2005 Mech FR

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(1)AP® Physics C: Mechanics 2005 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 4,700 schools, colleges, universities, and other educational organizations. Each year, the College Board serves over three and a half 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.. Copyright © 2005 by 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 Entrance Examination 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 Entrance Examination Board. PSAT/NMSQT is a registered trademark of the College Entrance Examination Board and National Merit Scholarship Corporation. Other products and services may be trademarks of their respective owners. Permission to use copyrighted College Board materials may be requested online at: http://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 and Pre-AP: apcentral.collegeboard.com..

(2) TABLE OF INFORMATION FOR 2005 CONSTANTS AND CONVERSION FACTORS 1 unified atomic mass unit,. UNITS. = 1.66 ¥ 10. 1u. -27. = 931 MeV/c Proton mass,. Name. PREFIXES. Symbol. 2. m p = 1.67 × 10 −27 kg. 10 9. giga. G. kilogram. kg. 10. 6. mega. M. 10. 3. kilo. k. centi. c. milli. m. micro. µ. nano. n. pico. p. second. s. Electron mass,. me = 9.11 × 10 −31 kg. ampere. A. Avogadro’s number, Universal gas constant,. e = 1.60 × 10. C. kelvin. N0 = 6.02 × 10 23 mol −1 R. = 8.31 J / ( mol ◊ K ). K. mole. mol. hertz. Hz. Boltzmann’s constant,. k B = 1.38 × 10 −23 J / K. Speed of light,. c = 3.00 × 10 8 m / s. newton. N. Planck’s constant,. h = 6.63 × 10 −34 J ⋅ s. pascal. Pa. = 4.14 × 10 −15 eV ⋅ s. hc = 1.99 × 10. −25. J⋅m. = 1.24 × 10 3 eV ⋅ nm. Vacuum permittivity, Coulomb’s law constant, Vacuum permeability, Magnetic constant, Universal gravitational constant, Acceleration due to gravity at the Earth’s surface, 1 atmosphere pressure,. ⑀ 0 = 8.85 × 10. −12. C / N⋅m 2. 2. k = 1 / 4π ⑀ 0 = 9.0 × 10 9 N ⋅ m 2 / C 2. µ 0 = 4π × 10 −7 (T ⋅ m) / A. k ' = µ 0 / 4π = 10 −7 (T ⋅ m) / A G. = 6.67. ¥. g = 9.8 m / s. 10. -11. 3. m / kg ◊ s. 2. 1 atm = 1.0 × 10 5 N / m 2 = 1.0 × 10 5 Pa. 1 electron volt,. Symbol. m. mn = 1.67 × 10 −27 kg. Magnitude of the electron charge,. Prefix. meter. Neutron mass,. −19. Factor. kg. 1 eV = 1.60 × 10 −19 J. 2. 10 −2 10 10. −3 −6. 10 −9 10. −12. VALUES OF TRIGONOMETRIC FUNCTIONS FOR COMMON ANGLES. joule. J. watt. W. θ. sin θ. cos θ. tan θ. coulomb. C. 0. 0. 1. 0. volt. V. ohm. Ω. 30. 1/2. 3 /2. 3 /3. henry. H. 37. 3/5. 4/5. 3/4. farad. F. tesla. T. 45. 2 /2. 2 /2. 1. degree Celsius. C. 53. 4/5. 3/5. 4/3. electronvolt. eV. 60. 3 /2. 1/2. 3. 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.

(3) ADVANCED PLACEMENT PHYSICS C EQUATIONS FOR 2004 and 2005 MECHANICS. =. u. + at. u0. 1 2 at 2 + 2 a ( x - x0 ). x = x0 + u0 t +. = u0 2 Â F = Fnet = ma. u. 2. dp dt J = Ú F dt = Dp p = mv F=. F fric £ mN W = Ú F dr 1 K = mu 2 2 dW P= dt P = Fⴢv DUg = mgh ∑. ac =. u. 2. =. r. 2. w r. t=r¥F Â t = t net = I a I = Ú r 2 dm = Â mr 2 rcm = Â mr Â m. = rw L = r ¥ p = Iw. u. K = w q. = =. 1 2 Iw 2 w0 q0. = = = = = = = = = 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. ∑. 1 2 at 2. q1 q 2 4 p⑀ 0 r 1. UE = qV = C= C=. Q V k⑀ 0 A. d. ∑i Ci. Cp =. ∑. 1 1 = Cs i Ci dQ dt 1 1 Uc = QV = CV 2 2 2 I=. R=. r. A V = IR Rs =. + at + w0t +. a F f h I J K k. ELECTRICITY AND MAGNETISM 1 q1 q 2 A = area F= 2 B = magnetic field 4 p⑀ 0 r C = capacitance F d = distance E= q E = electric field e = emf Q 养 E • dA = F = force ⑀0 I = current dV L = inductance E=− = length dr n = number of loops of wire qi 1 V= per unit length 4 p⑀ 0 i ri P = power. ∑ Ri. 1 = Rp. i. ∑ R1 i. i. Fs = - k x. P = IV. 1 Us = kx 2 2 2p 1 = T = f w. FM = qv × B 养 B • d 艎 = m0 I. m Ts = 2 p k. Bs = m0 nI. F = z I d艎 × B fm = z B • dA dfm. Tp = 2 p FG = UG. g Gm1m2 2. r Gm m =- 1 2 r. e=−. rˆ. dt dI e = −L dt 1 U L = LI 2 2. 3. Q= q = R = r = t = U= V = u = r = fm = k =. charge point charge resistance distance time potential or stored energy electric potential velocity or speed resistivity magnetic flux dielectric constant.

(4) ADVANCED PLACEMENT PHYSICS C EQUATIONS FOR 2004 and 2005 GEOMETRY AND TRIGONOMETRY Rectangle A = bh Triangle 1 A = bh 2 Circle A = pr 2 C = 2 pr Parallelepiped V = wh Cylinder V = pr 2 S = 2 pr + 2 pr 2 Sphere 4 V = pr 3 3 S = 4 pr 2 Right Triangle a 2 + b2 = c2 a sin q = c cos q =. b c. tan q =. a b. A= C= V= S = b = h = = w= r =. CALCULUS. area circumference volume surface area base height length width radius. c. d f d f du = dx du dx d n n -1 ( x ) = nx dx d x x (e ) = e dx d 1 (1n x) = dx x d (sin x) = cos x dx d (cos x) = - sin x dx 1 n +1 Ú xn dx = x , n π -1 n +1 Ú ex dx = ex dx Ú x = ln x Ú cos xdx = sin x. a 90°. q. Ú sin xdx = - cos x. b. 4.

(5) 2005 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS 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. 1. A ball of mass M is thrown vertically upward with an initial speed of u0 . It experiences a force of air resistance given by F = - k v , where k is a positive constant. The positive direction for all vector quantities is upward. Express all algebraic answers in terms of M, k, u0 , and fundamental constants. (a) Does the magnitude of the acceleration of the ball increase, decrease, or remain the same as the ball moves upward? ____ increases. ____ decreases. ____ remains the same. Justify your answer. (b) Write, but do NOT solve, a differential equation for the instantaneous speed u of the ball in terms of time t as the ball moves upward. (c) Determine the terminal speed of the ball as it moves downward. (d) Does it take longer for the ball to rise to its maximum height or to fall from its maximum height back to the height from which it was thrown? ____longer to rise. ____longer to fall. Justify your answer. (e) On the axes below, sketch a graph of velocity versus time for the upward and downward parts of the ball’s flight, where t f is the time at which the ball returns to the height from which it was thrown.. Copyright © 2005 by College Entrance Examination Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for AP students and parents).. GO ON TO THE NEXT PAGE. 5.

(6) 2005 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS Mech. 2. A student is given the set of orbital data for some of the moons of Saturn shown below and is asked to use the data to determine the mass M S of Saturn. Assume the orbits of these moons are circular. Orbital Period, T (seconds). Orbital Radius, R (meters). 8.14 ¥ 10 4. 1.85 ¥ 108. 1.18 ¥ 10 5. 2.38 ¥ 108. 1.63 ¥ 105. 2.95 ¥ 108. 2.37 ¥ 10 5. 3.77 ¥ 108. (a) Write an algebraic expression for the gravitational force between Saturn and one of its moons. (b) Use your expression from part (a) and the assumption of circular orbits to derive an equation for the orbital period T of a moon as a function of its orbital radius R. (c) Which quantities should be graphed to yield a straight line whose slope could be used to determine Saturn’s mass? (d) Complete the data table by calculating the two quantities to be graphed. Label the top of each column, including units. (e) Plot the graph on the axes below. Label the axes with the variables used and appropriate numbers to indicate the scale.. (f) Using the graph, calculate a value for the mass of Saturn. Copyright © 2005 by College Entrance Examination Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for AP students and parents).. GO ON TO THE NEXT PAGE. 6.

(7) 2005 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS. TOP VIEWS Mech. 3. A system consists of a ball of mass M 2 and a uniform rod of mass M 1 and length d. The rod is attached to a horizontal frictionless table by a pivot at point P and initially rotates at an angular speed w , as shown above 1 left. The rotational inertia of the rod about point P is M 1d 2 . The rod strikes the ball, which is initially at rest. 3 As a result of this collision, the rod is stopped and the ball moves in the direction shown above right. Express all answers in terms of M 1 , M 2 , w , d, and fundamental constants. (a) Derive an expression for the angular momentum of the rod about point P before the collision. (b) Derive an expression for the speed u of the ball after the collision. (c) Assuming that this collision is elastic, calculate the numerical value of the ratio M 1 M 2 .. (d) A new ball with the same mass M 1 as the rod is now placed a distance x from the pivot, as shown above. Again assuming the collision is elastic, for what value of x will the rod stop moving after hitting the ball?. END OF SECTION II, MECHANICS. Copyright © 2005 by College Entrance Examination Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for AP students and parents).. 7.

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