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Name: ________________________ Class: ___________________ Date: __________
ID: A
C_Rotation_2008 practice exam
Multiple Choice
Identify the choice that best completes the statement or answers the question.
I) a solid sphere
II) a hollow sphere
III) a flat disk
IV) a hoop
1. Objects I-IV, listed above, have the same mass and radius. They are rolled down a hill. Arrange them in the order they reach
bottom. (NOTE: It is possible to reason through this without specific formulas for rotational inertia).
A) I, II, III, IV
C) IV, II, III, I
E) I, IV, III, II
B) I, III, II, IV
D) IV, III, II, I
2. The angular speed in rad/s of the second hand of a watch is:
a) π/1800
A)
b) π/60
π
π
B)
1800
60
c) π/30
C)
d) 2π
π
e) 60
D) 2π
30
E)
60
3. A wheel is spinning at 27 rad/s but is slowing with an acceleration that has a magnitude given by 3t2, in rad/s2 for t in seconds.
It stops in a time of:
A) 1.7 s
B) 2.6 s
C) 3.0 s
D) 4.4 s
E) 7.3 s
4. A wheel rotates with a constant angular acceleration of π rad/s2. During the time interval from t1 to t2 its angular displacement
is π radians. At time t2 its angular velocity is 2π rad/s. Its angular velocity in rad/s at time t1 is:
A) 0
B) 1
D) π
C) π
2
E)
π 3
5. String is wrapped around the periphery of a 5.0-cm radius cylinder, free to rotate on its axis. If the string is pulled straight out
at a constant rate of 10 cm/s and does not slip on the cylinder, the angular velocity of the cylinder is:
C) 10 rad/s
E) 50 rad/s
D) 25 rad/s
A) 2.0 rad/s
B) 5.0 rad/s
5.0 N
2.0 m
30o
30o
4.0 m
5.0 N
6. A rod is pivoted about its center. A 5.0 N force is applied 4.0 m from the pivot and another 5.0 N force is applied 2.0 m from
the pivot, as shown above. The magnitude of the total torque about the pivot (in Nm) is:
0.0
B) 5.0
C) 8.7
D) 15.0
E)
A)
L/2
26.0
L/2
7. Three identical objects of mass M are fastened to a massless rod of length L as shown above. The rotational inertia about one
end of the rod of this array is:
A)
ML 2
2
B)
ML 2
C)
3ML 2
2
D)
1
5ML 2
4
E)
3ML 2
2
2MR
. Assume the earth is a sphere of uniform density. If the period of its rotation is given by T, then
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its rotational kinetic energy is given by
8. For a solid sphere, I =
A)
2MR 2
5
B)
2MR 2
10
C)
4π MR 2
5
D)
2π MR 2
5T
E)
4π 2 MR 2
5T 2
9. A grinding wheel, used to sharpen tools, is powered by a motor. A knife held against the wheel exerts a torque of 0.80 N m. If
the wheel rotates with a constant angular velocity of 20 rad/s, the work done on the wheel by the motor in 1.0 min is:
B) 480 J
C) 960 J
D) 1400 J
E) 1800 J
A) 0
10. A hoop (I = MR2) has mass of 200 g and a radius of 25 cm. It rolls without slipping along the ground at 500 cm/s. Its total
kinetic energy is:
A) 2.5 J
B) 5.0 J
E) need to know ω
C) 10.0 J
D) 250 J
4 m/s
6 kg
30o
12 m
O
11. A 6-kg particle moves to the right at 4 m/s as shown above. Its angular momentum in kg m2/s2 about point O is:
A) 144
B) 240
C) 260
D) 300
E) 388
12. The direction of the angular momentum about point O in the drawing above is:
A) Into paper
B) Out of paper C) ←
D) ↑
E) ↓
13. An ice skater with rotational inertia I0 is spinning with angular speed ωo. She pulls her arms in, decreasing her rotational inertia
to I0/3. Her angular speed becomes:
A)
ωo
3
B)
ωo
3
C) ωo
D)
3 ωo
E)
3ωo
14. A man, with his arms at his sides, is spinning on a light frictionless turntable. When he extends his arms
A) his angular velocity increases.
D) his rotational kinetic energy increases.
B) his angular velocity remains the same.
E) his rotational inertia decreases.
C) his angular momentum remains the same.
15. A top spinning on the floor precesses because the torque due to gravity, about the point of contact of the top with the floor, is:
A) parallel to the angular momentum vector.
B) parallel to the angular velocity vector.
C) parallel to the axis of rotation.
D) perpendicular to the floor.
E) parallel to the floor, in the direction of the precession.
16. An asteroid moves in an elliptical orbit with the Sun at one focus. Which of these quantities increases as the asteroid moves
from point P in its orbit to point Q?
A) speed
C) total energy
B) angular momentum
D) potential energy
E) kinetic energy
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17. Torque is the rotational analog of
A) mass
C) acceleration
B) linear momentum
D) force
18. Rotational inertia is the rotational analog of
A) mass
C) acceleration
B) linear momentum
D) force
E) kinetic energy
E) kinetic energy
A cylinder rotates with constant angular acceleration about a fixed axis. The cylinder’s moment of inertia about the axis is 4 kg
m2. At time t = 0, the cylinder is at rest. At time t = 2 seconds its angular velocity is 1 radian per second.
19. What is the angular acceleration of the cylinder described above between t = 0 and t = 2 second?
A) 0.5 rad/s2
B) 1 rad/s2
C) 2 rad/s2
D) 4 rad/s2
E) 5 rad/s2
20. For the cylinder described above, what is the angular momentum of the cylinder at time t = 2 seconds?
A) 1 kg m2/s
C) 3 kg m2/s
E) Need cylinder radius
2
B) 2 kg m /s
D) 4 kg m2/s
21. In which direction does the angular momentum point for the front wheel of a bicycle when the bicyclist is traveling due north?
A) North
B) South
C) East
D) West
E) down
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ID: A
C_Rotation_2008 practice exam
Answer Section
MULTIPLE CHOICE
1. ANS: B
The further from the center the mass is, the higher the rotational inertia and the more energy must go into rotation. That leaves
less energy available for translation, and a slower ride down the hill.
PTS: 1
DIF:
KEY: rotational inertia
2. ANS: C
π
2π
ω=
=
60s 30s
moderate
REF: 2006 C
TOP: conservation of energy with rotation
PTS: 1
3. ANS: C
Easy
REF: 2006 C
LOC: angular speed
∫
0
27
DIF:
t
t
0
0
ω = ∫ α dt = −∫ 3t 2 dt
−27 = −
3t 3
3
t = 3.0s
PTS: 1
DIF: Difficult
REF: 2006 C
TOP: Kinematics
KEY: Integral
4. ANS: D
This problem is really pretty easy, but the substitution makes it weird.
ω22 = ω21 + 2α ∆θ
(2π ) 2 = ω21 + 2π (π )
4π 2 = ω21 + 2π 2
ω21 = 2π 2
ω1 = π 2
PTS:
TOP:
5. ANS:
v
ω=
r
1
DIF:
Kinematics
A
0.10m / s
=
= 2.0 / s
0.05m
PTS: 1
DIF:
Moderate to Difficult
REF: 2006 C
Easy
TOP: Angular and Tangential parameters
REF: 2006 C
1
ID: A
6. ANS: D
Both torques have a common sense (causing rotation in a common direction), and both are applied at an angle of 30o relative to
the moment arm.
∑ τ = ∑ rF sin θ = (2m)(5N) sin(30 ) + (4m)(5N) sin(30 )
o
o
5 + 10 = 15Nm
PTS: 1
DIF: Moderate
7. ANS: D
ÊÁ L ˆ˜ 2
5ML 2
I = M ÁÁÁÁ ˜˜˜˜ + ML 2 =
4
Ë2¯
REF: 2006 C
TOP: Torque
PTS: 1
8. ANS: E
REF: 2006 C
TOP: Rotational Inerta of point masses
REF: 2006 C
TOP: Rotational Inertia
DIF:
K = Iω =
1
2
2
1
2
ÊÁ
ˆ
ÁÁ 2MR 2 ˜˜˜ ÊÁÁ 2π
ÁÁ
˜˜ ÁÁ
ÁÁ 5 ˜˜ ÁË T
Ë
¯
PTS: 1
9. ANS: C
DIF:
Easy
ˆ˜ 2 4π 2 MR 2
˜˜ =
˜˜
5T 2
¯
Easy
The work done by the motor must compensate for the negative work by friction; therefore
P=
W
= τ •ω
t
W = (τ • ω)t = (0.80)(20)(60) = 960J
PTS: 1
DIF: Moderate
REF: 3006 C
TOP: Power
KEY: Torque, angular velocity
10. ANS: B
v2 1
1
1
1
K = 2 Iω2 + 2 Mv 2 = 2 MR 2 2 + 2 Mv 2 = MR 2 = (0.200kg)(5.00m / s) 2 = 5.0J
R
PTS: 1
DIF: Easy to moderate
REF: 2006 C
TOP: Rotational Kinetic Energy
KEY: Rolling without slipping
11. ANS: A
l = r × p = rmv sin θ = (12m)(6kg)(4m / s) sin(30) = 144kg ⋅ m / s
PTS:
KEY:
12. ANS:
TOP:
1
DIF: Moderate
cross product, magnitude
A
PTS: 1
Angular momentum
REF: 2006 C
TOP: Angular Momentum
DIF: Easy
REF: 2006 C
KEY: cross product, direction
2
ID: A
13. ANS: E
I 0 ω0 = I 2 ω2
ω2 =
I o ωo I o ωo
=
= 3ωo
I2
I0
3
PTS: 1
14. ANS: C
DIF:
Easy
REF: 2006 C
TOP: Angular momentum conservation
Conservation of angular momentum -- c must be true. (By examination, all others are false)
PTS: 1
DIF:
KEY: rotational inertia
15. ANS: E
Easy
REF: 2006 C
TOP: Angular momentum conservation
The angular momentum vector follows the torque. This is called precession.
PTS: 1
DIF: Difficult
KEY: Angular momentum, torque
16. ANS: D
REF: 2006 C
TOP: Precession
Further from the sun, and gravitational potential energy rises.c
PTS: 1
DIF: Easy
KEY: Energy Conservation
17. ANS: D
REF: 2006 C
TOP: Angular Momentum Conservation
Definitional
PTS: 1
KEY: Force
18. ANS: A
DIF:
Easy
REF: 2006 C
TOP: Torque
PTS: 1
KEY: Force
19. ANS: A
∆ω 1
α=
=
∆t
2
DIF:
Easy
REF: 2006 C
TOP: Torque
PTS: 1
20. ANS: D
L = Iω = (4)(1) = 4
DIF:
Easy
REF: 2006 C
TOP: Angular acceleration
Definitional
PTS: 1
21. ANS: D
PTS: 1
TOP: Angular momentum direction
DIF: Easy
REF: 2006 C
KEY: compass directions
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