what distance must you focus a camera in order to photograph theobject? (based on HR 44.2)

5. An object is p centimeters in front of a concave mirror of focal length 20 cm. Where does the imageform and what is it like (magnification? type - real or virtual? erect or inverted?) if:

(a) p = +10 cm? (b) p = +30 cm? (HRW6e 35.10 a,c; see also YF12e 34.14)

6. A virtual image forms 10 cm from a spherical mirror with radius of curvature r = -40 cm.

(a) What type of mirror is this? (b) Where is the object? (HRW 6e 35.10e; see also YF12e 34.10)

8

7. An object is 10 cm from a thin lens with radii of curvature r1 = +30 cm and r2 = -30 cm. What type oflens is this? Where does the image form? What type of image is it (real - virtual - erect - inverted)? What isits magnification? Assume that the glass in the lens has index of refraction 1.5. (HRW 6e 35.24e; see also

YF12e 34. 28 & 80)

8. Repeat problem # 7 with r1 = -30 cm and r2 = -60 cm. (HRW 6e 35.24g)

9. An erect object is placed in front of a converging lens at a distance equal to twice the focal length f1 ofthe lens. On the other side of the lens is a concave mirror of focal length f2 separated from the lens by adistance 2(f1 + f2). (HRW 6e 35.52; see also YF12e 34.69, 89 & 94)

(a) Find the location, type, orientation, and lateral magnification of the final image, as seen by aneye looking toward the mirror through the lens and just past (to one side of) the object. (b) Draw a ray diagram to locate the image.

See Also Problem Summary: 10, 14, 28, 69, 80, 89, 94 (pp. 1199 ... 1204 in YF12e)

Brief Answers to Questions:

1. Object distances p > f for a concave mirror. Never for a convex mirror. Try simulations at:

http://wildcat.phys.northwestern.edu/vpl/optics/mirrors.html

2. Convex3. Yes; No

9

IF Assignments: Interference

Questions:

1. If you move from one bright fringe in a two-slit interference pattern to the next one farther out, (a) does the pathlength difference )L increase or decrease and (b) by how much does it change, in wavelengths 8? (HRW7e

35.6Q)

2. Does the spacing between fringes in a two-slit interference pattern increase, decrease, or stay the same if (a) theslit separation is increased, (b) the color of the light is switched from red to blue, and (c) the whole apparatus issubmerged in cooking sherry? (d) If the slits are illuminated with white light, then at any side maximum, does theblue component or the red component peak closer to the central maximum? (HRW7e 35.7Q)

3. The figure at right (Fig. 35-29 in HRW7e) shows the transmission of light through athin film in air by a perpendicular beam (tilted in the figure for clarity). (a) Did ray r3

undergo a phase shift due to reflection? (b) In wavelengths, what is the reflectionphase shift for ray r4 ? (c) If the film thickness is L, what is the path length differencebetween rays r3 and r4 ? (HRW7e 35.11Q)

4. Figure (a) at right (Fig. 35-30 in HRW7e) shows the cross section of a vertical thinfilm whose width increases downward because gravitation causes slumping. Figure (b)at right is a face-on view of the film, showing four bright (red) interference fringes thatresult when the film is illuminated with a perpendicular beam of red light. Points in thecross section corresponding to the bright fringes are labeled. In terms of thewavelength of light inside the film, what is the difference in film thickness between (a) points a & b and between points b & d ? (HRW7e 35.12Q)

5. A very thin soap film (n = 1.33), whose thickness is much less than a wavelength ofvisible light, looks black; it appears to reflect no light at all. Why? By contrast, an equally thin layer of soapywater on glass (index 1.50) appears quite shiny. Why is there a difference? (YF12e 35.14Q)

Problems:

1. The wavelength of yellow sodium light in air is 589 nm. What is (a) its frequency, (b) its wavelength inglass, and (c) its velocity in glass? Assume an index of refraction of 1.52 for the glass. (HRW6e&5e 36.1E)

2. Two waves of light in air, of wavelength 600.0 nm, are initially in phase. They then travelthrough plastic layers with L1 = 4.00 :m, L2 = 3.50 :m, n1 = 1.40, and n2 = 1.60. (a) Inwavelengths, what is their phase difference after they both have emerged from the layers? (b) If the waves later arrive at some common point, what type of interference do theyundergo? (HRW6e 36.9P)

3. In a lecture demonstration, laser light is used to illuminate two slits separated by 0.5 mm, and the interferencepattern is observed on a screen 5 m away. The distance on the screen from the centerline to the thirty-seventhbright fringe is 25.7 cm. What is the wavelength of the light? (T33.15; see also YF12e 35.11)

10

4. A double-slit experiment uses a helium-neon laser with a wavelength of 633 nm and a slit separation of 0.12mm. When a thin sheet of plastic is placed in front of one of the slits, the interference pattern shifts by 5.5 fringes. When the experiment is repeated under water, the shift is 3.5 fringes. Calculate (a) the thickness of the plasticsheet and (b) the index of refraction of the plastic sheet. (T33.67)

5. In a double-slit arrangement the distance between slits is 5.0 mm and the slits are 1.0 m from the screen. Twointerference patterns can be seen on the screen, one due to light of 480 nm and the other 600 nm. What is theseparation on the screen between the third-order interference fringes of the two different patterns? (HRW6e

36.19P)

6. Two radio beacons emit waves of frequency 2.0 x 105 Hz. The beacons are on a north-south line, separated by adistance of 3.0 km. The southern beacon emits waves 1/4 of a cycle later than the northern beacon. Find theangular directions for constructive interference. Measure angles relative to the east-west line and assume that thedistance between the beacons and the point of observation is large. (OH38.22)

7. In the figure at right, A and B are identical radiators of waves that are in phase and ofthe same wavelength 8. The radiators are separated by distance d = 3.00 8. Find thegreatest distance from A, along the x axis, for which fully destructive interferenceoccurs. (HRW 6e 36.20P ; see also YF12e 35.5 & 44)

8. A thin layer of a transparent material with an index of refraction of 1.30 is used as anonreflective coating on the surface of glass with an index of refraction of 1.50. Howthick should the material be in order for the glass to be nonreflecting for light ofwavelength 600 nm (in a vacuum)? (T33.7; see also YF12e 35.28)

9. White light reflected at perpendicular incidence from a soap bubble has an interference maximum at 600 nmand a minimum at the violet end of the spectrum, with no other minimum in between. If n = 1.33 for the film,calculate its thickness, assumed uniform. (HRW 6e 36.42P - modified; see also YF12e 35.33)

10. In the figure at right (top) a broad beam of light of wavelength 600 nm is sent directly downward through aglass plate (n = 1.5) that, with a plastic plate (n = 1.2), forms a thin wedge of air which acts as a thin film. Anobserver looking down through the top plate sees the fringe pattern shown in the figure (bottom), with dark fringescentered on ends A and B. (a) What is the thickness of the wedge at B? (b) Howmany dark fringes will the observer see if the air between the plates is replacedwith water (n = 1.33)? (HRW 6e 36.61P ; see also YF12e 35.29)

Brief Answers to Questions:

1. )L increases by one wavelength.2. (a) decreases; (b) decreases; (c) decreases; (d) blue is closer.3. (a) no; (b) zero; (c) 2L 4. (a) half a wavelength; (b) one wavelength5. Hint: Think about the phase changes for reflection.

See Also Problem Summary: YF12e - Ch. 35 - 5, 11, 28, 29, 33, 44

11

DF Assignments: Diffraction

Questions:

1. The figure at right shows the bright fringes that lie within the centraldiffraction envelope in two double-slit diffraction experiments using the samewavelength of light. Are (a) the slit width a, (b) the slit separation d, and (c)the ratio d/a in experiment B greater than, less than, or the same as those inexperiment A? (HRW7e 36.8Q)

2. The figure at the right shows a red line (left line) and a green line (rightline) of the same order in the pattern produced by a diffraction grating. Ifwe increased the number of rulings in the grating by removing tape thathad covered the outer half of the rulings, would (a) the widths of the linesand (b) the separation of the lines increase, decrease, or stay the same? (c)Would the lines shift to the right, shift to the left, or remain in place? (HRW7e 36.9Q)

3. Why can we readily observe diffraction effects for sound waves and wa5ter waves, but not for light? Is thisbecause light travels so much faster than these other waves? Explain. (YF12e Q36.1)

4. Light of wavelength 8 and frequency f passes through a single slit of width a. The diffraction pattern isobserved on a screen. Which of the following will decrease the width of the central maximum? (a) Decrease theslit width; (b) decrease the frequency f of the light; (c) decrease the wavelength 8 of the light. (YF12e Q36.4

modified)

5. In a diffraction experiment with waves of wavelength 8, there will be no intensity minima (that is, no darkfringes) if the slit width is small enough. What is the maximum slit width for which this occurs? (YF12e Q36.5)

Problems:

1. In a single slit diffraction pattern, the distance between the first minimum on the right and the first minimum onthe left is 5.2 mm. The screen on which the pattern is displaced is 80 cm from the slit and the wavelength is 546nm. Calculate the slit width. (HR 46.3)

2. A plane wave of wavelength 590 nm is incident on a slit with a = 0.40 mm. A thin converging lens of focallength +70 cm is placed between the slit and a viewing screen and focuses the light on the screen. (a) How far isthe screen from the lens? (b) What is the distance on the screen from the center of the diffraction pattern to thefirst minimum? (HRW6e 37.5E)

3. The telescope on Mount Palomar has a diameter of 200 inches. We want to use this telescope to resolve theimages of the two stars in a double star system that is 4 light-years distant from Earth. What minimum separationmust the two stars have? (Use 8 = 500 nm.) (T 33.45)

4. The Impressionist painter Georges Seurat used a technique called "pointillism," in which his paintings arecomposed of small, closely spaced dots of pure color, each about 2-mm in diameter. The illusion of the colorsblending together smoothly is produced in the eye of the viewer by diffraction effects. Calculate the minimumviewing distance for this effect to work properly. Use the wavelength of visible light that requires the greatestdistance, so that you're sure the effect will work for all visible wavelengths. Assume that the pupil of the eye has adiameter of 5-mm. (T 33.57)

12

5. How many complete fringes appear between the first minima of the diffraction envelope to either side of thecentral maximum for a double slit pattern if 8 = 550 nm, d = 0.15 mm and a = 0.030 mm? (HRW 6e 37.31a)

6. Light of wavelength 480-nm falls normally on four slits. Each slit is 2 :m wide and is separated from the nextby 6 :m. (a) Find the angle from the center to the first point of zero intensity of the single-slit diffraction patternon a distant screen. (b) Find the angles of any bright interference maxima that lie inside the central diffractionmaximum. (c) Find the angular spread between the central interference maximum and the first interferenceminimum on either side of it. (T33.55)

7. Sodium light of wavelength 589 nm falls normally on a square diffraction grating that is 2.0 cm on a side and isruled with 4000 lines per centimeter. The Fraunhofer diffraction pattern is projected onto a screen at 1.5 m by alens of focal length 1.5 m placed immediately in front of the grating. Find (a) the positions of the first twointensity maxima on one side of the central maximum, (b) the width of the central maximum, and (c) theresolution in the first order. (T 33.49)

8. Design a grating that will spread the first-order spectrum through an angular range of 20o if the spectrum rangesfrom 81 = 430 nm to 82 = 680 nm. (HRW6e 37.39P)

9. A square diffraction grating with an area of 25 cm2 has a resolution of 22,000 in the fourth order. At what angleshould you look to see a wavelength of 510-nm in the fourth order? (T33.59)

Brief Answers to Questions:

1. (a) Slit width is smaller in B than A from .

(b) Slit separation is larger in B than A from .

(c) Ratio d/a greater for B than for A.

2. (a) Width of lines decreases.(b) Separation of lines remains the same.(c) Remain in place.

3. No - it’s because the other waves have much larger wavelengths. Diffraction effects are not observed for awave unless the obstacle the wave encounters has dimensions comparable to the wavelength of the wave.

4. Answer (c) only because where m = 1 and .

5. Slit width less than or equal to the wavelength of the light.

See also problem summary: There are no see also problems for Ch. 36.

13

SR Assignments: Special Relativity

Questions:

1. You are standing on a train platform watching a high-speed train pass by. A light inside one of the train cars isturned on and then a little later it is turned off. (a) Who can measure the proper time interval for the duration ofthe light: you or a passenger on the train? (b) Who can measure the proper length of the train car, you or apassenger on the train? (c) Who can measure the proper length of a sign attached to a post on the train platform,you or a passenger on the train? (YF12e Q37.1)

2. A rocket is moving to the right at the speed of light relative to the earth. A light bulb in the center of a room

inside the rocket suddenly turns on. Call the light hitting the front end of the room event A and the light hitting theback of the room event B. Which event occurs first, A or B or are they simultaneous, as viewed by (a) anastronaut riding in the rocket and (b) a person at rest on the earth? (YF12e Q37.3)

3. Describe qualitatively (non-mathematically) how the length contraction effect is a consequence of thedisagreement between observers in two frames of reference about the synchronization of clocks in the two frames. Reason from the two fundamental postulates of Special Relativity.

4. When a monochromatic light source moves toward an observer, its wavelength appears to be shorter than thevalue measured when the source is at rest. Does this contradict the hypothesis that the speed of light is the samefor all observers? Explain. (YF12e Q37.12)

Problems:

1. The proper mean lifetime of a muon is 2 :s. Muons in a beam are traveling at 0.999c. (a) What is their meanlife-time as measured in the lab? (b) How far do they travel, on the average, before they decay? (T34.5; see also

YF12e 37.5 & 7)

2. The length of a spaceship is measured to be exactly half its rest length. (a) What is the speed of the spaceshiprelative to the observer’s frame? (b) By what factor do the spaceship’s clocks run slow, compared to clocks in theobserver’s frame? (HRW6e; see also YF12e 37.12)

3. In frame S, event B occurs 2 :s after event A and 1.5 km away from event A. How fast must an observer bemoving along the x-axis so that events A and B occur simultaneously? Is it possible for event B to precede event Afor some observer? (T 34.53)

4. A spaceship is moving east at speed 0.90c relative to the earth. A second spaceship is moving west at speed0.90c relative to the earth. What is the speed of one spaceship relative to the other? (T34.31; see also YF12e

37.19)

5. Two spaceships, each 100 m long when measured at rest, travel toward each other with speeds of 0.85c relativeto the earth. (a) How long is each ship as measured by someone on earth? (b) How fast is each ship traveling asmeasured by an observer on the other? (c) How long is one ship when measured by an observer on the other? (d) At time t = 0 on earth, the fronts of the ships are together as they just begin to pass each other. At what time onearth are their ends together? (T34.57; see also YF12e 37.22)

14

6. An armada of spaceships that is 1.00 ly long (in its rest system) moves with speed 0.800c relative to groundstation S. A messenger travels from the rear of the armada to the front with a speed of 0.950c relative to S. Howlong does the trip take as measured (a) in the messenger’s rest system, (b) in the armada’s rest system, and (c)by an observer in system S? (HRW6e 38.26P)

7. A distant galaxy is moving away from the earth with a speed that results in each wavelength received on earthbeing shifted such that 8 = 28o . Find the speed of the galaxy relative to the earth. (T 34.25)

8. Quasars are thought to be the nuclei of active galaxies in the early stages of their formation. A typical quasarradiates energy at the rate of 1041 W. At what rate is the mass of this quasar being reduced to supply this energy? (HRW6e 38.37P; see also YF12e 37.32)

9. The total energy of a particle is twice its rest energy. (a) Find v/c for the particle. (b) Show that its momentumis given by p = %3 mc. (T 34.40)

10. A spaceship of mass 106 kg is coasting through space when it suddenly becomes necessary to accelerate. Theship ejects 103 kg of fuel in a very short time at a speed of c/2 relative to the ship. (a) Neglecting any change in therest mass of the system, calculate the speed of the ship in the frame in which it was initially at rest. (b) Calculatethe speed of the ship using classical, Newtonian mechanics. (c) Use your results from part (a) to estimate thechange in the rest mass of the system. (T 34.68)

11. A particle of rest mass (1 Mev)/c2 and kinetic energy 2 Mev collides with a stationary particle of rest mass (2Mev)/c2. After the collision, the particles stick together. For the first particle, find (a) its speed before thecollision and (b) its total energy before the collision. For the system, find (c) its initial total momentum, (d) itstotal kinetic energy after the collision, and (e) its rest mass after the collision. (T 34.72)

Brief Answers to Questions:

1. (a) passenger; (b) passenger; (c) you2. (a) simultaneous; (b) B precedes A3. Consult lecture notes from Wed. May 19.

4. No - because the frequency increases in the equation to compensate.

See Also Problem Summary: YF12e 37.5, 7, 12, 19, 22, 32

15

MP Assignments: Modern Physics

Questions:

1. As a body is heated to a very high temperature and becomes self-luminous, the apparent color of the emittedradiation shifts from red to yellow to white and finally to blue as the temperature increases. Why does the colorshift? What other changes in the character of the radiation occur? (YF12e Q38.18)

2. Photon A is from an ultraviolet tanning lamp, and photon B is from a television transmitter. Which has thegreater (a) wavelength, (b) energy, (c) frequency, and (d) momentum? (HRW7e Q38.9)

3. In a photoelectric effect experiment, which of the following will increase the maximum kinetic energy of thephotoelectrons; (a) Use light of greater intensity; (b) use light of higher frequency; (c) use light of longerwavelength; (d) use a metal surface with a larger work function. (YF12e Q38.12)

4. If a proton and an electron have the same speed, which has the longer de Broglie wavelength? (YF12e Q39.2)

5. Why can an electron microscope have a greater magnification than an ordinary microscope? (YF12e Q39.15)

6. A hydrogen atom is in the third excited state. To what state (give the quantum number n) should it jump to (a)emit light with the longest possible wavelength, (b) emit light with the shortest possible wavelength, and (c) absorblight with the longest possible wavelength? (HRW7e Q39.17)

Problems:

1. Suppose that a 100-W source radiates light of wavelength 600 nm uniformly in all directions and that the eyecan detect this light if at least 20 photons per second enter a dark-adapted eye having a 7-mm diameter pupil. Howfar from the source can the light be detected under these rather extreme conditions? (T35.41; see a lso YF12e 38.4

& 54)

2. You wish to pick a substance for a photocell operable with visible light. Which of the following will do (workfunction in parenthesis): tantalum (4.2 eV), tungsten (4.5 eV), aluminum (4.2 eV), barium (2.5 eV), lithium (2.3eV)? (HRW6e 39.17E; see also YF12e 38.8)

3. Light of wavelength 200 nm falls on an aluminum surface. In aluminum 4.2 eV are required to remove anelectron. What is the kinetic energy of (a) the fastest and (b) the slowest emitted photoelectrons? (c) What is thestopping potential? (d) What is the cutoff wavelength for aluminum? (HRW6e 39.23E; see also YF12e 38.12, 13, 56)

4. If the de Broglie wavelength of a proton is 100 fm, (a) what is the speed of the proton and (b) through whatelectric potential would the proton have to be accelerated to acquire this speed? (HRW6e 39.54P; see also YF12e

39.8, 9, 11, 47)

5. A neutron has a kinetic energy of 10 Mev. What size object is necessary to observe neutron diffraction effects? Is there anything in Nature of this size that could serve as a target to demonstrate the wave nature of 10-Mevneutrons? (T 36.14)

6. A microscope using photons is employed to locate an electron in an atom to within a distance of 10 pm. Whatis the minimum uncertainty in a measurement of the momentum of the electron located in this way? (HRW 4e

44.41E; see also YF12e 39.48)

16

7. The Sun has a total radiation rate of approximately 3.9 × 1026 W. The Earth orbits the Sun at a mean distance of1.5 1011 m. Assume that the Earth absorbs all the solar radiation that falls on it. Assume that this heats the surfaceof the Earth to a uniform temperature T. Assume that the Earth radiates like a black body. And assume that inevery second the Earth radiates exactly as much energy as it receives from the Sun in one second; i.e. assume thatthe Earth is in thermal equilibrium. The calculate (a) the surface temperature of the Earth and (b) the region of

the electromagnetic spectrum in which the Earth has peak emissivity. (Based on HR 49.6; see also YF12e 38.45 & 48)

8. A hydrogen atom undergoes a transition from state n = 3 to n = 1. Find the:

a. energy of the emitted photon,b. momentum of the emitted photon, &c. wavelength of the emitted photon.d. What is the recoil speed of the atom?

(HRW7e 39.27; see also YF12e 38.24 & 63)

9. The binding energy of an electron is the minimum energy required to remove the electron from its ground stateto a large distance from the nucleus. What is the binding energy for the hydrogen atom? (T35.49; see also YF12e

Q38.16)

Brief Answers to Questions:

1. Color shifts because the peak of the emitted energy curve shifts to shorter wavelengths as the temperatureincreases. Intensity (brightness) also increases with temperature. (See the Mon. May 24 lecture notes & slides.)2. (a) B; (b) A; (c) A; (d) A.3. (b) only4. electron5. Electrons have a shorter (de Broglie) wavelength than for visible light, and so the electron microscope hashigher resolution - by Rayleigh’s criterion.6. (a) n = 3; (b) n = 1; (a) n = 4;

Summary of See Also problems:

Ch. 38 - 4, 8, 12, 13, 16, 24, 45, 48, 54, 56, 63Ch. 39 - 8, 9, 11, 47, 48