PCS 125—Waves & Fields
Midterm Exam
Winter, 2023
Version B
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Student Number:
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Page 1 of 17
Instructions
1. This exam consists of seventeen (17) multiple choice questions.
2. Your exam booklet should have seventeen (17) pages = nine (9) sheets of paper. Inform an invigilator
if you are missing any pages.
3. A formula sheet is attached. Any formulas not present in the formula sheet must be derived.
4. Neither the instructor nor any invigilator will tell you which formula to use from the formula sheet, nor
how it should be used.
5. Do not detach any pages from the exam booklet except the formula sheet.
6. You should fill in your answer choice (A, B, C, etc.) for each question on your Akindi bubble sheet. No
credit will be awarded for markings anywhere else.
7. You should select only one answer for each question. Questions with more than one answer bubbled in
will receive a score of zero.
8. There are several blank pages near the end of this document, which you may use for scratch work, in
addition to the space below each question.
9. For all questions, you must show your work in order to receive full credit. Instructors reserve the right
to deduct marks from questions without sufficient work shown, even if the answer choice is correct.
10. You have two hours to complete the exam.
Page 2 of 17
Multiple Choice Questions (1 point each)
1. If you blow across the top of an empty soft-drink bottle, you can produce sound by creating a
standing wave. At the bottom of the bottle, what is the correct description of the magnitudes of
the displacement of air molecules (|s|) and the corresponding pressure variation (|∆P |)?
(A)
(B)
(C)
(D)
(E)
|s| is at a maximum, and |∆P | is zero.
|s| and |∆P | are both zero.
|s| is zero, and |∆P | is at a maximum.
|s| and |∆P | are both at a maximum.
None of the above.
2. A sound source emits 20.0 W of acoustical power spread equally in all directions. What is the sound
intensity level 20.0 m from the source?
(A)
(B)
(C)
(D)
(E)
102 dB
115 dB
109 dB
92.5 dB
96.0 dB
Page 3 of 17
3. A mass of 200 g is attached to a spring with spring constant 10 N/m and released from rest at
x = +10.0 cm, where x denotes the displacement of the mass from equilibrium. How long does it
take for the mass to reach x = −10.0 cm for the first time?
(A)
(B)
(C)
(D)
(E)
0.89
0.07
0.22
0.14
0.44
s
s
s
s
s
4. A periodic transverse wave travels along the x axis with a speed of 4.0 m/s. The image below shows
the displacement y(t) of the element of the medium at a fixed position x = 0 as a function of time.
What is the wavelength of the wave?
(A)
(B)
(C)
(D)
(E)
0.40 m
1.6 m
0.80 m
0.16 m
0.05 m
Page 4 of 17
5. Each expression below represents the displacement from equilibrium x(t) of five different mass-ona-spring systems. If the mass has the same value in all five systems, which one has the largest total
energy (kinetic + potential)? All constants below are in a common set of arbitrary units appropriate
for each given quantity.
(A) 8 cos 2t − 2π
3
(B) 5 cos (−2t) (C) 9 cos −t + π2
(D) 3 cos 4t + π4
(E) 2 cos (3t)
6. Two identical train whistles each play a tone of frequency 155 Hz. When one train is moving away
from the station and the other is stopped, a commuter standing on the station platform hears beats
with a frequency of 5 Hz. What is the speed of the moving train? You may take the speed of sound
to be 340 m/s.
(A)
(B)
(C)
(D)
(E)
10.2
11.3
11.6
9.60
10.6
m/s
m/s
m/s
m/s
m/s
Page 5 of 17
7. A nylon clothesline of length 5 m and mass 76 g is under tension of magnitude 50 N. One end of
the line is wiggled up and down in simple harmonic motion with a frequency of 10 Hz, producing
a transverse wave of amplitude 2 cm. What is the power transferred from one end of the string to
the other?
(A)
(B)
(C)
(D)
(E)
1.69 W
1W
0.1 W
69 W
0.69 W
8. The plot to the right shows the displacement x of a mass on a spring as a function of time, t.
Which of the following statements is true at the time indicated
by the dashed vertical line? Select only one.
(A)
(B)
(C)
(D)
(E)
The
The
The
The
The
spring force is zero
spring force is negative
spring potential energy is zero
velocity is positive
velocity is negative
Page 6 of 17
9. Consider a spherical planet with radius R and mass M . Suppose a rock is released from rest at a
height 2R above the planet’s surface. With what speed will the rock strike the planet? You may
disregard friction and assume that the planet’s gravity is the only force acting on rock.
r
2GM
(A)
r R
GM
(B)
r R
GM
(C)
r 2R
2GM
(D)
r 3R
4GM
(E)
3R
10. In the diagram to the right, a ship travels parallel to the shore at a distance of d = 600 m.
The ship’s radio receives simultaneous signals of the same frequency and amplitude from two antennas A and B, which are
separated by distance L = 800 m. The signals interfere constructively when the ship is at point C, midway between A and
B. The ship’s communications officer notices that the total radio
signal vanishes for the first time when the ship reaches point D,
which is directly outward from the shore from point B. What is
the wavelength of the radio waves?
(A)
(B)
(C)
(D)
(E)
400
800
200
600
100
m
m
m
m
m
Page 7 of 17
11. Andromeda the closest big galaxy to our own (the Milky Way). Andromeda’s mass is about 2.8×1012
times the mass of our Sun, and is 2.42 × 1022 m away. The mass of the Milky Way is 2.1 × 1012 times
the mass of the Sun. Assume that the Milky Way experiences no other forces except Andromeda’s
gravity. Based on this, what is the magnitude of the Milky Way’s acceleration? The mass of the
sun is MS = 1.9891 × 1030 kg.
(A)
(B)
(C)
(D)
(E)
3.6 × 10−13
8.5 × 10−13
6.3 × 10−13
4.8 × 10−13
1.2 × 10−13
m/s2
m/s2
m/s2
m/s2
m/s2
12. Two planets orbit the same star in concentric circular orbits in the star’s equatorial plane. Of the
two, the planet that is farther from the star must have
(A)
(B)
(C)
(D)
(E)
The
The
The
The
The
larger period
larger universal gravitational constant
smaller period
larger mass
smaller mass
Page 8 of 17
13. Two waves traveling in opposite directions produce a standing wave. The individual wave functions
are:
y1 (t)
y2 (t)
= 4.0 sin(3.0x − 2.0t),
= 4.0 sin(3.0x + 2.0t).
Here, x and y are measured in centimeters and t is in seconds. What is the amplitude of the simple
harmonic motion of the element of the medium located at x = 2.3 cm?
(A)
(B)
(C)
(D)
(E)
6.0
2.3
6.5
3.2
4.6
cm
cm
cm
cm
cm
14. When you add damping to a simple harmonic oscillator (without a driving signal), what happens
to the amplitude and period of oscillations?
(A)
(B)
(C)
(D)
(E)
The
The
The
The
The
amplitude
amplitude
amplitude
amplitude
amplitude
decreases
decreases
decreases
decreases
decreases
exponentially over
exponentially over
linearly over time,
linearly over time,
exponentially over
time, and the period decreases.
time, and the period is unchanged.
and the period decreases.
and the period increases.
time, and the period increases.
Page 9 of 17
15. A flute has a length of 58 cm. If the speed of sound in air is 343 m/s, what is the fundamental
frequency of the flute, assuming it is a tube closed at one end and open at the other end?
(A)
(B)
(C)
(D)
(E)
591 Hz
296 Hz
444 Hz
148 Hz
None of the above
16. The displacement from equilibrium of an object undergoing simple harmonic motion is described by
x(t) = A cos (ωt + ϕ), where A = 5.0 cm and ω = 20 rad/s. Suppose that at t = 0, the object is at
x = 0, and traveling in the +x direction. Based on this, what is the value of the phase constant ϕ?
(A)
(B)
(C)
(D)
(E)
+π
− π2
+ π2
0
−π
Page 10 of 17
17. A standing wave having seven nodes is set up in a string fixed at both ends. If the frequency of the
wave is doubled, how many antinodes will there be?
(A)
(B)
(C)
(D)
(E)
7
13
12
6
14
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PCS 125 — Formula sheet
(Last modified: April 14, 2018)
Equations
ω=
ω=
r
s
k
m
ω02
s
ω=
−
b
2m
T
v=
µ
∆P
B=−
∆V /Vi
ρAω 2 s2max v
Poweravg =
2
1
Power = µω 2 A2 v
2
r
2
r
A= q
g
, ω=
L
r
mgd
I
F0 /m
2
(ω 2 − ω02 ) +
v = fλ
T =
2π
1
=
ω
f
bt
bω 2
m
s(x, t) = smax cos(kx − ωt)
Power
Power
=
A
4πr2
v + vO
f0 =
f
v − vS
I=
x(t) = Ae− 2m cos(ωt)
v=
s
B
ρ
∆Pmax = ρvωsmax
I=
2
∆Pmax
2ρv
TC
sin(a ± b) = sin a cos b ± cos a sin b
fbeat = |f2 − f1 |
1+
◦
273
C
I
a∓b
a±b
= (10 dB) log10
sin a ± sin b = 2 cos
sin
I0
2
2
Gm1 m2
GM m
kq1 q2
=
U
(r)
=
−
F
=
r2
r2
I r
σ
q
kq
1 q2
~ · dA
~ = in
=
ΦE = E
U (r) =
2ε0
0
r
Z B
dV
field
field
=−
UB − UA = q(VB − VA ) = −WA→B
WA→B
F~ · d~r
=
dx
A
~ · ~rAB = −El cos θ = −Ed
VB − VA = −E
v = (331 m/s)
β
F
~
E
E
Iavg = nqvD A
~ + q~v × B
~
F~ = q E
IB
VH =
nqt
∆V = RI
~ ×B
~
F~ = I L
Z
d~s × r̂
~ = µ0 I
B
4π
r2
Power = ∆V I
FB
µ0 I1 I2
=
L
2πa
µ0 I
(at a distance a from an infinitely long wire)
2πa
µ0 Iθ
B=
(at the centre of an arc wire of radius a subtended by an angle θ)
4πa
µ0 Ia2
B=
(at a distance x along the centre line axis of a circular loop of wire of radius a)
2 (a2 + x2 )3/2
B=
PCS 125 — formula sheet page 1 of 2
Constants
Grav. accel. on Earth
Speed of light in vacuum
Gravitational constant
Coulomb constant
Elementary charge
Electron volt
Mass of electron
Mass of proton
Permeability of vacuum
Permittivity of vacuum
Density of air
Threshold of hearing
g = 9.81 m · s−2
c = 3.00 × 108 m · s−1
G = 6.6742 × 10−11 N · m2 · kg−2
ke = 8.9876 × 109 N · m2 · C−2
e = 1.6022 × 10−19 C
eV = 1.6022 × 10−19 J
me = 9.1094 × 10−31 kg
mp = 1.6726 × 10−27 kg
µ0 = 4π × 10−7 T · m · A−1
ε0 = 8.8542 × 10−12 C2 · N−1 · m−2
ρair = 1.2 kg · m−3
I0 = 1 × 10−12 W · m−2
PCS 125 — formula sheet page 2 of 2