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Unformatted text preview: (15 pts) Problem 1: Multiple choice conceptual questions. Choose the best answer and ﬁll in the appropriate bubble on your bubble
sheet. You may also want to circle the letter of your top choice on this paper. 1.1. 1.2.. 1.3. 1.4. 1.5. 1.6. 1.7. 1.81. An extremely precise scale is used to measure an iron weight. It is found that in a room with the air sucked out, the mass of the weight is precisely 1.000000 kg. If you add the air back into the room, will the scale reading increase, decrease, or stay
the same? a. increase
(9;) decrease
0. stay the same mvtmu. The?“ (wt/“ﬂour! {och wryL. swam 5mm A. 3M mgbm‘ Water ﬂows from a little pipe into a big pipe with no friction or height change. No other pipes are connected. How many
kg/s ﬂow in the little pipe as compared to the big pipe? (kg/s is called the “mass ﬂow rate”) a. more kg/s ﬂow 1n the l1ttle p1pe “(MS is 6W“, H ﬂ M Cmvuvﬁwk OF b. more kg/s ﬂow in the big pipe . H g k \\
@ the same kg/s ﬂow in both pipes M55 0150 M ’61""*:’05h‘v\ “ML WV 3” 3"“ “R MM ‘ ll
A as under oes the c cle shown in the ﬁ ure. For A to B, is Won 35 ositive, ne ative, or I 13
g g y g g p 2%
zero?
a. positive 0 S U A) , w“); a. he. . '3; :42.
Q, negative q ’ ‘F 5 him“ ‘
c. zero Same ﬁgure. For the complete cycle, A —> B —> C —> A, how does the net Q compare to net
Why gas? (They are both positive quantities.)
a. net Q < net Why gas a?“ M GK «~13 EU. ?/O b. W as
G; 32:8:22EW:::.. we“ Miami If you ﬂip 13 coins, what is the probability of getting exactly 3 heads? 213 . 4i . b} e 3! i 13!
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d. W 31mm” Consider a transverse traveling wave of the form: y(x, t) = (2x ~10t)4 . (You may assume that the numbers have the appropriate units associated with them to make x, y, and tbe in standard SI units.) Is the wave moving in the +x or —x
direction? a. +x b”. —x c. it cannot be determined Nlrlmhu 3 v: 3”
Same situation. What is the wave’s speed? "‘1‘ + X
a. 2 m/s (fire LM 6. b. 4 “ f. 12
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d. 8 h. 40 m/s A car horn emits a 2000 Hz sound. Will you hear a higher frequency if the car approaches you at 30 mph while you are at
rest, or if you approach the car at 30 mph while the car is at rest? (a? Car approaches you I ., r \f '3 V4; _ 5%
b. You approach car /C r If N ‘Ué A“ CW1)“
c. Same shift for both cases \{ + 3b th 413 V
V V 1%qu Phys 123 Exam 1 —pg 2 T MK {M (‘5 I. TIM?! nu. vsvlo ‘; ’kafirl’f " a'ﬂ‘
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f(x, t) plotted as f(x) for t=0 f(x, t) plotted as f(x) for t=1 4,. E , Tl"
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1.9'. Which wave function f(x,t) is represented by the two graphs displayed? The leﬂhan graph is the wave function plo e or
t=0; the right hand graph is the wave function plotted for t=1. a. f(x,t) = 2 cos(3x  3t) f. f(x,t) = 4 cos(3x  3t) b. f(x,t) = 2 cos(3x  5t) . f(x,t) = 4 cos(3x  5t) ~ . c. f(x,t) = 2 cos(4x  4t) t/gh. ) f(x,t) = 4 cos(4x  4t)€ ~H’“ 5 ‘5‘ Tia
d. f(x,t) = 2 cos(SX  3t) i. f(x,t) = 4 cos(5x — 3t) Mb '1, ‘
e. f(x,t) = 2 cos(5x  5t) j. f(x,t) = 4 cos(5x  St) at“ h: A 1.10. If a transverse pulse travels down your slinky and reﬂects off of the end which is being held ﬁxed by a friend, will the
reﬂected pulse be inverted or noninverted? A r
inverted ’t Tm m3 ’9 vy‘r'ir’l \&
b. noninverted
c. it depends on the magnitude of the amplitude of the initial pulse 1.1.1. If you hang your slinky vertically and send a transverse pulse from the top towards the bottom, will the reﬂected pulse be
inverted or noninverted? _
a. inverted (\g'l‘“ Mull/N
C 11;) noninverted ' LﬁQSQ” QKS 4
c. it depends on the magnitude of the amplitude of the initial pulse 1.12. If a single violin produces a sound level of 60 dB, how loud would you expect two violins to be? (The violins are not
playing in phase with each other.) a. 60dB " +\ ( l e. 80 b. 62 m“ ‘M‘JM "3“"? f. 90
@ 63 w» “Bell’s “1(er g. 120dB
d. 70 1.13. In the “ladies belt demo” (the belt was like a “closedclosed” string), suppose the fundamental frequency is seen at
200 Hz. What frequency will have threeantinodes? a. 200 Hz d. 500 b. 230 ,3 e. 600
c. 300 435 “(l 1800Hz 1.14. Suppose a cellist tunes one of his strings to middle C. (That’s not commonly done by cellists, but then again inclined
planes aren’t usually frictionless either, and since when has that stopped physics professors? ©) He runs his bow across the
string and a middle C sounds. The cellist then plays the same string while touching the middle of the string to force a node
there. What note will the audience hear? a. The same note: a middle C ATEK {a MC MK Q3 SAC“A me‘i k,
b. One octave lower: the C below middle C I '
c Two octaves lower: the C below that
@ One octave higher: the C above middle C r t W \ , _ 5
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6. Two octaves higher. the C above that \g MA. 3 CDAB he, mswv "{ C641 5 ,L 1“?“ A aft Mme,
n39 k dim w “\7 , 0000 &.6~‘l’ 50:} 1.15. The sound of a trumpet playing a sustained note is qualitatively different than the sound of a ﬂute playing the same note. \~ ML.
Why is that? a. The two notes have different amplitudes. b. The two notes have different durations. O {Mix \J Ma SkQKQ‘K.
c. The two notes have different fundamental frequencies. ' ‘ \ ~ RS \6’
d. The two notes have different phases. e—§ angered“ 55*“? @ The two notes have different strengths of harmonics. I: ..
him/m9? "“ r). Phys 123 Exam 1 — pg 3 (12 pts) Problem 2. Suppose you are watching sinusoidal waves travel across a swimming pool. When you look at the water right in
front of you, you see it go up and down ten times in 3 seconds. At the peaks of the wave, the water is 1.0 cm below the edge of the
pool. At the lowest points of the wave the water is 6.0 cm below the edge of the pool. At one particular moment in time you notice
that although the water right in front of you is at its maximum height, at a distance 2 m away the water is at its minimum height. (This is the closest minimum to you.) (a) What is the frequency f for this wave? I 0 .1 "(W26
'3 sum9s (b) What is 0) for this wave (rad/s)? \F) i Z,“ (e) What is the amplitudeA of this wave? a qL ,)v {XML 3 S 0"“
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Ni'\ [ALV‘LUL 2 ’6 CW __bo\~\Q‘.+v>u\ >2,SCVV\ (f) What is the speed of water waves in this pool?
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i“ iii/Z. W «.1 3— Li ' {’30 5 V30 9’le‘onm/15hreri OX / Phys 123 Exam 1 — pg 4 (10 pts) Problem 3. (a) A Slinky is stretched to 2 m long. It has a mass of 0.2 kg. A transverse wave pulse is produced by plucking
one end of the Slinky. That pulse makes four round trips (down and back) along the cord in 5 seconds. What is the tension in the Slinky? alleﬁ a m ” 62/343 ’2 A lj
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Par) Phys 123 Exam 1 —pg 5 (10 pts) Problem 4. Add these cosine functions together and give the amplitude and phase of the resulting function (it will also have
a frequency of 11 rad/s): f1 (t) = 3 cos(11t+ 3)
f2 (t) = 260s(1 1t + 1) Phys 123 Exam 1 — pg 6 (6 pts) Problem 5. Suppose you have a light ray going from air into glass, with the light beam hitting the surface of the glass
perpendicularly. Light travels at 3.00 x 108 m/s in the air and at 1.90 x 108 m/s in the glass. What percent of the incident light power
will reflect off of the surface of the glass? (This is the same as the problem where a wave on a string reﬂects off an interface where the velocity abruptly changes from 3 to 1.9 m/s.) WW f‘“ , V2” vi ,5 EC} : A lug
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V» a V1» WW”) Phys 123 Exam 1 — pg 7 (8 pts) Problem 6. In the “mesmerizing” animated gif thatI created to show an example of phase and group velocities going in
different directions, this is the actual equation I programmed into Mathematica: f(x,t) = cos(kx+k‘2t) I then added up a bunch of different waves having k values equally spaced between 0.9 and 1.1, and plotted the sum at various times. (a) What was the “dispersion relation” of this wave, a) as a function of k? 7‘“ ; Casﬂx 'wi) My
1mg {ﬁphbﬁ N" i (b) What was the phase velocity? (You can assume all numbers are in SI units.) VJ ~‘/1LL . K 1
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b7ltwe ‘L kg, \g k I (c) What was the group velocity? 6
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“if” 9\\/\\r&\~9( ‘L 4V”! NB ox+ from) \r‘K’ “9‘7 2 M/s Phys 123 Exam 1 — pg 8 (9 pts) Problem 7. A jet airplane ﬂies with a speed of 500 m/s at a constant altitude that is 3600 m above where I’m standing. It
passes directly over my head. How soon after I see it pass directly above me will I hear the sonic boom? A”)
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1 Phys 123 Exam 1 — pg 9 (10 pts) Problem 8. A pipe open at both ends has a fundamental frequency of 500 Hz when the temperature is 0°C. (a) What is the length of the pipe? (b) What is the fundamental frequency at a temperature of 30°C? Assume that the displacement antinodes occur
exactly at the ends of the pipe. Neglect thermal expansion of the pipe. Hint: If you don’t have the equation for the speed of sound
vs. temperature written down, you can derive it by remembering that the speed is inversely proportional to the square root of the
density, remembering what the ideal gas law can tell you about how density relates to temperature, and noting the speed of sound at the standard temperature of 293K must come out to be 343 m/s. v N T‘ I ‘
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WWW) i: 'Jhiglbw‘xmf V —g\l°.0 w H Phys 123 Exam 1 — pg 10 (10 pts) Problem 9. A speaker at the front of a room and an identical speaker at the rear of the room are being driven at 400 Hz by
the same sound source. A student walks at a very uniform rate of 1.5 m/s away from one speaker and toward the other. How many
beats per second does the student hear? \f‘I v0 Weill“ {'3 t ffﬂ Phys 123 Exam 1 —pg 11 (10 pts) Problem 10. The function f(x), graphed on the right, is deﬁned as follows: f(x) = x2, forxbetween —l and +1
(repeated with a period of L = 2) I worked out the Fourier coefﬁcients for this function, and found the following: f(x) = 0.33 — 0.41 cos(7rx) + 0.1000s(27rx) — 0.045 cos(37rx) + 0.025 cos(47rx) + I’ve rounded all the numbers a bit. Plots of f(x) for increasing numbers of terms in the
summation are shown on the right. (a) Why are there no sin(nnx/L) terms in the series? (.1) .3 qvevx (b) Prove that the constant term in my expression is correct. as i.,if""“ l '1
2 AL S”‘ x 3%
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a 1 7 m
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\ho,re.(.,./q k,“ EWM‘H‘Wﬂijd \mU BK :: .2: .> FV‘ALFQ vi It“ CCM.(}m\$..X£ buy“ ["1 V“u£.'{‘ii}fL3 “4: W. ‘/ <3 (d) Set up an integral that you could, for example, plug into Mathematica to solve for
the mm cosine term of the series (n = 4). Phys 123 Exam 1 — pg 12 (ﬁrst 1 tcrm) 1_o , (ﬁrst 3 terms) (ﬁrst 50 terms) (12 pts) Problem 11. Suppose a string is tuned to exactly 220 Hz, the standard frequency for the A below middle C. The string is
then forced to vibrate in the third harmonic (2nd overtone), which is used to tune another string (the “second string” referred to
below). (a) What frequency (in Hz) is the second string being tuned to?
3,? kl'W‘L‘ : gX 22§Hr2 :— (b) What musical note is the second string being tuned to? (If you are not conﬁdent with musical scales, fit least put how many half—
‘Fi 7— 6 ‘Fv (VFW Q (ﬁx Valves) steps higher than the A this note is.)
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(c) If another string (the “third string”) were tuned to that same note using an equal temperament scale, what frequency would it be
vibrating at? Emmy m w; am we a; t  120 (d) How many beats would occur per second if the second string and the third string were played simultaneously? Ma new gems :7 Phys 123 Exam 1 — pg 13 (4 pts, 110 partial credit) Problem 12. According to the Dulong—Petit la , what should the speciﬁc heat of copper be? Note that the molar heat capacity C, in J/mol°C, is related to the speciﬁ heat c, in J/kg°C, via the molar mm Hint: You can compare your
answer to the measured speciﬁc heat of copper, given on g l of this exam. \” \ T‘leij l‘j‘ W1.“ (ﬂ. AU: \2,
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W> C23p\ Mﬁjkp Phys 123 Exam 1 — pg 14 (5 pts, no partial credit) Extra Credit. You may pick one of the following extra credit problems to do. (If you work more than one,
only the ﬁrst one will be graded.) (a) A light string of mass 15 g and length L = 3 m has its ends tied to two walls
that are separated by the distance D = 2 m. Two objects, each of mass M = 2 kg,
are suspended from the string as in the ﬁgure. If a wave pulse is sent from point A,
how long does it take to travel to point B? !. ' n T R 1
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\ML‘J‘M 1. \r/ 2» .mS/z 712:3 (b) Suppose that your shower stall is rectangular: 2.4 m tall, 1.2 m wide, and 1.0 m deep. As you sing in the shower, what is the
lowest frequency (in Hz) that will resonate? Do n_ot ignore side~to~side sound waves. Hint: As discussed in the HW extra credit “Dr.
Durfee milkshake problem”, the fact that the waves propagate threedimensionally means that the wavenumber k is really the sum of three components k = ,1ka + k),2 + kz2 , where kx = 27r/(xwavelength) and similarly for ky and kz. This results in an overall k for the fundamental mode which is larger than it would be if the side—toside standing waves could be neglected. ” a Z/li ; I
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(0) Suppose you want to create your own animated wave image with Mathematica, illustrating a situation where the group velocity is 5 m/s and the phase velocity is 20 m/s for your desired center k value of k = 3 rad/m. What is the equation for the wave function you
would need to use? Solving for a) as a function of k is sufﬁcient, because then (as in problem 6) your equation can be
cos(lcx ~— ﬁmction_of_k  t), summed over a bunch of different k values. Vrqu 1* Lg) vfirbvf \a Li C C k A NC’ JW
w owlr'wil l 2 ma l9 *
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(d) Work out the integral involved in the Fourier transform problem and ﬁnd a general expression for the cosine coefﬁcients. \ l
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_\ sung \,\.uv\(')/\)r “V EL)“ MR!“th hilt“ PJ‘J»., k‘ 1 : vi _ 1 Phys 123 Exam 1 —pg 16 ...
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