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Unformatted text preview: 1 996 The College Board Advanced Placement Examination
PHYSICS C
SECTION II TABLE OF INFORMATION CONSTANTS AND CONVERSION FACTORS I uniﬁed atomic mass unit. lu = 1.66 x 10'” kg = 931 MeV/c’
Rest mass ofthe proton. = 1.57 x 10‘“ kg
Rest mass of the neutron. = [.67 x 10'” kg
Rest mass ofthe electron. = 9_11 x 10‘“ kg
Magnitude ofthe electron charge, = 1.60 x 10"“c
Avogadm's number. = 6.02 x It'lln moi—1
Universal gas constant, = 8.31 J/ (mol ' K)
Boltzmann’s constant. = 1.33 x 10‘:3 1/]:
Speed of light, = 3.00 x 10' m/s Planck's constant. 6 63 X 10'3“
' VALUES 0F TRIGONOMETRIC FUNCTIONS 4.14 x 10'“ eV  5 FOR COMMON ANGLES 1.99x10'” Jm
1.24): 10’ eVnm Vacuum permittivity, 8.85 x 10‘” (32/ N  m1
Coulomb's law constant. 9.0x m9 N  rill/C2
Vacuum permeability. ' 41m to" Wb/ (A  11:)
Magnetic constant.  to" Wb/(A  m) Universal gravitational constnnt. 5.57 x 10'” “11/ kg  s2 Acceleration due to gravity
at the Earth's surface. 9.8 I'll/52 qgﬁma<n€h 1 annosphere pressure. 1.0): 10’ N/ml 1.0 x 10’ Pa
l electron volt,
1 angstrom. The following conventions are used in this examination.
1. Unless otherwise stated. the frame Vof reference of any problem is assumed to be inertial.
ll. The direction of any electric current is the direction of ﬂow of positive charge (conventional current). [I]. For any isolated electric charge, the electric potential is deﬁned as zero at an inﬁnite distance from the charge. Copyright © 1996 by College Entrance Examination Board and Educational Testing Service.
All rights reserved. 1 996
ADVANCED PLACEMENT PHYSICS c EQUATIONS MECHANICS ELECTRICITY AND MAGNETISM v=vo+at 1 2
s=so+vot+§at = 002 + 2a(s — so) a = acceleration F 2 force f 2 frequency h 2 height I = rotational inertia J = impulse K = kinetic energy I: = spring constant 8 = length L = angular momentum
m = mass N = normal force P = power p = momentum r = distance 5 = displacement T = period t = time U = potential energy v = velocity or speed
W = work x = displacement it = coefficient of friction
9 = angle 1: = torque to = angular speed a = angular acceleration A = area B = magnetic field strengh C = capacitance d = distance E = electric field strength 8 = emf F = force I = current L = inductance E = length n = number of loops of wire
per unit length P = power Q = charge 4 = point charge R = resistance r = distance r = time U = potential or stored energy V = electric potential I) = velocity or speed p = resistivity q; = magnetic flux K = dielectric constant GEOMETRY AND TRIGONONIETRY Rectangle
A = bh
Triangle 1
A—ibh Circle A = ﬁr C = 2m
Parallelepiped V = ewh
Cylinder V = rtrzg S = 21:)! + anz
Sphere 2 V=§Ttr3 S = 4m2
Right Triangle
a2 + b2 = £2 . a
Sine:— c059: tan9= CALCULUS .ﬂ£ixﬂ
m'maw n d .
a (5m x) — cos x d .
21; (cos x) — —51n x {Indxz 1x,“1
' n+1 jexdx=ex
9.x“: In lxl
x lcosxdx=sinx Isinxdx=—cosx A = area C = circumference
V = volume S = surface area b = base [1 = height 8 = length w = width r 2 radius 1996 1996 PHYSICS C
SECTION II, MECHANICS
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. l . »A thin, flexible metal plate attached at one end to a platform, as shown above, can be used to measure mass. When the free end of the plate is pulled down and released, it vibrates in simple harmonic motion
with a period that depends on the mass attached to the plate. To calibrate the force constant, objects of
known mass are attached to the plate and the plate is vibrated, obtaining the data shown below. Average Time
for Ten Vibrations (a) Fill in the blanks in the data table. GO ON TO THE NEXT PAGE 1996 PHYSICS C — MECHANICS (b) 0n the graph below, plot T2 versus mass. Draw on the graph the line that is your estimate of the
best straightline ﬁt to the data points. T262)
4.0  3.0 ———————————————————————————————————————————————————— “"_1___"T___ﬁ




 2.0 ————————————————————————————————————————————————————————— _L____I____..l...___L___J____L____I 1.0 ————— 1+—r—1r*“n*~—r1 —__ﬁ‘—_T__—_V’"T____F___1—_“T"W
__4—__ﬁ+$——L———4————L———J—___L_*MA
__~J____i____Lﬂw_4____L___J___L___J Mass
0.10 0.20 0.30 0.40 0.50 (kg) (c) An object whose mass is not known is vibrated on the plate, and the average time for ten vibra
tions ismeasured to be 16.1 s. From your graph, determine the mass of the object. Write your
answer with a reasonable number of signiﬁcant digits. (d)r Explain how one could determine the force constant of the metal plate. (e) Can this device be used to measure mass aboard the space shuttle Columbia as it orbits the Earth?
Explain brieﬂy. (f) If Columbia is orbiting at 0.3 x 105m above the Earth’s surface, what is the acceleration
of Columbia due to the Earth’s gravity? (Radius of Earth = 6.4 X 106 m, mass of
Earth 2 6.0 x 1024 kg) (g) Since the answer to part (f) is not zero. brieﬂy explain why objects aboard the orbiting Columbia
seem weightless. GO ON TO THE NEXT PAGE 1996 PHYSICS C — MECHANICS Meeh. 2. A 300—kg box rests on a platform attached to a forklift, as shown abOve. Starting from rest at time t = 0,
the box is lowered with a downward acceleration of 1.5 m/sz. (a) Determine the upward force exerted by the horizontal platform on the box as it is lowered. At time t = 0, the forklift also begins to move forward with an acceleration of 2 mls2 while lowering the
box as described above. The box does not slip or tip over. (b) Determine the frictional force on the box. (c) Given that the box does not slip, determine the minimum possible coefficient of friction between
the box and the platform. (d) Determine an equation for the path of the box that expresses y as a function of x (and n_ot of t),
assuming that, at time t = 0, the box has a horizontal position x = 0 and a vertical position
y = 2 m above the ground, with zero velocity. GO ON TO THE NEXT PAGE 1996 PHYSICS C — MECHANICS (c) On the axes below, sketch the path taken by the box. T____l'_—_7_'“”'T"'"" E
G
A
DI
TI
VA
E
N
E
H
T
D
T
N
0
D
G 1996 PHYSICS C — MECHANICS Mech. 3. Consider a thin uniform rod of mass M and length 9, as shown above. (a) Show that the rotational inertia of the rod about an axis through its center and perpendicular to its length
is Ml? 21'12 .  l
 i
 
 i
 i
 l  9 I The rod is now glued to a thin hoop of mass M and radius R = 9/2 to form a rigid assembly, as shown above. The centers of the rod and the hoop coincide at point P. The assembly is mounted on a horizontal
axle through point P and perpendicular to the page. (b) What is the rotational inertia of the rodhoop assembly about the axle?
Several turns of string are wrapped tightly around the circumference of the hoop. The system is at rest when a cat, also of mass M, grabs the free end of the string and hangs vertically from it without swinging as it nnwinds, causing the rodhoop assembly to rotate. Neglect friction and the mass of the
string. (c) Determine the tension T in the string.
(d) Determine the angular acceleration a of the rodhoop assembly.
(e) Determine the linear acceleration of the cat. (f) After descending a distance H = 59/3, the cat lets go of the string. At that instant, what is the
angular momentum of the cat about point P ? STOP END OF SECTION II, MECHANICS 1 996 PHYSICS C
SECTION II, ELECTRICITY AND MAGNETISM
Time—4S 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. E&M1. A solid metal sphere of radius a is charged to a potential V0 > 0 and then isolated from the charging
source. It is then surrounded by joining two uncharged metal hemispherical shells of inner radius [9 and
outer radius 2b, as shown above, without touching the inner sphere or any source of charge. (a) In terms of the given quantities and fundamental constants, determine the initial charge Q0 on
the solid sphere before it was surrounded by the outer shell. (b) Indicate the induced charge on the following after the outer shell is in place.
i. The inner surface of the shell ii. The outer surface of the shell GO ON TO THE NEXT PAGE E&M2. 1996 PHYSICS C — E 8: M (c) Indicate the magnitude of the electric ﬁeld as a function of r and the direction (if any) of the
ﬁeld for the regions indicated below. Write your ansWers on the appropriate lines. i. r < a Magnitude _._.__................._ Direction
ii. a < r < b Magnitude .e—__.____ Direction
iii. b < r < 219 Magnitude ______________ Direction
iv. 2b < r Magnitude _ DirectiOn (d) Does the inner sphere exert a force on the uncharged hemispheres while the shell is being
assembled? Why or why not? (e) Although the charge on the inner solid sphere has not changed, its potential has. In terms of
V0, a, and [1, determine the new potential on the inner sphere. Be sure to show your work. 5 R=ioon _L—'
Lg— S c,=4u1= c2 =12.tF Capacitors 1 and 2, of capacitance C. = 4 uF and C2 = 12 HF, respectively, are connected in a circuit
as shown above with a resistor of resistance R = 100 Q and two switches. Capacitor l is initially charged to a voltage V0 = 50 V, and capacitor 2 is initially uncharged. Both of the switches S are
then closed at time t = 0. (a) What are the ﬁnal charges on the positive plate of each of the capacitors 1 and 2 after
equilibrium has been reached? (b) Determine the difference between the initial and the ﬁnal stored energy of the system after
equilibrium has been reached. (c) Write. but do not solve, an equation that, at any time after the switches are closed, relates the
charge on capacitor C1, its time derivative (which is the instantaneous current in the circuit),
and the parameters V0, R. C,, and C2. The current in the resistor is given as a function of time by l = Ioe‘“", where 10 = 0.5A and
1 = 3 x 1W s. (d) Determine the rate of energy dissipation in the resistor as an expiicit ftinction of time. (e) How much energy is dissipated in the resistor from the instant the switch is closed to when
equilibrium is reached? GO ON TO THE NEXT PAGE E&M3. 1996 PHYSICS C — E 8; M According to Faraggy's law, the induced emf 8 due to a changing magnetic flux ‘i’m is given by
8 = ;E  d9 = “322, where E is the (induced) electric ﬁeld and d9 is a line element along the closed path of integration. A long, ideal solenoid of radius a is shown above. The magnitude of the
spatially uniform magnetic ﬁeld inside this solenoid (due to the current in the solenoid) is increasing
at a steady rate dB/dt. Assume that the magnetic ﬁeld outside the solenoid is zero. (a) For r < a. where r is the distance from the axis of the solenoid, ﬁnd an expression for the
magnitude E of the induced electric ﬁeld in terms of r and dBldt . (b) The ﬁgure below shows a cross section of the solenoid, with the magnetic ﬁeld pointing out of
the page. The electric ﬁeld induced by the increasing magnetic ﬁeld lies in the plane of the page. On the figure, indicate the direction of the induced electric ﬁeld at the three labeled points, Pl,
P2, and P3 . GO ON TO THE NEXT PAGE 1996 PHYSICS C —— E 8: M (c) For r > a, derive an expression for the magnitude E of the induced electric ﬁeld in terms
of r, a, and dBldt. (d) On the axes below, sketch a graph of E versus r for 0 S r 5 3a. STOP END OF SECTION II, ELECTRICITY AND MAGNETISM ...
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