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FACULTY OF SCIENCE
SAMPLE FINAL EXAMINATION
PHYSICS 198-101A (2000)
MECHANICS AND WAVES
Examiners:
INSTRUCTIONS:
Professor R.B. Moore
Date:
Thursday, Dec. 14, 1999
Professor Z. Altounian
Time:
9:00 - 12:00
Answer all 8 questions. All questions are worth 10 marks. The
marks for the parts of each question are indicated.
(80 marks are full marks for this exam.)
Note that marks are given separately for indications of the correct
method. Also, in some questions, marks are awarded for a
diagram.
Just stating the correct answer will not be enough for
full marks.
Formulae and other necessary data are at the back of the
examination paper.
Dictionaries are permitted
Any type of calculator may be used.
The examination questions are on two sheets. T h e
complete examination package comprises 5 s h e e t s ,
including this cover sheet.
You may keep the examination sheets if you wish.
Physics 198-101A Sample Final Examination (2000)
1.
page 1 of 2
A football player runs a pass-pattern which requires him to accelerate at a point 12
meters downfield, in a direction 30 degrees right of downfield, in 4 seconds, then,
without changing speed, to cut back in a direction 45 degrees left of downfield for
another 2.5 seconds at constant speed, whereupon he is to catch the ball in flight
at that point.
Draw a diagram of his path. (Include the distances.) (4 marks)
Assuming he accelerated at a constant rate, draw a graph of his x-velocity vs time. (x
is the direction to his right. Include numbers.) (4 marks)
Draw a graph of his y-velocity vs time. (y is the down field direction.) (2 marks)
2.
A student walks in a circle, taking steps of 80 cm. If at each step she changes
direction by 1.5 degrees, what is the diameter of her circle? (3 marks for diagram,
1 mark for method 1 mark for answer)
If her pace is 80 steps per minute, what is her angular velocity? (2 marks for
method 1 mark for answer)
What is her centripetal acceleration? (1 mark for method 1 mark for answer)
3.
The temperature of a chamber is found to vary with time according to the equation
T = 16.0 + 25.4 sin(0.00246t − 0.425) 0C
What is the amplitude of the temperature oscillation? (1 marks) What is its period?
(3 marks) What is its phase? (1 mark)
Sketch this function as a graph. (5 marks)
4.
A polarizing filter is placed in the light beam illuminating a microscope slide. Another
polarizing filter in the body of the microscope intercepting the light after it has
passed through the slide is oriented so that there is no light passing through to the
eye-piece. A specimen that is optically active is then placed on the slide.
EYEPIECE
ANALYZING POLARIZER
SPECIMEN
LIGHT POLARIZER
LIGHT SOURCE
If the optical rotation of this specimen allows an electric field vector at the eye that is
5% of that passed through the first filter, as measured by a light meter, by what
angle has the specimen rotated the plane of polarization? . (3 marks for drawing of
electric field vectors, 3 marks for method and 4 marks for answer.)
5.
A noisy piece of machinery is causing a 400 Hz tone at your ears, with a pressure
-4
amplitude of 2.5 × 10 Pa. There is a so-called "active noise suppression system in
place, which is essentially a sound amplifier which picks up the sound from the
machine and plays it back through a loudspeaker to your ears but with a phase
delay so that it tends to cancel the machine sound. If this loudspeaker gain is set so
-4
that it does produce 2.5 × 10 Pa at your ears but at a phase of 160 degrees
relative to the sound of the machine itself, what will be the sound pressure at your
ears? (3 marks for phasor diagram, 4 marks for method and 3 marks for correct
answer.)
Physics 198-101A Sample Final Examination (2000)
6.
page 2 of 2
You are sitting on a bathroom scale in an airplane. (Don't ask why you might be
doing this, perhaps you are a bathroom-scale fetiscist.) As the plane enters an "airpocket" you notice that the scale reads 45 kg. If you know that you weigh 80 kg,
what is the vertical acceleration of the plane ? (3 marks for diagram 2 marks for
method, 2 marks for answer)
If this acceleration persists, by how much has the plane risen (or fallen - specify
which) in 5 seconds? (2 marks for method, 2 marks for answer)
7.
At what altitude above the Earth's surface is the gravitational field 6.5 N/kg? (3
marks for method, 2 marks for answer) What is the gravitational potential at this
height? ? (3 marks for method, 2 marks for answer). Take the field at the Earth's
surface to be 9.8 N/kg and the potential to be zero at infinity. The mass of the Earth
and the gravitational constant can be found on the symbol sheet at the back of the
exam.
8.
A torque of 1.2 N-m2 is applied to a 12 kg hoop of 1.6 m diameter that is rotating
about its center. What is its angular acceleration? (2 marks for diagram, 2 marks
for method and 1 mark for correct answer). What would be the kinetic energy of
the hoop after 2 seconds of application of this torque? (1 mark for method, 1
mark for answer).
At this 2 seconds the torque is released and the hoop left to spin. An
electromagnetic clutch then locks the hoop to a disk on the same axis but with a
moment of inertia of 8 kg-m2. What will be the rotational speed of the locked
combination? (3 mark)
Physics 198-101A Symbols used in Formula Sheet (Revised Nov. 2000)
Quantity
Area
Base of triangle
Height of triangle
Radius
Volume
Horizontal coordinate
Vertical coordinate
Angle
Arc length
Angular speed
Velocity, speed
Angular acceleration
Acceleration
Time
Displacement
Displacement vector
Initial velocity
Final velocity
Average velocity
Initial angular velocity
Final angular velocity
Average angular velocity
Velocity vector
Acceleration vector
Centripetal acceleration
Density
Mass
Specific gravity
Pressure
Height, depth
Acceleration of gravity
Force
Torque
Force perpendicular. to r
Distance of line from pivot
Distance from center (pivot)
Spring constant
Stress
Strain
Young's Modulus
Shear stress
Shear strain
Shear Modulus
Gravitational constant
Major mass
Gravitational field
Gravitational potential
Distance from Earth center
Distance from Sun center
Radius of Sun
Radius of Earth
Mass of Sun
Mass of Earth
Quantity
Work
Symbol
A
b
h
r
V
x
y
θ
sarc
ω
v
α
a
t
x, y, z, s
s
vi
vf
vave
ωi
ωf
ω ave
v
a
aC
ρ
m
sg
p
h
g
F
τ
F
r
r
k
σ
ε
Y
σs
γ
S
G
M
g
φG
r
r
R
R
M
M
Symbol
W
Value, units
m2
m
m
m
m3
m
m
Radians (rad)
m
rad/s
m/s
m/s2
m/s2
s
m
m
m/s
m/s
m/s
rad/s
rad/s
rad/s
m/s
m/s2
m/s2
kg/m3
kg
Pascal (Pa)
m
9.8 m/s2
Newton (N)
N-m
N
m
m
N/m
Pa
Pa
Pa
Pa
6.67×10-11
N-m2/kg2
kg
N/kg
Joule (J)/kg
m
m
6.97×108 m
6.37×106 m
2.0 ×1030 kg
5.98×1024 kg
Value, units
J
Parallel comp. of l
Distance, length
Power
Potential energy
Potential energy of gravity
Potential energy of a spring
Kinetic energy
Kinetic energy of rotation
Moment of inertia
Linear momentum
Linear momentum vector
Weight
Tension
Angular Momentum
Wave distance
Wave, oscillation amplitude
Wave, oscillation frequency
Wave, oscillation period
Wavelength
Wavenumber
Angular frequency
Phase angle
Angle of incidence
Angle of reflection
Index of refraction
Angle of refraction
Speed of light
Refractive power (diopters)
Focal length
Distance of object from lens
Distance of image from lens
Height of image
Height of object
Magnification
Electric field strength
Two slit separation
Width of slit
Diameter of circular beam
Resolution
Numerical aperture
Diameter of a lens (aperture)
Decibels
Harmonic number
Fundamental frequency
Electrical current
Electrical potential
Electrical resistance
Number of kg-moles
Universal gas constant
Temperature
Diameter of tube
Viscosity
Amplitude of Sound pressure
Decibel relative
Decibel absolute
l
l
P
PE
PEG
PEk
KE
KEω
I
p
p
W
T
L
z
A
f
T
λ
κ
ω
φ
θi
θr
n
θt
c
D
f
so
si
hi
ho
M
E
s
w
d
R
NA
a
db
n
fo
I
V
R
n
R
T
d
η
pamp
dB
DBO
m
m
Watt (W)
J
J
J
J
J
kg-m2
kg-m/s
kg-m/s
N
N
kg-m2/s
m
m
Hz
s
m
m-1
rad/s
rad
rad
rad
rad
3.00 × 108 m/s
m-1
m
m
m
m
m
V/m
m
m
m
m
m
Hz
Ampere (A)
Volt (V)
Ohm ( Ω )
8314.41 J/kgmole/K
K
m
N-s/m2
Pa
Physics 198-101A Formula sheet (revised Nov 2000)
GEOMETRY
Area of a triangle
Area of a circle Surface area of a sphere Volume of a sphere
1
4
A = πr 2
A = 4πr 2
A = bh
V = πr 3
2
3
y
x
y
Right-angled triangle:
r 2 = x 2 + y 2 ; sin θ = ; cosθ = ; tan θ =
r
r
x
sarc
v
aT
θ2
Radian measures: θ =
Small angle limit: sin θ = tan θ = θ ; cosθ = 1 −
; ω= ; α=
r
r
r
2
KINEMATICS
1
Motion at constant velocity: s = vt; θ = ωt ( ω = 2πf ; f = )
T
1
1
vave = vi + v f ; v f = vi + at; s = vavet = vi t + at 2 ; v 2f = vi2 + 2 as
2
2
Motion at constant acceleration:
1
1
ω ave = ω i + ω f ; ω f = ω i + αt; θ = ω avet = ω i t + αt 2 ; ω 2f = ω i2 + 2αθ
2
2
v2
2
Vector addition of velocities: v AC = v AB + v BC Centripetal acceleration: ac = ω r =
r
STATICS
F
ρ
ρ
m
=
Density: ρ =
Specific gravity: sg =
; Weight: W = mg : Pressure: p = ;
ρwater 1000
V
A
(
(
)
)
Hydrostatic pressure: p = ρgh
Balance: ΣF = 0; Στ = 0; τ = F r = Fr = Fr sin θ ;
Spring formula: F = − kx
F
∆l
σ
Fs
σ
; Y = ; σs =
εs = γ ;S = s
Stress, strain: σ = ; ε =
A
l
ε
A
εs
GRAVITY: Law of gravitation
mM
FG = − G 2
r
DYNAMICS
Gravitational field
g = −G
Gravitational potential:
M
r2
φG = − G
M
= rg
r
1 2
1
1
kx ; KE = mv 2 ; KEω = Iω 2
2
2
2
dp
dL
1 2
2
Force, torque: F = ma =
; W = mg; p = mv; τ = Iα =
; L = Iω ; I = ∑ mi ri ; Idisk = mr
i
dt
dt
2
1
x = A sin(ωt + φ ); ω = 2πf ; T = ;
f
Mass - spring Pendulum
π
Simple Harmonic Motion: v x = Aω cos(ωt + φ ) = Aω sin(ωt + φ + );
g
k
2
ω=
ω=
l
m
2
2
2
ax = − Aω sin(ωt + φ ) = Aω sin(ωt + φ + π ) = −ω x
Work, energy: W = Fl = Fl cosθ ; P = Fv cosθ ; PEG = mgh; PEk =
GEOMETRICAL OPTICS
1
vi
1
1 1 1
; sin θ c = ; Lens refractive power: D = : Lens eqn:
+ =
vt
n
so si f
f
h
s
250 mm
1 1 1
Magnifying power of eyepiece =
: Close combined thin lenses = + ; D = D1 + D2 Magnification i = − i
ho
so
f
f1 f2
f
f
Compound microscope magnification: M = Mobjective × Meyepiece Telescope magnification: M = objective
feyepiece
Reflection: θ r = θ i ; Refraction: sin θ i = n sin θ t ; n =
WAVES
2π
Wave speed: v = fλ ;
λ
For light in a vacuum v = c ; For sound in normal air (20C) v = 343 m/s; Polarization: Et = Ei cosθ
Wave propagation y = A sin(ωt − kz ) ; k =
Interference and diffraction (central maximum-small angle):
λ
λ
λ
λ
Two-slit: θ = Single slit: θ = 2 Circular beam: θ = 2.44 Circle of confusion: θ = 1.22
w
s
d
d
λ
λ
a
Resolution of a microscope: R = 1.22
; NA =
Resolution of a telescope: θ = 1.22
f
NA
d
SOUND
 dBO 


p 
 p2 amp 
 20 
−5
p
=
2
.
8
×
10
×
10
;
; One decibel = 12.2% increase in pressure.
dB = 20 log10  2 ; dB0 = 20 log10 
amp

−5
p
2
.
8
×
10


 1
3
4
5
6
Harmonics: fn = nf0 ; Fifth = ; Fourth = ; Third = ; Minor Third = ; Semi-tone: 12 2 =1.0595 (5.95% increase)
2
3
4
5
MISCELLANEOUS
V
πd 4
p;
Electrical current: I = ;
Ideal Gas Law: pV = nRT ;
Viscous flow: v flow =
128ηl
R