Physics 221.
Final Exam
Summer 2003
The following situation refers to the next two problems:
A block of mass m = 3 kg is released from rest from a height h on frictionless incline. It reaches the
bottom of the incline with speed v = 10 m/s. (See figure 1).
m
m
h
h
Figure 1
m
h
Figure 2
Figure 3
101. Find the initial height of the block, h.
a.
b.
c.
d.
e.
h=1m
h=3m
h=5m
h=8m
h = 10 m
102. A disk and a ring, both of mass m (identical to the block) and radius R are released from
rest at height h from similar inclines that do exert friction, so the disk and the ring roll without
slipping (see figures 2 and 3). The three objects are released simultaneously. Rank the objects
from first to last to reach the bottom of the incline.
a.
b.
c.
d.
e.
disk, ring, box
ring, box, disk
ring, disk, box
box, disk, ring
They all reach the bottom at the same time.
Page 1 of 13
Physics 221.
Final Exam
Summer 2003
103. The object shown in the figure below is made of 10 identical, uniform flat squares of side a.
Find the x component of the position of the center of mass of the object. The origin is taken right at
the center of the empty square.
y
a
x
1
a. xCM = − a
2
1
b. xCM = − a
3
c. xCM = −a
2
d. xCM = − a
3
3
e. xCM = − a
2
104. A stone of mass m attached to a string is swirled around in vertical circles of radius R. Find an
expression for the tension on the string at the bottom of the trajectory, when the speed of the stone is
v.
a. T = mg
v2
b. T = m
R
 v2

c. T = m  − g 
R

2
v

d. T = m  + g 
R

 v2

e. T = m 
+ g
 2R

Page 2 of 13
Physics 221.
Final Exam
Summer 2003
The following situation refers to the next two problems:
A stone 1 of mass m1 = 1 kg is thrown from the ground at an angle θ with the horizontal. Two
G
seconds later, the velocity of the stone is v1 = 12 m/s iˆ . Right at this moment, it hits another stone 2
of mass m2 = 2 kg that was falling vertically at speed v2 = 10 m/s —at the moment of the collision
G
(see figure below). Immediately after the collision, the velocity of stone 2 is v2′ = 8 m/s iˆ .
m2
m1
v1
y
Right before
the collision
v2
θ
x
105. Determine the throwing angle θ.
a.
b.
c.
d.
e.
θ = 14°
θ = 25°
θ = 33°
θ = 59°
θ = 68°
106. Determine the velocity of stone 1 immediately after the collision.
G
a. v1′ = 8iˆ − 20 ˆj m/s
b.
c.
d.
e.
(
)
G
v ′ = ( −8iˆ − 20 ˆj ) m/s
G
v ′ = ( −4iˆ − 20 ˆj ) m/s
G
v ′ = ( 4iˆ − 10 ˆj ) m/s
G
v ′ = ( 4iˆ + 10 ˆj ) m/s
1
1
1
1
Page 3 of 13
Physics 221.
Final Exam
Summer 2003
The following situation refers to the next two problems:
A uniform disk of mass M = 20 kg and radius R = 1.2 m is set on a frictionless, horizontal surface
and is free to rotate about its axis. Three forces of identical magnitude F are applied as shown in the
figure below. The disk accelerates at 2 rad/s2.
F
R/2
R
y
F
z
F
x
107. Determine the magnitude of the forces.
a.
b.
c.
d.
e.
F=8N
F = 16 N
F = 24 N
F = 36 N
F = 48 N
108. If the disk was at rest at t = 0, what is the direction of the angular momentum of the disk at
t = 2 s?
iˆ
−iˆ
ĵ
d. kˆ
e. − k̂
.
a.
b.
c.
Page 4 of 13
Physics 221.
Final Exam
Summer 2003
109. A ring can rotate about three different axes 1, 2 and 3. All three axes as perpendicular to the
plane of the ring (which is also the plane of the page) and are indicated with an X on the figure
below. Rank the moments of inertia of the disk for rotations about the three axes.
×
×
×
Axis 1
a.
b.
c.
d.
e.
Axis 2
Axis 3
I1 < I2 < I3
I1 < I3 < I2
I2 < I1 < I3
I2 < I3 < I1
I3 > I1 > I2
110. A beam of mass M = 30 kg and length L = 2 m is hinged to a wall on one end and kept at an
angle θ = 60° with the wall with a perfectly horizontal string attached to the other end. Find the
magnitude of the tension on the string.
θ
a.
b.
c.
d.
e.
T = 255 N
T =390 N
T =420 N
T =510 N
T =595 N
Page 5 of 13
Physics 221.
Final Exam
Summer 2003
The following situation refers to the next two problems:
Three identical blocks are connected with ideal, massless strings and an ideal pulley as shown in the
figure below. Blocks 1 and 2 rest on a horizontal table. The coefficient of kinetic friction between
the blocks and the table is µk = 0.15, except for the last part of the table (near the pulley), where
friction can be neglected. Let T1 and T2 be the magnitudes of the tension on the strings (see figure).
M1
T1
M2
rough part
µk = 0.15
T2
M1 = M2 = M3 = 2.0 kg
smooth part
M3
111. Determine the acceleration of block 3 while blocks 1 and 2 are on the rough part of the table.
a.
b.
c.
d.
e.
a = 1.2 m/s2
a = 2.3 m/s2
a = 6.9 m/s2
a = 9.8 m/s2
a = 10.6 m/s2
112. When block 2 reaches the smooth, frictionless part of the table, T1 __________ and T2
__________.
a.
b.
c.
d.
e.
increases, increases
increases, decreases
decreases, increases
decreases, decreases
stays the same, decreases
Page 6 of 13
Physics 221.
Final Exam
Summer 2003
113. A car and a truck travel both at constant speed along parallel lanes on a straight road. The
figure below shows the snapshots of the motion at times t = 1 s to t = 3 s.
t=1s
t=2s
t=3s
West
East
t=1s
t=2s
t=3s
Consider the velocity of the truck relative to the car. At t = 1 s, this velocity _________________.
At t = 2 s, this velocity __________________.
a.
b.
c.
d.
e.
points West, points West.
points West, is zero.
points East, points West.
points East, points East.
points East, is zero.
114. A block sitting on a horizontal, frictionless table is attached to a spring fixed to a wall as shown
in the figure below. The block is pulled 20 cm to the right and released from rest from there. It
comes back to the same position after 2 s.
What is the speed of the block as it goes through the equilibrium position?
a.
b.
c.
d.
e.
v = 0.10 m/s
v = 0.23 m/s
v = 0.56 m/s
v = 0.63 m/s
v = 0.79 m/s
Page 7 of 13
Physics 221.
Final Exam
Summer 2003
115. A cube with side a is placed with one corner at the origin and its bottom face in the xz plane, as
shown in the figure below. A positive charge Q is placed right on the x-axis, at x > a. Choose the
area vectors pointing in the usual direction for closed surfaces (ie, pointing out). Let ΦB and ΦT be
the electric flux through the bottom and the top surfaces of the cube, respectively. Which of the
following is true?
y
a
Q
x
z
a.
b.
c.
d.
e.
ΦB = 0, ΦT = 0
ΦB = 0, ΦT > 0
ΦB = 0, ΦT < 0
ΦB > 0, ΦT > 0
ΦB < 0, ΦT < 0
The following situation refers to the next three problems:
Two point charges with masses m1 = 1.0 g and m2 = 2.0 g and charges q1 = 1 nC and q2 = 2 nC are
placed on the x-axis, at x = 0 and at x = a (a = 3 cm), respectively. Initially, the charges are kept
fixed in place. Gravity can be neglected.
y
q1 = 1 nC
q2 = 2 nC
a
m1
q1
a
m2
q2
×
P
x
m1 = 1.0 g
m2 = 2.0 g
a = 3 cm
116. Find the net electric potential due to the two charges at point P, located at x = 2a, on the x-axis,
assuming that the potential is zero at infinity.
a.
b.
c.
d.
e.
VP = 350 V
VP = 600 V
VP = 750 V
VP = 900 V
VP = 1100 V
(This problem continues in the next page)
Page 8 of 13
Physics 221.
Final Exam
Summer 2003
(Continued from previous page)
117. Which of the following diagrams best represents the electric field lines in the vicinity of the
two charges (as they are kept fixed)?
118. Both charges are then simultaneously released and allowed to move. The acceleration of q1 is:
G
a. a1 = 0
G
b. a1 = ( 0.02 ) iˆ m/s 2
G
c. a1 = − ( 0.02 ) iˆ m/s 2
G
d. a1 = ( 0.04 ) iˆ m/s 2
G
e. a1 = − ( 0.04 ) iˆ m/s 2
Page 9 of 13
Physics 221.
Final Exam
Summer 2003
The following situation refers to the next three problems:
A metal sphere with charge Q1 = +3 µC and radius R1 = 1 cm is surrounded by a concentric metal
shell of radii R2 = 2 cm and R3 = 3 cm and total charge Q2 = +3 µC.
Q1 = +3 µC
Q2 = +3 µC
R1
R2
Q1
R3
R1 = 1 cm
R2 = 2 cm
R3 = 3 cm
Q2
119. Find the potential of a point on the inner surface of the shell (r = R2), assuming that the
potential is zero at infinity.
a. V ( R2 ) = −1.4 ×106 V
b. V ( R2 ) = −0.9 ×106 V
c. V ( R2 ) = 0
d. V ( R2 ) = +0.9 ×106 V
e. V ( R2 ) = +1.4 ×106 V
120. Find the total charge Qin on the inner surface of the shell.
a.
b.
c.
d.
e.
Qinner = − 6 µC
Qinner = − 3 µC
Qinner = 0
Qinner = + 3 µC
Qinner = + 6 µC
121. The sphere and the shell are then connected with a thin conducting wire, allowing charge to
flow between them. What is the total charge on the shell after the new equilibrium is reached?
a.
b.
c.
d.
e.
Q2, new = − 6 µC
Q2, new = − 3 µC
Q2, new = + 0 µC
Q2, new = + 3 µC
Q2, new = + 6 µC
Page 10 of 13
Physics 221.
Final Exam
Summer 2003
The following situation refers to the next two problems:
A parallel plate capacitor C, a battery with emf ε and a switch are assembled into a circuit as shown
below.
C
C = 4.0 nF
ε = 100 V
ε
122. Find the energy U stored in the capacitor after the switch is closed and the system reaches
equilibrium.
a.
b.
c.
d.
e.
U = 15 µJ
U = 20 µJ
U = 30 µJ
U = 45 µJ
U = 60 µJ
123. After the capacitor is charged, the switch is opened. The plates are then pulled apart so the
distance d between them increases. As a consequence, the charge in the capacitor ____________ and
the capacitance of the system ______________.
a.
b.
c.
d.
e.
increases, increases
increases, decreases
decreases, remains the same
remains the same, increases
remains the same, decreases
Page 11 of 13
Physics 221.
Final Exam
Summer 2003
124. Two infinite plates with uniform charge densities +σ and −σ and an uncharged, inifinite metal
slab are placed parallel to each other as shown in the figure below. Let x1, x2, x3 and x4 indicate the
position of the plates and the sides of the slab.
−σ
+σ
Metal
slab
x=0
x1
x2
x3
x4
Which of the following diagrams best represents the magnitude of the electric field as a function of
position x?
E
E
x
x
x1
x2
x3
x1
x4
x2
x3
x4
B
A
E
E
x
x1
x2
x3
x
x4
x1
x2
C
x3
x4
D
E
x
x1
x2
x3
x4
E
Page 12 of 13
Physics 221.
Final Exam
Summer 2003
The following situation refers to the next three problems:
Four identical bulbs of resistance R are connected with ideal wires to an ideal battery. The switch is
initially open.
2
4
1
ε
3
125. Find the equivalent resistance of the circuit (with the switch open).
R
2
2R
b. Req =
3
R
c. Req =
3
d. Req = 3R
a. Req =
e. Req =
3R
2
126. Compare the magnitude of the potential drop across bulbs 1, 2 and 3 (with the switch open).
a.
b.
c.
d.
e.
V1 < V2 < V3
V1 = V2 = V3
V1 < V2 = V3
V1 > V2 = V3
V1 > V2 > V3
127. What happens to the brightness of bulbs 1 and 2 when the switch is closed?
a.
b.
c.
d.
e.
They both glow dimmer.
Bulb 1 glows dimmer, bulb 2 glows brighter.
Bulb 1 glows dimmer, bulb 2 stays the same.
Bulb 1 stays the same, bulb 2 glows brighter.
They both stay the same.
Page 13 of 13