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Unformatted text preview: FACULTY OF SCIENCE
FINAL EXAMINATION
PHYSICS 198131A MECHANICS AND WAVES Examiner: Prof. R. Harris Wednesday December 8th, 2004
Associate Examiner: Prof. M. Kilfoil 9:00 am. — 12 noon Answer all questions:
Part I: 15 questions at 1 point each for a total of 15 points
Part II: 5 questions at 5 points each for a total of 25 points INSTRUCTIONS: Answer Part I by ﬁlling out the pink computer sheet. Be sure to write your name, student ID and version number on the sheet, and
also enter them on the sheet at the left, so that they can be machineread.
The Examination Security Monitor Program detects pairs of students with unusually similar answer patterns on multiplechoice exams such as this one. Data generated by this program can be used as admissible evidence, either to initiate or corroborate an investigation or a charge of cheating under Section 16 of the Code
of Student Conduct and Disciplinary Procedures. Answer Part II in the booklet provided. Calculators are permitted. Formulae are provided on the last page.
You may keep the question paper.
This exam comprises 10 pages, with questions on pages numbered 1 through 7. PHYS 131 1 Part I: Questions Each question is worth 1 point.
Mark your answers on the pink computer cards. 1. The Montreal—Ottawa train takes 2 hours to travel between the two cities. Starting from
rest, the train steadily increases speed for the ﬁrst 15 minutes, then travels at a constant
speed of 80 km/ hr, then steadily reduces speed for the last 15 minutes, ﬁnally coming to
rest again. What is its average acceleration during the entire journey? A 80 km / hr UU ( l
( l
(C) 160 liIIl/l'lI‘2.
(D) 320 km/hrB. 2. A foam—rubber ball is thrown into the air. A physics student realises that air—resistance is
very important for the description of its motion The magnitude of its acceleration as it travels upwards is A B sometimes greater, sometimes smaller than 9. smaller than g Q ( J
( l
( ) equal to g
(D) greater than 9 3. At a particular instant, the velocity of a body moving in a straight line is negative. The
slope of the velocity—time plot at this instant is also negative. See the diagram. The
mathematical deﬁnition of the acceleration means that it ) has a sign opposite to the velocity, and is decreasing. B) has a sign the same as the velocity, and is increasing. C) has a sign opposite to the velocity, and is increasing.
) has a sign the same as the velocity, and is decreasing. PHYS 131 2 4. The freebody diagram shows the four forces acting on a wooden block. The forces are:
W, the force of gravity; N, the normal force of the ground; P, the push of a child; and
"f, the force of friction. The block is not moving. Which pair of forces are equal and opposite because of Newton’s Third Law? (A) None of them. i (B) P and Ff. (C) w and N. ﬂ (D) W and P. “Ff
lN An inexperienced mover tries to push a stove across a level kitchen floor. The stove weighs
5000 Newtons, and the mover pushes with a force of 1500 Newtons. The stove does not move. 0'! The coefﬁcient of static friction is A) smaller than or equal to 0.3. (
(B) unknown: need more information.
(G) greater than or equal to 0.3. l D) exactly 0.3. 6. Two carts are at rest on an air track. One has mass m, the other 2m. You push each of
them in turn with the same force for the same interval of time. The change of momentum of the heavier mass is therefore (A) half that of the lighter mass. (B) four times that of the lighter mass.
(C) twice that of the lighter mass. (D) equal to that of the lighter mass.
(E) a quarter that of the lighter mass. 7. A student is given two carts, X and Y, to place on an air—track. They look identical, but
they may be constructed of different materials. He places X at rest on the track, and gives
Y a (positive) velocity towards X, so that there is a collision. After the collision, he claims
that both carts move in the same direction, but that the (magnitude of the) velocity of
Y is greater than it was before. He is unclear as to whether the ﬁnal velocities are both positive or both negative! Do you conclude that (A) both carts are identical? (B) cart X is less massive than Y?
(C) cart Y is less massive than X?
C l the student must be mistaken? 8. 10. 11. JLIILJ lul "’ A mass m is attached to a long string, and acts as a pendulum. It starts its swing at a
height h above a frictionless horizontal surface. At the bottom of its swing it collides with
a second mass which is at rest on the surface. It bounces back, with its velocity reduced,
and the second mass carries away 25% of its kinetic energy. To what height will it bounce
back? Ignore the effects of air resistance. (A) Exactly h/4. (B) Less than or equal to h/4
(C) Less than or equal to Biz/4
(D) Exactly Biz/4. A tennis ball is tied by a long string to a ﬁxed pole. It is not moving. A second ball
collides with the ﬁrst ball at right angles to the string, as shown, causing the ball on the
string to rotate around the pole with angular velocity am. If the experiment is repeated, with all conditions identical except that the string is twice
as long, is the new angular velocity (A) none of the above? (— . ‘—
(B) w/Q? w ‘
(C) 2W? \ Strlﬂg
(D) u)? Pole A ﬁgure skater stands on one spot on the ice (assumed frictionless) and spins around with
her arms extended. When she pulls in her arms, she reduces her moment of inertia, so
that her angular velocity increases. Compared to her initial rotational kinetic energy, her
new rotational kinetic energy is (A) higher.
(B) lower.
(C) the same. (D) not determined, because more information is required. Just as described in the midterm exam, passengers are inside a rotating ring at the amuse—
ment park. The axis of rotation is horizontal. After an accident, when the motor failed,
and passengers fell from the top of the ring, the operators were required to install safety—
harnesses. At the bottom of the ring, which of the following statements is correct? N is the normal
force of the ring, and H the restraining force of the harness. The (magnitude of the) total force on a passenger of mass m, moving with speed ’0, is A — mg + H 03 N )N
)
)mg
)
) O (
(
(
(D
(E PHYS 131 4 12. 13. 14. The change in gravitational potential energy required to put a particular satellite in a
circular orbit of radius R is G'rrLME(1/RE — 1 / R) : 3.14 X 1011 joules. The symbols have
their usual meanings. However, during the launch, the total energy available is only 3.15 X 1011 joules. The
designers therefore decide to direct the launch vertically upwards. What happens? (All
the energy is available for the satellite, but the satellite itself has no means of propulsion
or steering.) A The satellite goes into an orbit of radius greater than R. C The satellite reaches beyond R, but then falls back to the ground. ( l
(B) The satellite escapes entirely from Earth’s gravitational ﬁeld.
( l
(D) The satellite goes successfully into orbit with radius R. A 10 kg mass is hanging in equilibrium on a strong spring. It is then about 10cm above
some eggs lying on the ground, as shown in the ﬁgure. It is pulled down — extending the
spring — until it just touches the eggs. It is then released, so that the contracting spring
pulls it upwards. Eventually, of course, the force of gravity pulls it back downwards, and it
oscillates about its equilibrium position. Friction and air resistance have negligable effects
on the motion. / f 5
At the lowest point of its motion, the mass will 5 / f
(A) just touch the eggs. 9
(B) be less than 10 cm below the equilibrium point. 10 cm
(C) be above the equilibrium point. _._ (D) smash the eggs. A string is stretched between two ﬁxed points, and vibrates with a transverse standing
wave. At a particular instant of time, the string has its maximum transverse displacement,
as shown in the diagram. The positive direction of transverse displacements is also shown. y A .. At time shown, the velocity of points along the string is ) everywhere negative.
) everywhere positive.
C) everywhere zero. positive in some places, negative elsewhere. PHYS 131 UK 15. A sinusoidal traveling wave on the surface of a pond has the equation
y = Acos [27r(:c/A — t/T)J
A duck is moving up and down on the wave at position a = A/ 2. A second duck is moving up and down at position a: I 3M4, and a third duck at :1: = 29AM. Which one of the following observations is false? (A) Ducks 1 and 2 never have the same displacement at the same time.
(B) The three ducks move with the same frequency. (C) Ducks 1 and 3 reach their maximum heights at times T/4 apart. ( l D Ducks 2 and 3 move exactly out of phase with each other. Part II: Problems Each question is worth 5 points.
W’rite your answers in the booklets provided.
You must give reasons for your answers. Credit will not be given for answers with no
supporting reasoning, even if they are numerically correct.
In all questions, take the acceleration due to gravity to be
10 111/52.
. A stone—age hunter places a 1.0 kg rock in a sling, and swings it in a horizontal circle
around his head on a length of vine (string). The vine is 1.0 m long. The angular velocity of the rock is 100 rpm. (a) Draw a free body diagram for the rock, clearly showing all the forces that act on it.
(Neglect friction and air resistance.) (1)) Calculate the tension in the vine. The tension that breaks the Vine is 200 N. (c) What is the maximum possible angular velocity of the rock? PHYS 131 6 2. In Northern Quebec, dog—sleds are still sometimes used to transport goods and people
between isolated communities. A wooden sled, with rider and payload, has a mass of 200
kg. Starting from rest, when the snow surface is horizontal, it takes a dog—team 15 metres
to reach their “cruising speed” of 5.0 m/s. Two ropes are attached to the sled, one on each side of the dog team. The ropes pull
upwards at 10° to the horizontal. The coefﬁcient of kinetic friction between the sled and the snow is 0.06. (a) What is the acceleration of the sled during the time to reach cruising speed? Assume
that it is uniform. (b) Draw a free—body diagram for the sled which clearly identiﬁes the horizontal and
vertical forces which act on it. (c) Write the equations which demonstrate Newton’s 2nd Law applied to the horizontal
and vertical motions. (d) Calculate the tension in each of the ropes. 3. A mass of 0.01 kg is attached to a spring of spring constant 4.0 N /m. The “system” is set
in motion, and oscillates back and forth on a horizontal surface. There is no friction or
air—resistance. The initial velocity at zero displacement is 0.5 m/s. (a) What is the period of oscillation?
(b) What is the maximum displacement (the amplitude)? (c) What is the total energy of the “system”? Now consider the same system, but with friction: the mass is sliding on a horizontal surface
with coefﬁcients of friction n, = 0.9 (static) and #1: = 0.6 (kinetic). Once again, the initial
velocity at zero displacement is 0.5 m/s. (d) How far does the mass travel before it comes instantaneously to rest? (e) After coming to rest at this point, does it start to move again? Give reasons for your
answer. PHYS 131 7 4. A solid disk has a diameter of 20 cm and a mass of 2 kg. It is rotating freely about its
axis at 200 rpm. (There is no motor attached to it.) A circular hoop, also of diameter 20
cm, but of mass 1 kg is dropped straight down onto the rotating disk. See the diagram. rThere is friction between the disk and the hoop, which causes the hoop to (angularly)
accelerate until it is moving at the same (angular) velocity as the disk.
(a) What is the ﬁnal angular velocity of the disk and hoop?
(b) What is the change in the kinetic energy of the disk?
(c) How much work is done by the frictional torque to bring the system to the ﬁnal
velocity?
The time taken to reach the ﬁnal velocity is 10 seconds. ((1) Therefore, what is the magnitude of the average frictionai torque? 5. Two loudspeakers are facing each other from opposite sides of a room. The distance
between the loudspeakers is 50 metres, and the wavelength of the sound is 1 metre. Standing waves are set up.
(a) How many nodes are there between the two walls? How many alitiniodes? Your friend runs from one wall to the other, at 5 m/ s. She hears sound which periodically
rises and falls in intensity. (b) How many times per second does she hear the rise and fall of intensity? From your friend’s point of view, the travelling waves coming from the two speakers do
not have the same frequency, because of the Doppler effect. (c) What are the two frequencies that she associates with the two speakers?
((1) What is the beat frequency between these? (6) For two bonus points: all or nothing! Is this the same frequency as in part (b)?
Why, or why not? FORMULAE For angular motion, replace A3: Av 93, v and a by 6, w and a <v>=— <a>=—— At ’ At vzvi+at, v2=vf+2am l 2
xtmi+vit+—at 2
R:vfsin26'/g
f—1 w—Qarf
_T’ _
w2 z i, [U22 _9'_
m l
$=Acoswt, v:—wAsinwt
2
(LEE{cm
r 1
FR i —bv, FR : —;2—DpAv2
W: F s : [F d3
1 1 2
KB — 5m?)
AW
P = — 2 F
At ”
:— —k33
l
E: §k$2
AU 7 mgh s=r6,v:rw, (17:70 F : —G'M'm/cr‘2, U r —G'Mm/T respectively. T=FTJ_:ICI
W=T6=de9 I : CMR2 2 MW
hoop: c: 1; disk : c: 1/2 IIICM+MD2
Lxlw Ezmvrzpr KE— 1I 2
wzzDMg/I $0M = Zmiﬂii/ 2m:
vfo, '1) :1/T/u
k227r/A y 2 Aces (in: :i: wt) 3; t A sin k9: sin wt 3:10l0gI/Ig
fb =f1w f2
_ v—%
f” fay .._ v3
—b:t\/b2—4ac
x132 = 2a Useful constants: 9 =10 m/s2 2 10 N/kg G : 6.67 x 10—11Nm2/kg2 ...
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