Physics 140 Winter 2004 Midterm 3
1. (P) The figure below shows an 18.0kg crate of salami sliding down a ramp on a
(massless) rope that is wrapped around a disk of radius 0.25m and mass 6.0kg. The
coefficient of kinetic friction between the crate and ramp is µk=0.24. What is the
magnitude of the disk’s angular acceleration?
a)
b)
c)
d)
e)
3.1 rad/s2
6.7 rad/s2
9.8 rad/s2
11.4 rad/s2
13.7 rad/s2
2. (P) A uniform solid sphere rests suspended from a cord anchored at point P along an
inclined plane. Which of the following statement(s) is(are) true?
P
C
a)
b)
c)
d)
e)
The torque about point P due to the weight of the sphere is zero.
The torque about point P due to the tension in the cord is zero.
The torque about point C due to the normal force of the incline on the sphere is zero.
both b) and c)
all of a), b) and c)
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Physics 140 Winter 2004 Midterm 3
3. (P) A rigid assembly consists of a thin hoop of radius R and mass m attached to a thin
rod also of mass m and length 2R rests in the vertical position shown in the figure
below. Given a tiny nudge (involving negligible energy) from behind, the assembly
rotates about the horizontal axis shown. What is its angular speed as it passes through
the inverted (upside-down) orientation?
hoop
rod
rotation
axis
a)
b)
c)
d)
e)
(96g/65R)1/2
(33g/15R)1/2
(48g/17R)1/2
(17g/12R)1/2
(8mg/15R)1/2
4. Kepler’s second law of planetary motion – the observation that the orbits of planets
sweep out equal areas around the Sun in equal times – mainly reflects the
conservation of which physical quantity?
a)
b)
c)
d)
e)
mechanical energy.
angular momentum.
linear momentum.
gravitational potential energy.
mass.
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Physics 140 Winter 2004 Midterm 3
5. Three coplanar bodies A, B and C with masses m, 2m, and 8m, respectively, are
shown below. At what (x,y) location should a fourth body D of mass 8m be placed
such that the total gravitational force on body A is zero?
y
B
d
C
x
A
2d
a)
b)
c)
d)
e)
( –21/4 d, 21/4 d )
(21/4 d, d/21/4)
(21/4 d, –21/4 d )
( d/21/4, –d/21/4)
( d/21/4, –21/4 d )
6. A 2kg box of pepper-encrusted salamis sits 0.5m from the right edge of a uniform 4kg
beam of length 2m. The markings on the beam are at quarter-length intervals. Both
the beam and the salamis rest atop a pair of rollers A and B, located at the center and
right edge of the beam. What is the magnitude of the normal force acting on the
beam from roller A?
A
a)
b)
c)
d)
e)
B
18 N
28 N
38 N
49 N
62 N
3
Physics 140 Winter 2004 Midterm 3
7. A puck tied to a string is revolving on horizontal, frictionless ice in a circle centered
on an anchor point O. If the string breaks suddenly, then just after the break the
angular momentum of the puck about the anchor point O will
a)
b)
c)
d)
e)
be smaller than before the break.
be the same as before the break.
be larger than before the break.
be zero.
be the same magnitude as before the break, but with opposite sign.
8. (P) The figure below is hinged about point P. The four forces applied at each corner
have equal magnitude. Rank the forces according to the magnitude of the torque
they create about point P, greatest to least.
1
P
•
4
2
3
a)
b)
c)
d)
e)
1234
4312
2134
3214
2314
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Physics 140 Winter 2004 Midterm 3
9. (P) Starting from rest, a small hollow, uniform sphere of mass m=1.5kg and radius
r=0.75cm rolls without slipping down a ramp and executes a loop-the-loop. The
initial height of the sphere’s center of mass above the bottom of the loop is H=25cm
and the loop radius is R=7.5cm. What is the center-of-mass velocity of the sphere at
the top of the loop? (Take into account the finite size of the ball when computing the
vertical displacement.)
a)
b)
c)
d)
e)
0.75 m/s
0.99 m/s
1.08 m/s
1.12 m/s
1.24 m/s
10. (P) A uniform block of width W and height H rests on a rough inclined board with
coefficients µs and µk of static and kinetic friction, respectively. As the angle θ made
by the board with respect to the horizontal increases, the block will
W
H
a)
b)
c)
d)
e)
always slide before tipping over.
always tip over before sliding.
tip first if µs > W/H.
slide when tan θ = µk.
tip first if µs < W/H.
θ
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Physics 140 Winter 2004 Midterm 3
11. (P) Consider the following thought experiment that could be performed in deep space.
A very small “test mass” m is released from rest a distance d from the edge of a thin
shell of mass M and radius R. Gravity attracts the test mass toward the shell, and a
small hole drilled in the outer shell allows it to enter the interior. What is the velocity
of the test mass as it passes through the center of the shell?
d
R
•m
small hole
M
a)
b)
c)
d)
e)
[GMR / (R+d)]1/2
[2GMd / (R+d)]1/2
[2GM / (R+d)]1/2
[2GMd / R(R+d)]1/2
[2GMR / d(R+d)]1/2
12. (P) What is the total angular momentum about the origin O for the three-particle
system shown below?
y
12m/s
2kg
1m
3kg
6m/s
a)
b)
c)
d)
e)
O
2m
2kg
x
1.5m
24 kg m/s
12 kg m/s
8 kg m/s
0 kg m/s
–12 kg m/s
6
4m/s
Physics 140 Winter 2004 Midterm 3
13. In the birds-eye view below, the smiley-face marks the location of a child of mass
40kg standing at rest on the edge of a horizontal merry-go-round that can be treated
as a disk of mass 60kg and radius1.5m. A ball of mass 0.5kg is thrown with velocity
3m/s at the angle shown. After the child catches the ball, and assuming a frictionless
axle, how long will it take the child/ball/merry-go-round system to rotate 1 radian?
35º
a)
b)
c)
d)
e)
2s
13 s
45 s
88 s
123 s
14. (P) Imagine drilling a very thin hole straight down from the surface to the center of a
spherical moon of mass M and radius R and uniform mass density. Further imagine
lowering a small mass m into the hole until it is a distance r from the center (r<R). If
you let the mass drop from rest, what will be its kinetic energy as it reaches the
moon’s center?
a)
b)
c)
d)
e)
GMmr / R3
–GMmr2 / R3
GMmr / R2
GMmr2 / 2R3
GMmr / 3R3
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Physics 140 Winter 2004 Midterm 3
15. In the static configuration below, box A of mass mA=11kg lies on a frictionless ramp
inclined at an angle θ=30º while box B of mass mB=7kg and hangs from a vertical
cord. What is the magnitude of the tension in the upper cord C?
C
A
θ
B
a)
b)
c)
d)
e)
48 N
67 N
106 N
235 N
471 N
16. Consider a can of tomato soup rolling up an inclined ramp without slipping. Which
one of the following statements about the motion is false?
vcom
•O
•
C
a)
b)
c)
d)
e)
The magnitude of the can’s angular velocity is decreasing.
The ratio Krot/Ktrans of kinetic energies in rotation and translation is constant.
The torque about the contact point C due to the weight of the can is non-zero.
The torque about the can’s center of mass O due to the normal force of the ramp is zero.
The force of static friction at the contact point acts in the direction opposite the center of
mass velocity vcom.
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Physics 140 Winter 2004 Midterm 3
17. Comets A, B and C have equal mass and orbit a common star as shown below.
Comets A and B have orbits with the same semi-major axis. Which of the following
statement(s) about their orbits is(are) true?
B
C
A
a)
b)
c)
d)
e)
C’s orbit has lower angular momentum than B’s.
A’s orbit has a more negative mechanical energy than B’s.
C’s orbit has a more negative mechanical energy than A’s.
A’s orbit has higher angular momentum than B’s.
A’s orbit has the same period as B’s.
18. (P) Two uniform rods, with center-of-mass locations CM shown below, are connected
to tables via frictionless pivots at their ends. Rod B is longer than rod A, but both
are made of the same material with fixed mass per unit length. Both are initially
released from rest at the same angle θ. Which of the following statements about the
motion is false?
a)
b)
c)
d)
The initial linear accelerations of the CM locations of both rods are the same.
The initial angular accelerations of both rods are the same.
The angular accelerations of the rods both increase in time.
At some time after release but before hitting the table, the kinetic energy of rod B is
larger than that of rod A.
e) The linear accelerations of the CM location of the rods initially point in the same
direction.
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Physics 140 Winter 2004 Midterm 3
19. The gyroscope below – an aluminum wheel of mass m and radius r, spinning with
angular velocity ω – is observed to revolve around the pivot with period T. If the
lever arm d from the pivot were doubled, what change could you make to the wheel
to keep the precession period T fixed? (Assume all other variables are held fixed
during the change.)
d
P
a)
b)
c)
d)
e)
halve its mass.
halve its radius.
double its radius.
double its angular velocity.
double its mass.
20. A solid disk and a ring roll without slipping down an incline. When will the disk
accelerate more quickly down the incline than the ring?
a)
b)
c)
d)
e)
Only if the disk is more massive than the ring.
Only if the disk has larger radius than the ring.
In all cases.
In no case.
Only if the ramp is inclined at an angle greater than 30 degrees.
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Physics 140 Winter 2004 Midterm 3
Answer Key
correct /partial credit
1. c /e
2. d /b,c
3. a /c
4. b
5. c
6. d
7. b
8. e /c
9. d /c
10. c /e
11. d /c
12. b /e
13. e
14. d /c
15. c
16. e
17. e
18. b
19. d
20. c
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