Physics 221 Spring 2006 Exam 2
PHYSICS 221
Spring 2006
EXAM 2: March 30 2006 8:00pm—10:00pm
Name (printed): ____________________________________________
ID Number: ______________________________________________
Section Number: __________________________________________
INSTRUCTIONS:
Each question is of equal weight, answer all questions. All questions are multiple choice.
Choose the best answer to each question.
Before turning over this page, put away all materials except for pens, pencils, erasers,
rulers, your calculator and “aid sheet”. An “aid sheet” is one two sided 8½×11 page of
notes prepared by the student. There is also a list of possibly useful equations at the end
of the exam.
"In general, any calculator, including calculators that perform graphing numerical
analysis functions, is permitted. Electronic devices that can store large amounts of text,
data or equations are NOT permitted." If you are unsure whether or not your calculator
is allowed for the exam ask your TA.
Examples of allowed calculators: Texas Instruments TI-30XII/83/83+/89, 92+
Casio FX115/250HCS/260/7400G/FX7400GPlus/FX9750 Sharp EL9900C.
Examples of electronic devices that are not permitted: Any laptop, palmtop, pocket
computer, PDA or e-book reader.
In marking the multiple choice bubble sheet use a number 2 pencil. Do NOT use ink. If
you did not bring a pencil, ask for one. Fill in your last name, middle initial, and first
name. Your ID is the middle 9 digits on your ISU card. Special codes K to L are your
recitation section, for the Honors section please encode your section number as follows:
H1⇒02; H2⇒13 and H3⇒25.
If you need to change any entry, you must completely erase your previous entry. Also,
circle your answers on this exam. Before handing in your exam, be sure that your
answers on your bubble sheet are what you intend them to be.
It is strongly suggested that you circle your choices on the question sheet. You
may also copy down your answers on a piece of paper to take with you and compare
with the posted answers. You may use the table at the end of the exam for this.
When you are finished with the exam, place all exam materials, including the bubble
sheet, and the exam itself, in your folder and return the folder to your recitation
instructor. No cell phone calls allowed. Either turn off your cell phone or leave it at home.
Anyone answering a cell phone must hand in their work; their exam is over.
Total number of questions is 28. Question 58 is “extra credit”
Best of luck, David Atwood and Paula Herrera-Siklody
Page 1 of 20
Formula sheet
2 –2006
Phys
2212 – Spring
PhysicsExam
221 Spring
Exam
Vectors and math
G G
G
2
2
2
A
⋅ B = AB cos θ = Ax Bx + Ay By + Az Bz
A = Ax + Ay + Az
G G
G G
A × B = AB sin θ
A × B = ( Ay Bz − Az By ) iˆ + ( Az Bx − Ax Bz ) ˆj + ( Ax By − Ay Bx ) kˆ
ax 2 + bx + c = 0
2006
−b ± b 2 − 4ac
2a
d
d
sin x = cos x
cos x = − sin x
dx
dx
⇒
x=
d n
x = nx n −1
dx
Geometry
perimeter circle: 2π R
area circle: π R 2
z
area sphere: 4π R 2
4
volume sphere: π R 3
3
1 revolution = 2π radians = 360D
Conversion factors (for barbaric units)
k̂
iˆ
ĵ
y
x
1 yard = 3 foot = 36 inches
1 inch = 2.54 cm
1 mile = 1.609 km
1 lb = 4.448 N
1 gallon = 3.788 liters
Physical constants
g = 9.81 m/s 2
G = 6.67 ×10−11 Nm 2 /kg 2
1
N m2
C2
−12
e = 1.60 ×10−19 C
= 8.99 × 109
8.85
10
ε
=
×
0
2
2
4π ε0
C
Nm
General kinematics
G
G
G
G
G
G dr
∆r
∆v
G dv
G
v average =
v=
aaverage =
a=
∆t
dt
∆t
dt
Constant acceleration
v 02
G G
1G2
G G G
G G G
2
2
R
=
sin 2θ
v
−
v
=
2
⋅
∆
a
r
r = r0 + v 0t + at
v = v 0 + at
0
g
2
2
1
v x = v 0 x + axt
v x − v 02x = 2ax ∆x
x = x0 + v 0 x t + ax t 2
2
ke =
dθ
dω
s = Rθ
v = Rω atan = Rα
α=
dt
dt
G
G G
G
dv
v2
2
Circular motion
a = arad + atan
atan =
arad =
= Rω
dt
R
1 2π 2π R
T= =
=
v
f
ω
1
ω = ω0 + α t
Constant α: θ = θ 0 + ω 0t + α t 2
ω 2 − ω 02 = 2α∆θ
2
ω=
Relative motion
G
G
G
G
G
G
rA relative to C = rA relative to B + rB relative to C
v A relative to C = v A relative to B + v B relative to C
G
G
G
aA relative to C = aA relative to B + aB relative to C
Page 2 of 20
10−15
10−12
10−9
10−6
10−3
103
106
109
1012
femto- (f)
pico- (p)
nano- (n)
micro- (ì)
milli- (m)
kilo- (k)
mega- (M)
giga- (G)
tera- (T)
Physics 221 Spring 2006 Exam 2
Forces
G
G
∑ F = ma
G
G
G
Fg (≡ W ) = mg
fs ≤ µs N
fk = µk N
Work and energy
G G
1 2 p2
Wnet = ∆KE
W = ∫ F ⋅ dl
KE = mv =
2
2m
G
G
r G
G
G
U (r ) − U (r0 ) = − ∫G F ⋅ dl
Wconservative = −∆U
1 2
kx + C
2
Pinst =
U = mgy + C
( When only conservative forces do work: ∆E = 0 )
∆E = Wnon-conservative
E = KE + U
Momentum, impulse. Systems of particles.
G
G
G
G
G
J = ∆p = ∫ Fdt
p = mv
G
G
mi ri
miv i
∑
∑
G
G
G
rCM = i
v CM = i
aCM =
∑ mi
∑ mi
i
i
mA − mB
v0x
mA + mB
v Bx =
G
G
G
∆p J
Fave =
=
∆t ∆t
G
∑m a
∑m
i i
KElab = KECM + KErelative to CM
i
G
G
dptotal
G
= mtotal aCM
Fnet =
dt
G
G
ptotal = mtotalv CM
v Ax =
dW G G
= F ⋅v
dt
∂U
( Fx = −
, etc)
∂x
W
∆t
G
G
F = −∇U
Pave =
r0
U=
FHooke = − k ∆x
i
i
G
( When F
net
2mA
v0x
mA + mB
G
G
= 0, ptotal,i = ptotal,f
)
v A,i,x − v B,i,x = −(v A,f,x − v B,f,x )
Rigid-body motion
KEtranslation =
KEtotal = KEtranslation + KErotation
I = ∑ mi ri 2
G
I = I CM + md 2
i
G
τ net
G
dL
=
dt
G
G
KErotation =
G
τ net = Iα
τ = r×F
1
I CMω 2
2
G G G
L=r×p
G
G
L = Iω
G
G
(When τ net = 0, Ltotal,i = Ltotal,f )
2 2
mr
5
2
I hollow sphere = mr 2
3
I solid sphere =
I solid cylinder =
I rod =
L
G
1 2
mv CM
2
1
mL2
12
1 2
mr
2
b
a
Page 3 of 20
I hollow cylinder = mr 2
with thin walls
I rectangle =
1
m ( a 2 + b2 )
12
Physics 221 Spring 2006 Exam 2
Simple harmonic motion
d 2x
+ω2x = 0
2
dt
T=
1 2π
=
f
ω
G
G
Fdamping = −bv
A=
x = A cos(ω t + ϕ )
ω=
k
m
x = A(t ) cos(ω ′t + ϕ )
v = − Aω sin(ωt + ϕ )
ω=
κ
ω=
I
A(t ) = Ae
−
b
t
2m
Fmax
m
 bω 
(ω − ω ) +  d 
 m 
2
0
2
2 2
d
Page 4 of 20
a = − Aω 2 cos(ωt + ϕ )
g
l
ω=
mgd
I
 b 
ω′ = ω − 

 2m 
2
0
2
Physics 221 Spring 2006 Exam 2
Please be sure you mark the answers in spaces 31-58 on the bubble sheet
[31] A 2kg block is attached to a spring with force constant k=100N/m. The block is
initially moving at a speed of 3m/s and the spring is compressed 0.3m from its relaxed
length. What is the total mechanical energy of this system taking the potential energy of
the spring to be 0 when the spring is relaxed?
(A) 4.5J
(B) 9.0J
(C) 13.5J
(D) 18.0J
(E) 22.5J
[32] Particles P, Q, R and S are moving in the +x direction. The momenta and kinetic
energies of these particles are given in the table below. Which is the correct ranking of
the speeds of these particles?
Particle
P
Q
R
S
Kinetic Energy
K0
2K0
K0
2K0
Momentum
p0
p0
2p0
2p0
(A) vR > vP > vS > vQ
(B) vR > vS > vP > vQ
(C) vQ > vP = vS > vR
(D) vQ > vS > vP > vR
(E) None of the above
[33] If a collision between two particles is elastic, which of the following statements is
false?
(A) The total kinetic energy is conserved.
(B) The total momentum is conserved.
(C) The velocity of the center of mass of the system does not change.
(D) The total kinetic energy is reduced by the collision.
(E) The total angular momentum of the system is conserved.
Page 5 of 20
Physics 221 Spring 2006 Exam 2
This graph applies to questions 34, 35 and 36
t=−4s
t=−2s
t=0s
t=2s
t=4s
t=6s
t=8s
t=10s
t=12s
x=2m
x=0m
x=−2m
[34] The graph above is a sketch of the position versus time graph for a mass attached to
a spring undergoing simple harmonic motion. What is the period of this motion?
(A) 6s
(B) 11s
(C) 12s
(D) 24s
(E) 0s
[35] In the graph above, what is the phase angle φ if we write the solution in the form
x(t ) = A cos(ω t + φ ) ?
(A) −
π
2
(B) −
π
3
(C) 0
(D) +
π
3
(E) +
π
2
[36] In the graph above, if the spring constant is k=10N/m, what is the mechanical
energy of the system?
(A) 10J
(B) 20J
(C) 40J
(D) 80J
(E) The mechanical energy changes with time.
Page 6 of 20
Physics 221 Spring 2006 Exam 2
[37] A car with mass 1000kg is driving along a road. It collides with a stationary truck
with mass 2000kg. The two vehicles fuse together and the moment after the collision, the
fused wreckage of the two vehicles is moving at a speed of 10m/s. What was the initial
speed of the car?
(A) 10m/s
(B) 15 m/s
(E) None of the above.
(C) 17 m/s
(D) 25 m/s
[38] A particle with mass m=5kg resting on a frictionless horizontal surface is connected
to an ideal spring with force constant k=100N/m. At t=0s, the mass is at the equilibrium
position and has a kinetic energy of 10J. What is the amplitude of the subsequent
oscillation?
k=100 N/m
5kg
(A) 45cm
(B) 89cm
Kinitial=10J
(C) 63cm
Page 7 of 20
(D) 32cm
(E) 98cm
Physics 221 Spring 2006 Exam 2
[39] A 3m long plank of weight 60N and uniform density rests on two scales which
measure force in Newtons. Scale #1 supports the left hand end of the plank while scale #2
supports a point 1m from the right end of the plank. What are the readings of the two
scales?
3m
1m
2m
Plank weight=60N
Scale 1
Scale 2
(A) Scale 1 reads 45N; Scale 2 reads 15N
(B) Scale 1 reads 40N; Scale 2 reads 20N
(C) Scale 1 reads 30N; Scale 2 reads 30N
(D) Scale 1 reads 20N; Scale 2 reads 40N
(E) Scale 1 reads 15N; Scale 2 reads 45N
[40] A rock of mass 200kg is traveling through space at a speed of 10m/s in the +x
direction. It collides with a stationary rock of mass 400kg. The 200kg rocks bounces off
at an angle of 10º with respect to the x-axis. What is the velocity of the center of mass of
the two rock system after the collision?
(A) (10m/s) iˆ
(B) (3.3m/s) iˆ
(C) (20m/s) iˆ
(E) Cannot be determined from the information given.
Page 8 of 20
(D) (30m/s) iˆ
Physics 221 Spring 2006 Exam 2
[41] Consider the configuration of two thin rods shown
below. Each rod has a mass of 5kg uniformly distributed
over a length of 2m. If it rotates at an angular velocity of
ω=3rad/s about the axis shown, what is the rotational
kinetic energy of the system?
(A) 120J
(D) 40J
(B) 240J
(E) 135J
Axis
2m; 5kg
(C) 27J
G
2m; 5kg
[42] If a particle has momentum p = (−i − ˆj + kˆ)Ns and radius
G
G
vector R = (3i + 2 ˆj + kˆ)m , what is the angular momentum vector L of the particle about
the origin?
G
(A) L = (−3,+4,+1) Js
G
(D) L = (+3,+4,−1) Js
G
(B) L = (+3,−4,−1) Js
G
(E) L = (−3,−6,+1) Js
Page 9 of 20
G
(C) L = (−3,−4,+1) Js
Physics 221 Spring 2006 Exam 2
[43] A physics book (2 kg) is dropped from the top of ISU Campanile (about 94 m high).
What is the magnitude of the average force it exerts on the ground if it is in contact with
the ground for 0.1s before coming to a complete stop? Neglect air resistance.
(A) 86N
(B) 196N
(C) 860N
(D) 980N
(E) 1960N
[44] Sarah throws a 50g snowball at John who is in a tree. Sarah releases the snowball at
an angle of 60º to the horizontal at a speed of 20m/s from a point 1.5m above the ground.
The snowball strikes John who is 8m above the ground. How fast is the snowball
traveling when it strikes John? Neglect air resistance.
(A) 16.5m/s
(B) 15.6m/s
(C) 18.3m/s
(D) 13.1m/s
(E) The snowball cannot reach John given the information in the problem.
[45] A rocket engine operates by ejecting exhaust gasses at a speed of 300m/s. If it
produces exhaust at a rate of 3kg/s, what is the magnitude of force that the rocket engine
generates?
(A) 300N
(B) 600N
(C) 900N
Page 10 of 20
(D) 1350N
(E) 45000N
Physics 221 Spring 2006 Exam 2
This figure applies to questions 46 and 47
y-axis
Axis
4m
1kg
2m
1kg
2m
1kg
2kg
1kg
x-axis
1kg
4m
2m
[46] Consider the system of point masses shown above. What is the x-coordinate of the
center of mass of this system?
(A) 1.7m
(B) 2.0m
(C) 0.0m
(D) 4.0m
(E) 2.3m
[47] Consider the system of point masses shown above. What is the moment of inertia of
this system of masses about the indicated axis that passes through the origin at an angle
of 45° to the x-axis?
(A) 5 kg m²
(B) 10 kg m²
(C) 12 kg m²
Page 11 of 20
(D) 20 kg m²
(E) 24 kg m²
Physics 221 Spring 2006 Exam 2
[48] The angular position of a wheel with moment of inertia I=2kg m² is given as a
function of time by θ = Ae − Bt where A=4 radians and B=3s−1. At t=0s, what is the
magnitude of the angular momentum of the wheel?
(A) 72 Js
(B) 18 Js
(C) 24 Js
(D) 144 Js
(E) 54Js
[49] A 2kg block is sliding to the right along a frictionless horizontal surface. Initially the
kinetic energy of the block is 16J. It encounters an incline that slopes upwards at an angle
of 30º. The coefficient of kinetic friction between the block and the incline is µ k = 0.58 .
What is the maximum distance, d, which the block slides up the ramp before coming to a
stop?
2kg
µ k = 0.58
d=?
K.E.=16J
2kg
30º
Frictionless
(A) 41cm
(B) 82cm
(C) 163cm
Page 12 of 20
(D) 71cm
(E) 52cm
Physics 221 Spring 2006 Exam 2
[50] A small 0.5-kg ball is pressed against a vertical spring of negligible mass as shown
below. The spring is compressed x= 7.0 cm from its relaxed position. When the system is
released from rest, the ball is shot up in the air and reaches a maximum height of h = 1.1
m above its initial position. Determine the force constant k of the spring. Neglect air
resistance.
h
x
(A) 500 N/m
(B) 1100 N/m
(E) The ball can never reach that height.
(C) 1500 N/m
Page 13 of 20
(D) 2200 N/m
Physics 221 Spring 2006 Exam 2
[51] In the figure at right, a horizontal rod of
uniform density with weight 40N and length
2m is attached to a wall with a frictionless
ideal hinge. An ideal massless string connects
the end of the rod to the wall at a point 2m
above the hinge. A box with weight 20N is
hung from the end of the rod with an ideal
massless string. If the system is in
equilibrium, what is the tension, T0, in the
string connecting the end of the rod to the
wall?
2m
T0
40N
2m
Hinge
20N
Wall
(A) 57N
(B) 85N
(C) 40N
(D) 60N
(E) 42N
[52] In the figure at right, a horizontal rod of uniform
density with weight 40N and length 2m is attached to a
wall with a frictionless ideal hinge. If the rod is released at
rest, what is the angular acceleration of the rod about the
hinge just after it is released?
(A) 4.90 rad/s²
(B) 7.35 rad/s²
(C) 8.40 rad/s²
(D) 9.80 rad/s²
(E) 14.70 rad/s²
40N
2m
Hinge
Wall
Page 14 of 20
Physics 221 Spring 2006 Exam 2
[53] A disk of uniform density with mass 2kg and radius 1m can rotate without friction
about a vertical axle. It is initially rotating with a period of 4s. A sticky ball of mass 1kg
(which can be treated as a point mass) is dropped vertically downwards. The ball sticks to
the rim of the disk. After the ball sticks, what is the period of rotation of the disk?
Disk
radius = 1m
mass=2kg
Vertical axle
Ball
Mass=1kg
(edge view)
(A) 2s
(B) 4s
(C) 6s
(D) 8s
(E) 12s
[54] A 1.5 kg rock initially at rest undergoes an explosion into three fragments of mass
0.5kg each. Fragment #1 has a kinetic energy of 25J; Fragment #2 has a kinetic energy of
16J and fragment #3 has a kinetic energy of 100J. What is the angle between the velocity
vectors of fragment #1 and fragment #2?
(A) 49°
(B) 42°
(D) There is not enough information given to determine this angle
(E) The scenario described is not physically possible.
Page 15 of 20
(C) 90°
Physics 221 Spring 2006 Exam 2
[55] Consider the graphs of potential energy of a 4kg particle versus position shown
below. If the particle is released at rest from the position x=6m, what is the maximum
kinetic energy that the particle assumes thereafter?
Potential
(J)
6
4
2
0
2
4
6
8
−2
−4
(A) 0J
(B) 1J
(C) 2J
(D) 3J
Page 16 of 20
(E) 4J
Position
(m)
Physics 221 Spring 2006 Exam 2
[56] In the three systems below, the inclines have the same angle with the horizontal and
the objects are released from the same height h.
1. Block of mass M on a
frictionless incline.
h
θ
2. Solid cylinder of mass M
and radius R rolling down
without slipping.
h
θ
3. Solid cylinder of mass M
and radius 2R rolling down
without slipping.
h
θ
How do their speeds at the bottom of the incline compare?
A.
B.
C.
D.
E.
v1 < v2 = v3
v1 < v2 < v3
v1 > v2 > v3
v1 > v3 > v2
v1 > v2 = v3
Page 17 of 20
Physics 221 Spring 2006 Exam 2
[57] A 250-g particle attached to a spring with k = 150 N/m is also subject to a damping
force F = − bv, where v is the velocity of the particle and b = 1.5 kg/s. Which of the
graphs shown below best represents the kinetic energy of the particle as a function of
time?
Page 18 of 20
Physics 221 Spring 2006 Exam 2
[58] (Note: Extra Credit Problem)
The figure below depicts a position dependent non-conservative force acting on a
particle. To the right of the dashed line the force is constant while to the left it varies as
shown. P, Q, R and S are closed path direction indicated. Note that R is a “figure eight”
path that crosses itself. In which case is the work done by the field on a particle moving
around the closed path not zero.
(A) All of the paths
(B) Paths P and R
(C) Paths Q and S
(D) Paths P and Q
(E) Path P only
Page 19 of 20
Physics 221 Spring 2006 Exam 2
You may record your answers here and take this page with you to compare with the posted answers.
31
41
51
32
42
52
33
43
53
34
44
54
35
45
55
36
46
56
37
47
57
38
48
58
39
49
40
50
Page 20 of 20