2006 C Mech Fr

2006 C Mech Fr - AP® Physics C Mechanics 2006...

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Unformatted text preview: AP® Physics C: Mechanics 2006 Free-Response Questions The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded in 1900, the association is composed of more than 5,000 schools, colleges, universities, and other educational organizations. Each year, the College Board serves seven million students and their parents, 23,000 high schools, and 3,500 colleges through major programs and services in college admissions, guidance, assessment, financial aid, enrollment, and teaching and learning. Among its best-known programs are the SAT®, the PSAT/NMSQT®, and the Advanced Placement Program® (AP®). The College Board is committed to the principles of excellence and equity, and that commitment is embodied in all of its programs, services, activities, and concerns. © 2006 The College Board. All rights reserved. College Board, AP Central, APCD, Advanced Placement Program, AP, AP Vertical Teams, Pre-AP, SAT, and the acorn logo are registered trademarks of the College Board. Admitted Class Evaluation Service, CollegeEd, connect to college success, MyRoad, SAT Professional Development, SAT Readiness Program, and Setting the Cornerstones are trademarks owned by the College Board. PSAT/NMSQT is a registered trademark of the College Board and National Merit Scholarship Corporation. All other products and services may be trademarks of their respective owners. Permission to use copyrighted College Board materials may be requested online at: www.collegeboard.com/inquiry/cbpermit.html. Visit the College Board on the Web: www.collegeboard.com. AP Central is the official online home for the AP Program: apcentral.collegeboard.com. TABLE OF INFORMATION FOR 2006 and 2007 CONSTANTS AND CONVERSION FACTORS UNITS 1 unified atomic mass unit, 1 u = 1.66 × 10−27 kg = 931 MeV c Proton mass, Name meter kilogram m p = 1.67 × 10−27 kg Electron mass, me = 9.11 × 10−31 kg e = 1.60 × 10−19 C ampere Boltzmann’s constant, Speed of light, Planck’s constant, N 0 = 6.02 × 1023 mol−1 kelvin R = 8.31 J (moli K) k B = 1.38 × 10 −23 c = 3.00 × 10 m/s = 4.14 × 10 eVis −25 i hc = 1.99 × 10 Jm Vacuum permittivity, Vacuum permeability, 0 1 atmosphere pressure, 1 electron volt, G = 6.67 × 10 m nano n pico p VALUES OF TRIGONOMETRIC FUNCTIONS FOR COMMON ANGLES 1 atm = 1.0 × 10 N m 5 = 1.0 × 105 Pa 1 eV = 1.60 × 10−19 J 2 W θ sin θ cos θ coulomb C 0 0 1 0 V 30 1/2 3 /2 3 /3 W henry H 37 3/5 4/5 3/4 F 45 2 /2 2 /2 1 tesla g = 9.8 m s 2 kg is 2 watt farad m 3 J ohm µ0 = 4π × 10−7 (T i m) A Magnetic constant, k ′ = µ0 / 4π = 10−7 (T ⋅ m) A Universal gravitational constant, Acceleration due to gravity at Earth’s surface, Pa 10 m micro -1 2 c milli -6 k centi -3 10 -9 Hz N kilo -2 M volt = 9.0 × 109 Nim 2 C 2 −11 10 mega 3 joule = 1.24 × 103 eV i nm −12 2 C Nim2 0 = 8.85 × 10 Coulomb’s law constant, k = 1 4π 10 mol pascal −15 10 K newton h = 6.63 × 10−34 J is 10 A hertz J/K 9 106 s mole 8 10 kg second Prefix Symbol giga G Factor m mn = 1.67 × 10−27 kg Avogadro’s number, Universal gas constant, Symbol 2 Neutron mass, Electron charge magnitude, PREFIXES T 53 4/5 3/5 4/3 degree Celsius C 60 3 /2 1/2 3 90 1 0 • electronvolt eV tan θ 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 + 12 at 2 u 2 = u0 2 + 2a ( x - x0 )  F = Fnet = ma dp dt F= J = Ú F dt = Dp p = mv F fric £ m N W= ÚF K= 12 mu 2 P= dW dt ∑ dr P=Fv 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= 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 =  t = t net = I a Ú r dm =  mr 2 2 rcm =  mr  m L = r ¥ p = Iw 12 Iw 2 T= 12 kx 2 E= F q ÚE Tp = 2p UG = 12 at 2 r2 Gm1m2 r q  rii 1 4p 0 V= i UE = qV = 1 q1q2 4p 0 r Q V C= k C= 0A d  Ci Cp = i 1 1 = Cs Ci i dQ dt r A V = IR Rs = ˆ r  Ri 1 = Rp 1 ÂR i P = IV FM = qv ¥ B 3 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= ÚB ∑ d = m0 I dB = m0 I d ¥ r 4p r3 ÚI d ¥B Bs = m0 nI fm = Ú B ∑ d A d fm dt e i i n = = = = = = = = = = = = F= I = Neud A g Gm1m2 dV dr E = rJ m k Ts = 2 p F I J L 0 E=- A B C d E e Q dA= ∑ R= 2p 1 = w f FG = - w = w0 + at q = q0 + w0 t + 1 q1q2 4p 0 r 2 1 1 Uc = QV = CV 2 2 2 u = rw K= F= I= 2 t=r¥F I= ELECTRICITY AND MAGNETISM =- e = -L UL = dI dt 12 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 area circumference volume surface area base height length width radius df d f du = dx du dx dn ( x ) = nxn -1 dx dx (e ) = e x dx d (1n x ) = 1 dx x d (sin x ) = cos x dx d (cos x ) = - sin x dx V = pr2 Úx S = 2p r + 2p r 2 Úe Sphere V= 43 pr 3 S = 4p r 2 c Ú a 90° q a c b c tan q = x dx = e x dx = ln x x Ú sin x dx = - cos x a 2 + b2 = c2 cos q = dx = Ú cos x dx = sin x b Right Triangle sin q = 1 x n + 1 , n π -1 n +1 n a b 4 2006 AP® 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 small block of mass M B = 0.50 kg is placed on a long slab of mass M S = 3.0 kg as shown above. Initially, the slab is at rest and the block has a speed u0 of 4.0 m/s to the right. The coefficient of kinetic friction between the block and the slab is 0.20, and there is no friction between the slab and the horizontal surface on which it moves. (a) On the dots below that represent the block and the slab, draw and label vectors to represent the forces acting on each as the block slides on the slab. At some moment later, before the block reaches the right end of the slab, both the block and the slab attain identical speeds u f . (b) Calculate u f . (c) Calculate the distance the slab has traveled at the moment it reaches u f . (d) Calculate the work done by friction on the slab from the beginning of its motion until it reaches u f . © 2006 The College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for students and parents). GO ON TO THE NEXT PAGE. 5 2006 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS Mech 2. A nonlinear spring is compressed various distances x, and the force F required to compress it is measured for each distance. The data are shown in the table below. x (m) F (N) 0.05 4 0.10 17 0.15 38 0.20 68 0.25 106 Assume that the magnitude of the force applied by the spring is of the form F ( x) = Ax 2 . (a) Which quantities should be graphed in order to yield a straight line whose slope could be used to calculate a numerical value for A ? (b) Calculate values for any of the quantities identified in (a) that are not given in the data, and record these values in the table above. Label the top of the column, including units. (c) On the axes below, plot the quantities you indicated in (a) . Label the axes with the variables and appropriate numbers to indicate the scale. (d) Using your graph, calculate A. The spring is then placed horizontally on the floor. One end of the spring is fixed to a wall. A cart of mass 0.50 kg moves on the floor with negligible friction and collides head-on with the free end of the spring, compressing it a maximum distance of 0.10 m. (e) Calculate the work done by the cart in compressing the spring 0.10 m from its equilibrium length. (f) Calculate the speed of the cart just before it strikes the spring. © 2006 The College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for students and parents). GO ON TO THE NEXT PAGE. 6 2006 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS Mech 3. A thin hoop of mass M, radius R, and rotational inertia MR 2 is released from rest from the top of the ramp of length L above. The ramp makes an angle q with respect to a horizontal tabletop to which the ramp is fixed. The table is a height H above the floor. Assume that the hoop rolls without slipping down the ramp and across the table. Express all algebraic answers in terms of given quantities and fundamental constants. (a) Derive an expression for the acceleration of the center of mass of the hoop as it rolls down the ramp. (b) Derive an expression for the speed of the center of mass of the hoop when it reaches the bottom of the ramp. (c) Derive an expression for the horizontal distance from the edge of the table to where the hoop lands on the floor. (d) Suppose that the hoop is now replaced by a disk having the same mass M and radius R. How will the distance from the edge of the table to where the disk lands on the floor compare with the distance determined in part (c) for the hoop? ____ Less than ____ The same as ____ Greater than Briefly justify your response. END OF EXAM © 2006 The College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for students and parents). 7 ...
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