phy1 - AP® Physics C: Mechanics 2010 Free-Response...

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Unformatted text preview: AP® Physics C: Mechanics 2010 Free-Response Questions The College Board 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 College Board is composed of more than 5,700 schools, colleges, universities and other educational organizations. Each year, the College Board serves seven million students and their parents, 23,000 high schools, and 3,800 colleges through major programs and services in college readiness, college admission, guidance, assessment, financial aid and enrollment. Among its widely recognized programs are the SAT®, the PSAT/NMSQT®, the Advanced Placement Program® (AP®), SpringBoard® and ACCUPLACER®. 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. © 2010 The College Board. College Board, ACCUPLACER, Advanced Placement Program, AP, AP Central, SAT, SpringBoard and the acorn logo are registered trademarks of the College Board. Admitted Class Evaluation Service is a trademark 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 2010 and 2011 CONSTANTS AND CONVERSION FACTORS Proton mass, m p = 1.67 ¥ 10 -27 kg Neutron mass, mn = 1.67 ¥ 10 -27 kg Electron mass, me = 9.11 ¥ 10 -31 kg Avogadro’s number, N 0 = 6.02 ¥ 1023 mol-1 Universal gas constant, Electron charge magnitude, e = 1.60 ¥ 10 -19 C 1 electron volt, 1 eV = 1.60 ¥ 10 -19 J Speed of light, Universal gravitational constant, Acceleration due to gravity at Earth’s surface, c = 3.00 ¥ 108 m s G = 6.67 ¥ 10 -11 m 3 kgis2 R = 8.31 J (mol iK) g = 9.8 m s2 Boltzmann’s constant, k B = 1.38 ¥ 10 -23 J K 1 unified atomic mass unit, Planck’s constant, Vacuum permittivity, Coulomb’s law constant, k = 1 4 p Vacuum permeability, 1 u = 1.66 ¥ 10 -27 kg = 931 MeV c 2 h = 6.63 ¥ 10 -34 J is = 4.14 ¥ 10 -15 eV is hc = 1.99 ¥ 10 -25 J im = 1.24 ¥ 103 eV i nm 0 0 = 8.85 ¥ 10 -12 C2 N im 2 = 9.0 ¥ 109 N im 2 C2 m0 = 4 p ¥ 10 -7 (T im) A Magnetic constant, k ¢ = m0 4 p = 1 ¥ 10 -7 (T im) A 1 atmosphere pressure, meter, kilogram, second, ampere, kelvin, m kg s A K mole, hertz, newton, pascal, joule, mol Hz N Pa J 1 atm = 1.0 ¥ 105 N m 2 = 1.0 ¥ 105 Pa watt, coulomb, volt, ohm, henry, W C V W H farad, tesla, degree Celsius, electron-volt, F T ∞C eV UNIT SYMBOLS PREFIXES Factor 10 9 VALUES OF TRIGONOMETRIC FUNCTIONS FOR COMMON ANGLES Symbol G M k c m Prefix giga mega kilo centi milli micro nano pico q sin q cos q tan q 0 30 37 45 53 60 90 0 1 0 12 32 33 35 45 34 22 22 1 45 35 43 32 1 0 106 103 10 -2 10 -3 10 -6 10 -9 10 -1 2 12 3 • m n p The following conventions are used in this exam. 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 2010 and 2011 MECHANICS ELECTRICITY AND MAGNETISM u = u0 + at x = x0 + u0 t + 12 at 2 u 2 = u0 2 + 2 a ( x - x0 )  F = Fnet = ma F= dp dt a F f h I J K k J = Ú F dt = Dp p = mv Ffric £ m N W= K= P= Ú F i dr 1 mu 2 2 dW dt P = Fiv DUg = mgh = = = = = = = = = 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 F= E= 1 q1q2 4p 0 r 2 F q Q 0 A B C d E Ú E i dA = E=V= dV dr 1 4p 0  rii i q = = = = = e= F= I= J= L= = n= N= UE = qV = C= C= Q V k 0A 1 4p 0 q1q2 r d Cp =  Ci i 1 1 = Cs i Ci dQ I= dt Uc = 1 1 QV = CV 2 2 2 P= Q= q= R= r= t= U= V= u= r= 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 fm = magnetic flux k = dielectric constant ac = u2 = w2r r Fs = - kx t=r¥F  t = t net = I a I = Ú r 2 dm =  mr 2 Ú Bid dB = F= = m0 I Us = T= 12 kx 2 R= 2p 1 = w f r A m0 I d ¥ r 4p r3 ¥B E = rJ I = Neud A ÚI d rcm =  mr  m u = rw Ts = 2 p Tp = 2 p FG = - m k g ˆ r Bs = m0 nI fm = Ú B i dA V = IR Rs = L = r ¥ p = Iw K= 12 Iw 2  Ri i e i = Gm1m2 r2 1 = Rp ÂR i 1 Ú Eid dI dt 12 LI 2 =- d fm dt e = -L w = w0 + at q = q0 + w0 t + 12 at 2 UG = - Gm1m2 r P = IV FM = qv ¥ B UL = -3- ADVANCED PLACEMENT PHYSICS C EQUATIONS FOR 2010 and 2011 GEOMETRY AND TRIGONOMETRY Rectangle A = bh Triangle A= C= V= S= b= h= = w= r= area circumference volume surface area base height length width radius CALCULUS df d f du = dx du dx dn ( x ) = nx n -1 dx dx (e ) = e x dx d (ln x ) = 1 dx x d (sin x ) = cos x dx d (cos x ) = - sin x dx A= Circle 1 bh 2 A = pr 2 C = 2pr Parallelepiped V = wh Cylinder V = pr 2 S = 2pr + 2pr 2 Sphere Úx Úe n dx = 1 x n + 1 , n π -1 n +1 x dx = e x V= 43 pr 3 Ú dx = ln x x S = 4pr 2 Right Triangle Ú cos x dx = sin x Ú sin x dx = - cos x c q b 90° a a 2 + b2 = c2 sin q = cos q = tan q = a c b c a b -4- 2010 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. Students are to conduct an experiment to investigate the relationship between the terminal speed of a stack of falling paper coffee filters and its mass. Their procedure involves stacking a number of coffee filters, like the one shown in the figure above, and dropping the stack from rest. The students change the number of filters in the stack to vary the mass m while keeping the shape of the stack the same. As a stack of coffee filters falls, there is an air resistance (drag) force acting on the filters. (a) The students suspect that the drag force FD is proportional to the square of the speed u : FD = Cu 2 , where C is a constant. Using this relationship, derive an expression relating the terminal speed uT to the mass m. The students conduct the experiment and obtain the following data. Mass of the stack of filters, m (kg) Terminal speed, uT ( m s ) (b) 1.12 ¥ 10 -3 0.51 2.04 ¥ 10 -3 0.62 2.96 ¥ 10 -3 0.82 4.18 ¥ 10 -3 0.92 5.10 ¥ 10 -3 1.06 (i) Assuming the functional relationship for the drag force above, use the grid below to plot a linear graph as a function of m to verify the relationship. Use the empty boxes in the data table, as appropriate, to record any calculated values you are graphing. Label the vertical axis as appropriate, and place numbers on both axes. © 2010 The College Board. Visit the College Board on the Web: www.collegeboard.com. GO ON TO THE NEXT PAGE. -5- 2010 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS (ii) Use your graph to calculate C. A particular stack of filters with mass m is dropped from rest and reaches a speed very close to terminal speed by the time it has fallen a vertical distance Y. (c) (i) Sketch an approximate graph of speed versus time from the time the filters are released up to the time t = T that the filters have fallen the distance Y. Indicate time t = T and terminal speed u = uT on the graph. (ii) Suppose you had a graph like the one sketched in (c)(i) that had a numerical scale on each axis. Describe how you could use the graph to approximate the distance Y. (d) Determine an expression for the approximate amount of mechanical energy dissipated, DE , due to air resistance during the time the stack falls a distance y, where y > Y . Express your answer in terms of y , m, uT , and fundamental constants. © 2010 The College Board. Visit the College Board on the Web: www.collegeboard.com. GO ON TO THE NEXT PAGE. -6- 2010 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS Mech. 2. A bowling ball of mass 6.0 kg is released from rest from the top of a slanted roof that is 4.0 m long and angled at 30∞ , as shown above. The ball rolls along the roof without slipping. The rotational inertia of a sphere of 2 mass M and radius R about its center of mass is MR 2 . 5 (a) On the figure below, draw and label the forces (not components) acting on the ball at their points of application as it rolls along the roof. (b) Calculate the force due to friction acting on the ball as it rolls along the roof. If you need to draw anything other than what you have shown in part (a) to assist in your solution, use the space below. Do NOT add anything to the figure in part (a). (c) Calculate the linear speed of the center of mass of the ball when it reaches the bottom edge of the roof. (d) A wagon containing a box is at rest on the ground below the roof so that the ball falls a vertical distance of 3.0 m and lands and sticks in the center of the box. The total mass of the wagon and the box is 12 kg. Calculate the horizontal speed of the wagon immediately after the ball lands in it. © 2010 The College Board. Visit the College Board on the Web: www.collegeboard.com. GO ON TO THE NEXT PAGE. -7- 2010 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS Mech. 3. A skier of mass m will be pulled up a hill by a rope, as shown above. The magnitude of the acceleration of the skier as a function of time t can be modeled by the equations a = amax sin =0 pt T (0 < t < T ) (t ≥ T ), where amax and T are constants. The hill is inclined at an angle q above the horizontal, and friction between the skis and the snow is negligible. Express your answers in terms of given quantities and fundamental constants. (a) Derive an expression for the velocity of the skier as a function of time during the acceleration. Assume the skier starts from rest. (b) Derive an expression for the work done by the net force on the skier from rest until terminal speed is reached. (c) Determine the magnitude of the force exerted by the rope on the skier at terminal speed. (d) Derive an expression for the total impulse imparted to the skier during the acceleration. (e) Suppose that the magnitude of the acceleration is instead modeled as a = amax e - p t 2T for all t > 0 , where amax and T are the same as in the original model. On the axes below, sketch the graphs of the force exerted by the rope on the skier for the two models, from t = 0 to a time t > T . Label the original model F1 and the new model F2 . END OF EXAM © 2010 The College Board. Visit the College Board on the Web: www.collegeboard.com. -8- ...
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This note was uploaded on 01/10/2011 for the course PHYS 100 taught by Professor Razasuleman during the Spring '10 term at University of Engineering & Technology.

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