PHY 213
Exam 3: Rotational Motion
November 9, 2020
Name ____________________________________________________________
INSTRUCTIONS:
• This examination consists of 6 multiple choice questions (4 points each), two short answer (5 points
each), a rank task/TIPER question (6 points), and 3 workout problems (20 points each).
• You may use a non-graphing calculator for the exam. You may not have your phone out during the
exam.
• You may use g=10 m/s2 to simplify your calculations if you wish.
TO EARN CREDIT YOU HAVE TO SHOW ALL YOUR WORK
Do not write below this line
6 Multiple Choice
(24 pts)
Short Answer
(10 pts)
_________
Ranking Task
(6 pts)
_________
Workout 1
(20 pts)
Workout 2
(20 pts)
Workout 3
(20 pts)
Extra Credit
(5 pts)
TOTAL
(100 pts)
_________
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PHY 213
Exam 3: Rotational Motion
November 9, 2020
2
PHY 213
Exam 3: Rotational Motion
November 9, 2020
Some possibly useful formulas and constants:
!
!"
𝜐()* =
+! $+"
,! $,"
%
x # = nx #$%
g ≈ 10 m/s '
𝑎()* =
-! $-"
,! $,"
∫ x # dx = #&% x #&% + constant
G ≈ 6.67 × 10$%% N ∙ m' /kg '
'
𝜐+' − 𝜐.+
= 2𝑎+ (𝑥 − 𝑥. )
𝐹345 = 𝑚𝑎
𝐹 = −𝑘𝑥
%
𝐾 = ' 𝑚𝜐 '
𝑎/01, =
𝑓6 = 𝜇6 𝑁
<latexit sha1_base64="s6IA3dw6VNYl/wpHGoT80gFLLwQ=">AAACBnicbVBNS8NAEN3Ur1q/oh5FWCyCp5JUQS9CURCPFWwrNKFsNpN26WYTdjeFUnry4l/x4kERr/4Gb/4bt20O2vpg4e17M8zMC1LOlHacb6uwtLyyulZcL21sbm3v2Lt7TZVkkkKDJjyRDwFRwJmAhmaaw0MqgcQBh1bQv574rQFIxRJxr4cp+DHpChYxSrSROvZhC19ijwmNvQFQfIM9Gib5J1Qdu+xUnCnwInFzUkY56h37ywsTmsUgNOVEqbbrpNofEakZ5TAueZmClNA+6ULbUEFiUP5oesYYHxslxFEizTMLTdXfHSMSKzWMA1MZE91T895E/M9rZzq68EdMpJkGQWeDooxjneBJJjhkEqjmQ0MIlczsimmPSEK1Sa5kQnDnT14kzWrFPa1U787Ktas8jiI6QEfoBLnoHNXQLaqjBqLoET2jV/RmPVkv1rv1MSstWHnPPvoD6/MHhqSXQw==</latexit>
Z
)
Wtot = ∆𝐾𝐸
∆𝑝⃗
𝐹⃗345 = V∆𝑡
𝑝⃗ = 𝑚𝜐⃗
∑ B"
$
𝜏345 = 𝐼𝛼
𝜏 = 𝑟C 𝐹 = 𝑟𝐹C = 𝑟𝐹𝑠𝑖𝑛𝜃
𝜔 = 𝜔. + 𝛼𝑡
𝜃 = 𝜃. + 𝜔. 𝑡 + ' 𝛼𝑡 '
∆𝑝⃗ = 𝐹⃗()* ∆𝑡
∑ B " +"
𝐼'#!*+* = , 𝑚𝑟 $
%
𝐹8 .1 : = −𝐹: .1 8
𝑈*2()=,> = 𝑚𝑔ℎ
𝑥?@ =
𝐼%&'( = $ 𝑚𝑟 $
𝑎 = 𝑟𝛼
~
F~ · ds
%
∆𝐸 = 𝑊3?
𝜐 = 𝑟𝜔
2
𝑓7 ≤ 𝜇7 𝑁
𝑈;<2=1* = ' 𝑘𝑥 '
cm moments of inertia: 𝐼!""# = 𝑚𝑟 $
-#
𝑊 = 𝐹+ ∆𝑥 = 𝐹𝑐𝑜𝑠𝜃∆𝑥
W =
𝐸 = 𝐾 + 𝑈*2()=,> + 𝑈;<2=1*
%
𝑥 = 𝑥. + 𝜐.+ 𝑡 + ' 𝑎+ 𝑡 '
𝜐+ = 𝜐.+ + 𝑎+ 𝑡
%
)
𝐼+"% = )$ 𝑚𝐿$
%
'
𝐾 = ' 𝑚𝜐?@
+ ' 𝐼?@ 𝜔'
𝜔' − 𝜔.' = 2𝛼(𝜃 − 𝜃. )
𝜔!"# =
$- %$.
&- %&.
𝐼 = 𝐼?@ + 𝑀ℎ'
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PHY 213
Exam 3: Rotational Motion
November 9, 2020
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PHY 213
Exam 3: Rotational Motion
November 9, 2020
Multiple Choice Questions:
1. A woman stands on the very end of a 50 kg uniform board of length 12 ft, which is supported at 3 ft from
one end and is balanced. What is the mass of the woman?
a)
b)
c)
d)
e)
40 kg
50 kg
60 kg
80 kg
100 kg
2. In both cases the beam has the same mass and length and is attached to the wall by a hinge. In which of the
static cases shown is the tension in the supporting wire bigger?
a) Case 1
b) Case 2
c) Same
3. A disc spins on an axis and is observed to have a constant angular velocity, then it has:
a) a zero net torque
b) a constant net torque
c) an increasing net torque
d) a constant angular acceleration
e) an increasing angular acceleration
4
A wheel which is initially at rest starts to turn with a constant angular acceleration. After 3 seconds it has
made 2 complete revolutions. How many total revolutions has it made after 6 seconds?
a) 2
b) 4
c) 6
d) 8
e) 12
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PHY 213
Exam 3: Rotational Motion
November 9, 2020
5. A wrench has a mass of 0.5 kg, and a moment of inertia around its center-of-mass of ICM=20.0 kg cm2. If it
is rotated about an end which is 10 cm from the center of mass, what is the moment of inertia about this
axis?
a. 5 kg cm2
b. 20 kg cm2
c. 25 kg cm2
d. 50 kg cm2
e. 70 kg cm2
6. On a digital video disc (DVD), video and audio data are stored in a series of tiny pits that are evenly spaced
along a long spiral that spans most of the surface of the disc and extends from the inner edge of the DVD to
the outer edge. The scanning laser in a DVD player reads this information at a constant rate. As the player
reads information recorded closer and closer to the outer edge of the DVD, the disc should
a) rotate faster
b) rotate slower
c) rotate at the same speed
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PHY 213
Exam 3: Rotational Motion
November 9, 2020
Short Answer (5 points each):
1a. When reducing the mass of a racing bike, the greatest benefit is realized from reducing the mass of the tires
and wheel rims. Why does this allow a racer to achieve greater accelerations than would an identical reduction
in the mass of the bicycle’s frame?
1b. Suppose a piece of food is on the edge of a rotating microwave oven plate. Does it experience nonzero
tangential acceleration, centripetal acceleration, or both when: (a) the plate starts to spin faster? (b) The plate
rotates at constant angular velocity? (c) The plate slows to a halt? If the food experiences these accelerations,
make sure to give their direction.
2. Calculate the torques of the five forces acting on the metal bar below. Each force has a magnitude of 10 N
and the bar has a length of 2.5m. Don’t forget to include direction.
________________________
____________________
________________________
____________________
_________________________
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PHY 213
Exam 3: Rotational Motion
November 9, 2020
Ranking Task/TIPER Question (6 pts):
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PHY 213
Exam 3: Rotational Motion
November 9, 2020
Workout Question #1: A solid disk (mass = 5 kg, radius = 20 cm, height = 5 cm) has four 0.5-kg masses attached
around the outside edge as shown. The disk is mounted on an axle at its center and a thin rope wrapped around
the outer edge. You pull on the rope with a constant force. After 10 seconds, the masses are moving with a
tangential speed of 5 m/s.
a) What is the moment of inertia of the system (disk and four masses)?
b) After 5 seconds, what is the angular velocity of the disk?
c) What is the magnitude of the torque you exert?
d) What is the magnitude of the force you are pulling with?
e) What is the change in kinetic energy of the disk during these 10 seconds?
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PHY 213
Exam 3: Rotational Motion
November 9, 2020
Workout Question #2: A solid sphere of uniform density starts from rest and rolls without slipping down an inclined
plane with angle 𝜃 = 20˚. The sphere has mass M = 2.5 kg and radius R = 0.50 m. The coefficients of kinetic and static
friction between the sphere and the plane are 𝜇( = 0.15, 𝜇' = 0.25.
a) Draw a free-body diagram for the sphere.
b) Calculate the magnitude of the acceleration of the center of mass of the
sphere.
c) What is the force of friction on the sphere?
Now, the ball is started from a different condition. It is given a push down the incline so that it has no initial rotational
speed, and is initially slipping.
d) Calculate the magnitude of the acceleration of the center of sphere.
e) Calculate the magnitude of the angular acceleration of the sphere.
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PHY 213
Exam 3: Rotational Motion
November 9, 2020
Workout Question #3: Sir Lancelot rides slowly out of the castle at Camelot and onto the L = 12.0 m long
drawbridge that passes over the moat. Unbeknownst to him, his enemies have secretly partially severed the
drawbridge cable holding up the front end of the drawbridge (which makes an angle of q=25.9° with the
drawbridge as show below) so that it will break under a tension of 1.2x104 N. The drawbridge has a mass of 200
kg. Sir Lancelot, his lance, his armor, and his horse together have a combined mass of 600 kg.
a) Will the cable break before Sir Lancelot reaches the end of the drawbridge? If so, how far from the castle end
of the bridge will the Sir Lancelot be when the cable breaks?
b) What is the force that the hinge exerts on the drawbridge at (depending on your answer to part a)) either the
moment when the drawbridge cable is about to break or when Lancelot reaches the end of the drawbridge
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PHY 213
Exam 3: Rotational Motion
November 9, 2020
Extra Credit [5 points]: One end of a post weighing 400N and with total length L rests on a rough horizontal
surface with s = 0.30. The upper end is held by a rope fastened to the surface and making an angle of 36.98 with
the post. A horizontal force F is exerted on the post as shown at a height h. In this problem we’re interested in
the maximum force F = Fmax that can be applied to the post before it begins to slip.
a) Draw a free-body diagram for the post.
b) Using the conditions for rotational statics, write down equations for SFx, SFy, and St.
c) Find the normal force acting on the post.
d) Solve for the tension in the rope in terms of the force Fmax.
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