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# AP physics C section 2-2007

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```AP&reg; Physics C: Mechanics
2007 Free-Response Questions
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TABLE OF INFORMATION FOR 2006 and 2007
CONSTANTS AND CONVERSION FACTORS
UNITS
1 unified atomic mass unit,
1 u = 1.66 &yen; 10 -27 kg
Name
= 931 MeV c 2
meter
m
kilogram
kg
Proton mass,
m p = 1.67 &yen; 10 -27 kg
Neutron mass,
mn = 1.67 &yen; 10 -27 kg
Electron mass,
me = 9.11 &yen; 10 -31 kg
Electron charge magnitude,
Avogadro’s number,
Universal gas constant,
Boltzmann’s constant,
Speed of light,
Planck’s constant,
10
kelvin
K
10
R = 8.31 J (mol iK)
mole
mol
k B = 1.38 &yen; 10 -23 J /K
hertz
Hz
h = 6.63 &yen; 10
-34
J is
⑀ 0 = 8.85 &yen; 10 -12 C2 N im 2
Coulomb’s law constant,
k = 1 4 p ⑀0 = 9.0 &yen; 109 N im 2 C2
m0 = 4 p &yen; 10 -7 (T im) A
k &cent; = m0 /4 p = 10 -7 (T ◊ m) A
G = 6.67 &yen; 10 -11 m 3 kgis2
g = 9.8 m s
2
1 atm = 1.0 &yen; 105 N m 2
= 1.0 &yen; 105 Pa
1 electron volt,
10
s
N 0 = 6.02 &yen; 1023 mol -1
c = 3.00 &yen; 108 m s
1 eV = 1.60 &yen; 10
-19
J
newton
N
pascal
Pa
joule
J
watt
W
9
106
10
Vacuum permittivity,
1 atmosphere pressure,
10
A
= 1.24 &yen; 103 eV i nm
Universal gravitational constant,
Acceleration due to gravity
at Earth’s surface,
second
Prefix Symbol
giga
G
Factor
ampere
hc = 1.99 &yen; 10 -25 J im
Magnetic constant,
Symbol
e = 1.60 &yen; 10 -19 C
= 4.14 &yen; 10 -15 eVis
Vacuum permeability,
PREFIXES
3
-2
-3
-6
10 -9
10
-12
mega
M
kilo
k
centi
c
milli
m
micro
m
nano
n
pico
p
VALUES OF
TRIGONOMETRIC
FUNCTIONS FOR COMMON
ANGLES
q
sin q
cos q tan q
0
1
0
coulomb
C
0
volt
V
30
1/2
3/2
3/3
ohm
W
henry
H
37
3/5
4/5
3/4
farad
F
45
2/2
2/2
1
tesla
T
53
4/5
3/5
4/3
degree
Celsius
∞C
60
3/2
1/2
3
electronvolt
eV
90
1
0
•
The following conventions are used in this examination.
I. Unless otherwise stated, the frame of reference of any problem is assumed to be inertial.
II. The direction of any electric current is the direction of flow of positive charge (conventional current).
III. For any isolated electric charge, the electric potential is defined as zero at an infinite distance from the charge.
-2-
ADVANCED PLACEMENT PHYSICS C EQUATIONS FOR 2006 and 2007
MECHANICS
u = u0 + at
x = x0 + u0 t +
1 2
at
2
u 2 = u02 + 2a ( x - x0 )
&Acirc; F = Fnet = ma
F=
dp
dt
J = &Uacute; F dt = Dp
p = mv
F fric &pound; m N
W =
&Uacute;F
K =
1 2
mu
2
P=
dW
dt
∑
dr
P = Fⴢv
DUg = mgh
ac =
a
F
f
h
I
J
K
k
=
=
=
=
=
=
=
=
=
L =
m=
N =
P =
p =
r =
r =
T =
t =
U=
u =
W=
x =
m =
q =
t =
w =
a =
ELECTRICITY AND MAGNETISM
acceleration
force
frequency
height
rotational inertia
impulse
kinetic energy
spring constant
length
angular momentum
mass
normal force
power
momentum
radius or distance
position vector
period
time
potential energy
velocity or speed
work done on a system
position
coefficient of friction
angle
torque
angular speed
angular acceleration
u
= w2 r
r
Fs = - k x
Us =
&Acirc; t = t net = I a
I = &Uacute; r dm = &Acirc; mr
2
2
rcm = &Acirc; mr &Acirc; m
E=
F
q
&Uacute;E
T =
1 2
kx
2
UE
L = r &yen; p = Iw
1 2
Iw
2
FG = -
w = w0 + at
UG = 1 2
at
2
r2
Gm1m2
r
1 q1q2
= qV =
4 p⑀0 r
C =
Q
V
C =
k⑀0 A
d
&Acirc; Ci
Cp =
i
1
1
=&Acirc;
Cs
C
i i
dQ
dt
r
A
V = IR
Rs =
rˆ
&Acirc; Ri
1
=
Rp
i
1
&Acirc;R
i
i
P = IV
FM = qv &yen; B
-3-
n
=
=
=
=
=
=
=
=
=
=
=
=
area
magnetic field
capacitance
distance
electric field
emf
force
current
current density
inductance
length
number of loops of wire
per unit length
number of charge carriers
per unit volume
power
charge
point charge
resistance
distance
time
potential or stored energy
electric potential
velocity or speed
resistivity
magnetic flux
dielectric constant
N =
P
Q
q
R
r
t
U
=
=
=
=
=
=
=
V=
u =
r =
fm =
k =
&Uacute;B
∑
d ᐉ = m0 I
dB =
F=
I = Neud A
g
Gm1m2
i
E = rJ
m
k
Tp = 2 p
q
&Acirc; rii
1
4 p⑀0
V =
F
I
J
L
⑀0
dV
dr
E =-
R=
2p
1
=
f
w
Ts = 2 p
dA=
∑
A
B
C
d
E
e
Q
1
1
Uc = QV = CV 2
2
2
u = rw
q = q0 + w0t +
1 q1q2
4 p⑀ 0 r 2
I =
2
t=r&yen;F
K =
F =
m0 I d ᐉ &yen; r
4p r 3
&Uacute; I dᐉ &yen; B
Bs = m0 nI
fm = &Uacute; B ∑ d A
d fm
dt
e
=-
e
= -L
UL =
dI
dt
1 2
LI
2
ADVANCED PLACEMENT PHYSICS C EQUATIONS FOR 2006 and 2007
GEOMETRY AND TRIGONOMETRY
Rectangle
A = bh
Triangle
A=
1
bh
2
Circle
A = pr2
C = 2p r
Parallelepiped
V = wh
Cylinder
A=
C=
V=
S =
b =
h =
=
w=
r =
CALCULUS
df
d f du
=
dx
du dx
area
circumference
volume
surface area
base
height
length
width
radius
d n
( x ) = nxn - 1
dx
d x
(e ) = e x
dx
d
(1n x ) = 1
dx
x
d
(sin x ) = cos x
dx
d
(cos x ) = - sin x
dx
V = pr2
S = 2p r + 2p r 2
Sphere
V =
4 3
pr
3
a
c
tan q =
a
b
&Uacute;e
x
dx = e x
dx
= ln x
x
&Uacute; sin x dx = - cos x
a 2 + b2 = c 2
b
c
dx =
&Uacute; cos x dx = sin x
Right Triangle
cos q =
n
&Uacute;
S = 4p r 2
sin q =
1
x n + 1 , n π -1
n +1
&Uacute;x
c
a
90&deg;
q
b
-4-
2007 AP&reg; PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
PHYSICS C: MECHANICS
SECTION II
Time—45 minutes
3 Questions
Directions: Answer all three questions. The suggested time is about 15 minutes for answering each of the questions,
which are worth 15 points each. The parts within a question may not have equal weight. Show all your work in the
pink booklet in the spaces provided after each part, NOT in this green insert.
Mech. 1.
A block of mass m is pulled along a rough horizontal surface by a constant applied force of magnitude F1 that acts at
an angle q to the horizontal, as indicated above. The acceleration of the block is a1 . Express all algebraic answers
in terms of m, F1 , q , a1 , and fundamental constants.
(a) On the figure below, draw and label a free-body diagram showing all the forces on the block.
(b) Derive an expression for the normal force exerted by the surface on the block.
(c) Derive an expression for the coefficient of kinetic friction m between the block and the surface.
(d) On the axes below, sketch graphs of the speed u and displacement x of the block as functions of time t if the
block started from rest at x = 0 and t = 0.
(e) If the applied force is large enough, the block will lose contact with the surface. Derive an expression for the
magnitude of the greatest acceleration amax that the block can have and still maintain contact with the ground.
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-5-
2007 AP&reg; PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
Mech. 2.
In March 1999 the Mars Global Surveyor (GS) entered its final orbit about Mars, sending data back to Earth.
Assume a circular orbit with a period of 1.18 &yen; 102 minutes = 7.08 &yen; 103 s and orbital speed of 3.40 &yen; 103 m s .
The mass of the GS is 930 kg and the radius of Mars is 3.43 &yen; 106 m .
(a) Calculate the radius of the GS orbit.
(b) Calculate the mass of Mars.
(c) Calculate the total mechanical energy of the GS in this orbit.
(d) If the GS was to be placed in a lower circular orbit (closer to the surface of Mars), would the new orbital period
of the GS be greater than or less than the given period?
_______Greater than
_________Less than
Justify your answer.
(e) In fact, the orbit the GS entered was slightly elliptical with its closest approach to Mars at 3.71 &yen; 105 m
above the surface and its furthest distance at 4.36 &yen; 105 m above the surface. If the speed of the GS at
closest approach is 3.40 &yen; 103 m s , calculate the speed at the furthest point of the orbit.
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GO ON TO THE NEXT PAGE.
-6-
2007 AP&reg; PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
Mech. 3.
The apparatus above is used to study conservation of mechanical energy. A spring of force constant 40 N/m is held
horizontal over a horizontal air track, with one end attached to the air track. A light string is attached to the other
end of the spring and connects it to a glider of mass m . The glider is pulled to stretch the spring an amount x from
equilibrium and then released. Before reaching the photogate, the glider attains its maximum speed and the string
becomes slack. The photogate measures the time t that it takes the small block on top of the glider to pass through.
Information about the distance x and the speed u of the glider as it passes through the photogate are given below.
Trial #
Extension of the Spring
x (m)
Speed of Glider
u (m/s)
Extension Squared
x2 m2
Speed Squared
u 2 m 2 s2
1
0.30 &yen; 10 -1
0.47
0.09 &yen; 10 -2
0.22
2
0.60 &yen; 10 -1
0.87
0.36 &yen; 10 -2
0.76
3
0.90 &yen; 10 -1
1.3
0.81 &yen; 10 -2
1.7
4
1.2 &yen; 10 -1
1.6
1.4 &yen; 10 -2
2.6
5
1.5 &yen; 10 -1
2.2
2.3 &yen; 10 -2
4.8
( )
(
)
(a) Assuming no energy is lost, write the equation for conservation of mechanical energy that would apply to this
situation.
(b) On the grid below, plot u 2 versus x 2 . Label the axes, including units and scale.
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-7-
2007 AP&reg; PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS
(c)
i. Draw a best-fit straight line through the data.
ii. Use the best-fit line to obtain the mass m of the glider.
(d) The track is now tilted at an angle q as shown below. When the spring is unstretched, the center of the glider is
a height h above the photogate. The experiment is repeated with a variety of values of x.
i. Assuming no energy is lost, write the new equation for conservation of mechanical energy that would
apply to this situation.
ii. Will the graph of u 2 versus x 2 for this new experiment be a straight line?
_____ Yes
_____ No
Justify your answer.
END OF EXAM
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