Download Instructions Grading Scheme

Physics 2210
NAME: _____________________________________________
Spring Semester 2017
UID NUMBER: _______________________________________
Midterm 3, Version A
Wednesday, April 5, 2017
Academic Integrity: Giving assistance to or receiving assistance from another student or using unauthorized
materials during a University Examination can be grounds for disciplinary action, up to and including dismissal
from the University.
Honor pledge: On my honor as a University of Utah student, I have neither given nor received unauthorized
assistance on any part of this exam.
______________________________
Name (please print)
______________________________
Signature
____________
Date
Instructions
11. Use a pencil to fill in the circles corresponding to your name and UID on the answer sheet.
12. In the space above, write your name and UID#.
13. Sign and date the honor pledge above.
14. Use a pencil to fill in the circles on the answer sheet that match your answer choices.
15. Each problem has only one correct answer, but you can select more than one answer for partial credit on
questions that have 5 choices (MC5): see the grading scheme below.
16. Circle your answers in the exam booklet ALSO as a backup to the answer sheet.
17. Use the space provided in the exam booklet to work out your problems. If you need more space, we can
provide scratch paper.
18. Bring your completed exam booklet and answer sheet to the front of the classroom along with your UID
card. Your UID card must match your face and the name on your exam to receive a grade.
Grading Scheme
MC5: multiple-choice-five-answer questions, each worth 8 points.
Partial credit will be granted as follows:
(a) If you mark only one answer and it is the correct answer, you earn 8 points.
(b) If you mark two answers, one of which is the correct answer, you earn 4 points.
(c) If you mark no answers, or more than two, you earn 0 points.
MC3: multiple-choice-three-answer questions, each worth 4 points.
No partial credit. Choose only one answer!
Either/Or (T/F): either/or and true/false questions, each worth 2 points.
No partial credit. Choose only one answer!
Either/Or Section: each question worth 2 points.
1. Two solid balls with different masses and different radii are released from rest simultaneously at the top of a ramp.
The balls then roll down the ramp without slipping (neglect air resistance). The two balls reach the bottom of the
ramp at the same time.
a. True
b. False
2. A wheel is spinning clockwise and slowing down. The wheel’s angular velocity and angular acceleration vectors
point in opposite directions.
a. True
b. False
3. A wheel is spinning with constant angular acceleration. The net torque on the wheel is zero.
a. True
b. False
4. Two cars of equal mass are traveling at the same speed. One car crashes into a wall and stops suddenly. The other
car applies its brakes and gradually slows to a stop. The impulse imparted to each car while stopping is the same in
magnitude.
a. True
b. False
5. Consider two parallel rotation axes, only one of which passes through the center of mass of a particular object. The
object’s moment of inertia around the center-of-mass axis is larger than its moment of inertia around the other axis.
a. True
b. False
6. Two balls of equal mass are dropped from the same initial height and bounce off the ground to the same final height.
During the bounce, the first ball is in contact with the ground for twice as long as the second. The average force
exerted by the ground is twice as much for the first ball than the second.
a. True
b. False
7. Newton’s cradle is a toy with five steel balls suspended vertically by strings in a single line. Collisions between the
balls are completely elastic. When one ball is lifted to the side and released two balls swing outward on the other side
after the collision.
a. True
b. False
8. A bomb that is initially stationary will have zero net momentum immediately after it explodes.
a. True
b. False
9. A bus with large mass and a bullet with small mass have the same kinetic energy (not equal to zero). The bus has
more momentum.
a. True
b. False
10. Before a collision, the total kinetic energy of two isolated objects is greater than zero. After the objects collide, the
total kinetic energy cannot be zero.
a. True
b. False
Scenario 1 (20 points):
A bicycle racer propels his bike by cranking the pedals, which in turn exerts a torque on
the rear wheel. The bicycle wheels are 70 cm in diameter and each has a mass of 3 kg.
The total mass of the rider + bike is 70 kg.
11. [MC3: Select only 1 answer] If the bicycle accelerates in the forward direction (to
the left), which one of the statements below best represents the situation?
a. The force of friction from the ground acts to the right on both wheels.
b. The force of friction from the ground acts to the left on the front wheel and to the right on the rear wheel
c. The force of friction from the ground acts to the right on the front wheel and to the left on the rear wheel
12. [MC5: Select up to 2 answers] As the rider accelerates to the left with acceleration !, what is the magnitude of the
friction force exerted by the ground on the front wheel? (Each wheel has radius " and moment of inertia #$%&&' , and
the total weight of the bicycle+rider is equally distributed between the two wheels.)
a. () = 0
,
b. () = .) /010 2
fs R = Iwheel ↵
Iwheel a
=) fs =
R2
-
c. () =
345667 9
d. () =
345667 9
e. () =
345667 9
:;
-: ;
<:
13. [MC5: Select up to 2 answers] The wheels can be considered simple hoops, each with mass 3 kg and radius 35 cm.
The total mass of the rider+bike (including the wheels) is 70 kg. If the bike is rolling without slipping, what fraction
of its total kinetic energy is from the rotation of the wheels?
a.
8%
b. 3%
A010 = A0B9C),EFG& + 2AB10,$%&&'
,
,
M
= /010 J - + 2 K #$%&&' L$%&&'
c. 5%
,
-
-
,
-
= /010 J + #$%&&'
d. 1%
N;
:;
= /010 J + O$%&&' "-
e. It is not possible to answer without knowing the bike’s speed.
-
,
= K /010 + O$%&&' M J
N;
:;
-
-
-PQRS
PSRS
=
T45667 N ;
U
K VSRS WT45667 MN ;
;
= 0.078 ≈ 8%
=
U
;
T45667
K VSRS WT45667 M
Scenario 2 (20 points):
A top spins with angular speed L along its axis with the bottom point of its axle resting in a
shallow frictionless depression that allows it to spin without resistance. When it is spun at a
slight angle from vertical it will precess: the bottom point of contact will remain in place,
but the top of the axle will rotate along a circular path that lies in a horizontal plane. For
clarity, the direction of the angular velocity vector, L, has been shown at a particular
moment in time when the axle is tilted to the right by an angle [ relative to the vertical. The
central axle of the toy has length ℓ, and there is a disk of radius " located a distance ℓ/3
from the bottom. The mass of the disk is much larger than the mass of the axle, so the axle
does not contribute appreciably to the moment of inertia.
L
b⃗
[
ℓ
14. [MC3: Select only 1answer] If you were to observe this situation by looking down from above (aerial view), which
statement below best describes what you would see?
a. The disk spinning counter-clockwise (Q) and the top of the axle precessing clockwise (P).
b. The disk spinning counter-clockwise (Q) and the top of the axle precessing counter-clockwise (Q).
c. The disk spinning clockwise (P) and the top of the axle precessing clockwise (P).
15. [MC5: Select up to 2 answers] Consider the situation where the mass of the disk is / = 0.02 kg, its radius is " =
1.4 cm (0.014 m), the tilt angle is [ = 60°, and the length of the axle is ℓ = 3.6 cm (0.036 m). In this case, what is
the angular speed of the disk needed for the top to precess once per second? (**None of the answers below are
exact. Choose the one that is closest to the exact answer**)
a. L = 20 rad/s
b. L = 60 rad/s
deB&f
c. L = 100 rad/s
d. L = 180 rad/s
e. L = 300 rad/s
ℓ
/2 K M
/2g
2 2ℓ
2 (9.81)(0.036)
rad
3
=
=
⟹L=
=
= 191
#hF)G L 1 /"- L
3 " d 3 (0.014) (2m)
s
2
Note: If you used deB&f =
r6sS
tSRu
, then you would calculate:
ℓ
/2 K M sin(180° − [)
rad
3
L=
= 191 sin(120°) = 165
.
1
s
/"- d
2
Either way, you would choose L = 180 rad/s.
16. [MC5: Select all that apply] What factors does the precession frequency depend upon? (**Select all that apply**)
a. The mass of the disk.
b. The distance from the bottom of the axle to the disk.
c. The radius of the disk.
d. The angular speed of the disk.
e. None of the above.
Scenario 3 (20 points):
When a tennis ball is hit by a racquet, the racquet strings and ball distort, providing a restoring
force much like a spring. Thus the collision is approximately elastic. Regulation tennis balls have
a mass of approximately 0.06 kg, and modern racquets have a mass of approximately 0.6 kg and
are about 70 cm long.
17. [MC3: Select only 1 answer] If the collision is treated as perfectly elastic, and considering
the difference in mass between the ball and racquet (and mass of the person holding the
racquet) which of the following statements best describes the situation?
a. The racquet’s velocity does not change much during impact, but the ball’s velocity changes significantly.
b. The velocity of both the ball and racquet change significantly during impact.
c. The racquet’s velocity changes significantly during impact, but the ball’s velocity does not change much.
18. [MC5: Select up to 2 answers] If the velocity of the ball just before impact is yE,F and its velocity immediately after
impact is yE,z and the ball contacts the racquet for time {|, what is the average force on the ball during the collision?
a. }9~ =
b. }9~ =
c. }9~ =
d. }9~ =
e. }9~ =
TÄ (~Ä,Å W~Ä,Ç )
É0
TÄ (~Ä,Å Ñ~Ä,Ç )
É0
TÄ (~Ä,Å W~Ä,Ç )
-É0
TÄ (~Ä,Å Ñ~Ä,Ç )
-É0
TÄ ~Ä,Å
É0
19. [MC5: Select up to 2 answers] The ball is thrown straight upward by the player during a serve and the racquet hits
the ball at the peak of its trajectory. The ball has a mass of 0.06 kg and the racquet has a mass of 0.6 kg (and is being
held by a player of mass 70 kg). What is the approximate speed of the racquet head just before it hits the ball if the
ball’s speed just after impact is 65 meters/second?
a. JB,F ≈ 70 meters/sec
b. JB,F ≈ 130 meters/sec
c. JB,F ≈ 100 meters/sec
vb,f = Vr,i + vb,i + Vr,f
d. JB,F ≈ 35 meters/sec
= Vr,i + 0 + Vr,i
e. JB,F ≈ 15 meters/sec
mb
vb,f
Mr
✓
◆
mb
vb,f
1+
=
2
Mr
= 2Vr,i
=) Vr,i
Mr Vr,i = Mr Vr,f + mb vb,f
mb
Vr,f = Vr,i
vb,f
Mr
mb
vb,f
Mr
Scenario 4 (20 points):
A T-shaped object is made by combining two identical, uniform
rods of equal mass / and length Ö. The thickness of each rod is
negligible. The object can be rotated around the four different
axes shown by the dashed lines.
~g
The mass of each rod is Ü = á. à kg and each has a length of â = à. ä m.
20. [MC3: Select only 1 answer] Rank in increasing order the moments of inertia, #, to #< :
a. #- < #, < #< < #å
b. #, < #- < #< < #å
c. #, < #- < #å < #<
I1 =
1
M L2
12
I3 =
1
4
M L2 + M L2 = M L2
3
3
I2 =
1
M L2
3
I4 =
1
7
M L2 + M (L/2)2 =
M L2
3
12
21. [MC5: Select up to 2 answers] Consider the situation depicted above in diagram #3. What is the moment of inertia
of the object around the axis shown?
a. #å = 5.1 kg m2
b. #å = 6.1 kg m2
c. #å = 8.1 kg m2
d. #å = 7.4 kg m2
1
I3 = Itop,3 + Ivert,3 = 0 + M D2 + M L2
3
1
4
I3 = M L2 + M L2 = M L2 1
3 L2 + M L2 = 4 M L2
D3 = L =) I3 = M
3
3
4
= (2.1 kg)(1.7 m)2 = 8.1 kg m2
3
e. #å = 4.0 kg m2
22. [MC5: Select up to 2 answers] The T-shape in diagram #2 above is flipped upside down. At “Time 1” it is released
from rest. It then rotates freely around the axis indicated by the dashed line (gravity points downward as indicated).
At “Time 2” it has rotated into the configuration shown. What is the angular speed of the T-shape at “Time 2”?
(**None of the answers below are exact. Choose the one that is closest to the exact answer**)
a. L = 0 rad/s
b. L = 5 rad/s
c. L = 10 rad/s
d. L = 20 rad/s
e. L = 25 rad/s
1 #L
2
1 1
1
/2{é = è /Ö- ê L- = /Ö- L2 3
6
çC&0 = {A =
t
t
<
-
éëV = , so {é =
L=í
ì
-t
=í
ì∙ï.ñ,
-(,.ó)
= 4.2 rad/s
Time 2
Time 1
~g
Scenario 5:
A stick which can rotate around a stationary vertical rod can be used to measure the speed
of a bullet. The stick is mounted onto the rod by drilling a hole near the end of the stick of
the same diameter as the rod so that when the stick rotates, the two surfaces slide against
each other. If the coeffecient of kinetic friction between these two surfaces is assumed to
be constant, then there is a constant dissipative torque acting on the stick while it rotates.
During the collision, the bullet embeds completely into the end of the stick.
23. [MC3: Select only 1 answer] Consider the situation described above. Which of the following statements best
describes the motion?
a. Angular momentum is conserved during the collision. After the collision, the angular speed of the stick+bullet
system is constant for some time and then decreases rapidly to zero.
b. Angular momentum is not conserved during the collision. After the collision, the angular speed of the
stick+bullet system decreases at a constant rate to zero.
c. Angular momentum is conserved during the collision. After the collision, the angular speed of the stick+bullet
system decreases at a constant rate to zero.
24. [MC5: Select up to 2 answers] Find an expression for the angular speed, LF , of the stick+bullet system just after the
collision. The bullet has mass O and its speed before the collision is y. The stick has mass / and length Ö. The total
moment of inertia of the stick+bullet system is #010 .
a. LF =
T~t
b. LF =
T Ä ~Ä t
c. LF =
rQRò
d. LF =
e. LF =
3SRS
å3SRS
t
Vô WTÄ t;
3SRS rQRò
(TWV)~t
,-3SRS
25. [MC5: Select up to 2 answers] Find an expression for the number of rotations of the stick+bullet system after the
collision if its initial angular speed is ωõ and the dissipative torque from the rod on the stick is úB1h :
a. ù =
b. ù =
c. ù =
d. ù =
e. ù =
3SRS ûÇ;
-rQRò
<ü; 3SRS ûÇ;
rQRò
-ü3SRS ûÇ;
rQRò
3SRS ûÇ
-rQRò
3SRS ûÇ;
<ürQRò
Wnet =
1
Itot !i2
2
⌧rod ✓ =
Wnet =
K
K=
!f2
!i2 = 2↵ ✓
✓
◆
⌧rod
=2
(2⇡N )
Itot
Itot !i2
=) N =
4⇡⌧rod
1
Itot !i2
2
Itot !i2
N=
4⇡⌧rod
⌧rod (2⇡N ) =
⌧rod (2⇡N )