2017 Mech FR
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(2) ADVANCED PLACEMENT PHYSICS C TABLE OF INFORMATION 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. 6.02 10 23 mol 1. Avogadro’s number, N 0. R. Universal gas constant,. 8.31 J (mol <K). 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,. 3.00 108 m s. c. 6.67 10 11 N<m 2. G. kg 2. 9.8 m s2. g. 1.38 10 23 J K. Boltzmann’s constant, k B. 1 unified atomic mass unit,. 1u. 1.66 10 27 kg. 931 MeV c 2. Planck’s constant,. h. 6.63 10 34 J <s. 4.14 10 15 eV <s. hc. 1.99 10 25 J <m. 1.24 103 eV < nm. e0. 8.85 10 12 C2 N < m 2. Vacuum permittivity,. 1 4 pe0. Coulomb’s law constant, k. m0. Vacuum permeability,. m0 4 p. Magnetic constant, k 1 atmosphere pressure,. UNIT SYMBOLS. e. Electron charge magnitude,. meter, kilogram, second, ampere, kelvin,. PREFIXES Factor Prefix Symbol. m kg s A K. mole, hertz, newton, pascal, joule,. 1 atm mol Hz N Pa J. 9.0 109 N< m 2. C2. 4 p 10 7 (T <m) A 1 10 7 (T< m) A 1.0 105 N m 2. watt, coulomb, volt, ohm, henry,. W C V W H. 1.0 105 Pa farad, tesla, degree Celsius, electron volt,. F T C eV. VALUES OF TRIGONOMETRIC FUNCTIONS FOR COMMON ANGLES D q 0 30D 37D 45D 53D 60D 90D. 109. giga. G. sin q. 0. 12. 35. 2 2. 45. 3 2. 1. 106. mega. M. cos q. 1. 3 2. 45. 2 2. 35. 12. 0. 103. kilo. k. tan q. 0. 3 3. 34. 1. 43. 3. . 10 2. centi. c. 10 3. milli. m. 6. micro. m. 10 9. nano. n. 10 12. pico. p. 10. The following assumptions are used in this exam. I. The frame of reference of any problem is inertial unless otherwise stated. II. The direction of current is the direction in which positive charges would drift. III. The electric potential is zero at an infinite distance from an isolated point charge. IV. All batteries and meters are ideal unless otherwise stated. V. Edge effects for the electric field of a parallel plate capacitor are negligible unless otherwise stated.. -2-.
(3) ADVANCED PLACEMENT PHYSICS C EQUATIONS MECHANICS. Ãx. Ãx 0 a x t. x. x0 Ãx 0 t . Ãx2. G a. G F G J. G p. 2. Ãx 0 2 a x x x0. m G dp dt G Ô F dt. G m FN. G G Ô F <dr. W. K. 1 2 mà 2. dE dt G G F <v. DUg. ac. G t G a. I. mg Dh. Ã2 r. G L. K. w2r. Us. G G r F G G t net Çt I I 2. Ô r dm. Ç mr. acceleration energy force frequency height rotational inertia impulse kinetic energy spring constant length angular momentum mass power momentum radius or distance period time potential energy velocity or speed work done on a system position coefficient of friction angle torque angular speed angular acceleration phase angle G k D x 1 k Dx 2. x. xmax cos( wt f. T. 2p w. Ts Tp. 2p 2p. rw G G r p. G Iw. 1 2 Iw 2. w. w0 at. q. q0 w0 t . 2. G FE. G FG. UG. 1 2 at 2. A B C d E. G. G. area magnetic field capacitance distance electric field e emf F force I current J current density inductance L A length n number of loops of wire per unit length N = number of charge carriers per unit volume P = power Q = charge q = point charge R = resistance r = radius or distance t = time U = potential or stored energy V = electric potential v = velocity or speed r = resistivity F = flux k = dielectric constant G G G FM qv B G G B Ô < d A m0 I. Q e0. vÔ E < dA Ex. DV V. dV dx G G Ô E <dr. . q. 1 4pe0. UE. qV. DV. Q C. Ç rii i. 1 q1q2 4pe0 r. C. k e0 A d. Cp. Ç Ci. 1 Cs. ÇC. I. i. 1 i. i. dQ dt. = = = = = = = = = = = =. v. G m0 I d A r 4p r 2 G G I d Ô AB. 1 f. R. m k. G E. A g. I. Nevd A. FB. I. DV R. e. G G E vÔ < d A. e. L. UC. r2 Gm1m2 r. Rs 1 Rp. P. -3-. Ç Ri i. 1. ÇR i. I DV. i. 1 C DV 2. G dB. 1 QDV 2 rA A G rJ. Gm1m2. . 1 q1q2 4pe0 r 2 G FE q. G E. 2. Ç mi xi Ç mi. x cm. Ã. G Dp. G mv. DE. P. G Fnet m. G. ÇF. G Ff. P. 1 2 at 2 x. a = E= F = f = h = I = J = K = k = A = L = m= P = p = r = T = t = U= v = W= x = m = q = t = w = a = f = G Fs. ELECTRICITY AND MAGNETISM. 2. G F. Bs. UL. m0 nI. Ô. G G B < dA. dI dt. 1 2 LI 2. . d FB dt.
(4) ADVANCED PLACEMENT PHYSICS C EQUATIONS GEOMETRY AND TRIGONOMETRY Rectangle A bh. 1 bh 2. Circle. A. pr2. C. 2p r. s rq Rectangular Solid V wh Cylinder V. pr . S. 2p r 2p r. df dx. A = area C = circumference V = volume S = surface area b = base h = height = length w = width r = radius s = arc length q = angle. Triangle. A. CALCULUS. S. r. d ax e dx. aeax. q 2. @. a cos ax. d >cos ax dx. @. a sin ax. Ôe. n. dx. ax. dx. dx. Ôxa. 4 3 pr 3 4p r. 2. a 2 b2. 1 x. d >sin ax dx. Ôx. Right Triangle. 1 n 1 x , n 1 n 1 1 ax e a ln x a. Ô cos ax dx. 1 sin ax a. Ô sin ax dx. 1 cos ax a. c2. sin q. a c. cos q. b c. tan q. nx n 1. d ln ax dx. Sphere. V. d n x dx. s. 2. d f du du dx. a b. c q. VECTOR PRODUCTS A B AB cos q AB AB sin q. a 90°. b. -4-.
(5) 2017 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 this booklet in the spaces provided after each part.. 1. An Atwood’s machine consists of two blocks connected by a light string that passes over a frictionless pulley of negligible mass, as shown in the figure above. The masses of the two blocks, M1 and M 2 , can be varied. M 2 is always greater than M1 . (a) On the dots below, which represent the blocks, draw and label the forces (not components) that act on the blocks. Each force must be represented by a distinct arrow starting on and pointing away from the appropriate dot. The relative lengths of the arrows should show the relative magnitudes of the forces.. (b) Using the forces in your diagrams above, write an equation applying Newton’s second law to each block and use these two equations to derive the magnitude of the acceleration of the blocks and show that it is given by the equation:. a=. ( M 2 - M1 ) g ( M1 + M 2 ). © 2017 The College Board. Visit the College Board on the Web: www.collegeboard.org.. -5-. GO ON TO THE NEXT PAGE..
(6) 2017 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS The magnitude of the acceleration a was measured for different values of M1 and M 2 , and the data are shown below. M1 (kg) M 2 (kg). a (m/s2). 1.0 2.0 3.02. 2.0 3.0 1.82. 5.0 12.0 4.21. 6.0 8.0 1.15. 10.0 14.0 1.71. (c) Indicate below which quantities should be graphed to yield a straight line whose slope could be used to calculate a numerical value for the acceleration due to gravity g. Vertical axis: Horizontal axis: Use the remaining rows in the table above, as needed, to record any quantities that you indicated that are not given. (d) Plot the data points for the quantities indicated in part (c) on the graph below. Clearly scale and label all axes including units, if appropriate. Draw a straight line that best represents the data.. (e) Using your straight line, determine an experimental value for g.. © 2017 The College Board. Visit the College Board on the Web: www.collegeboard.org.. -6-. GO ON TO THE NEXT PAGE..
(7) 2017 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS. The experiment is now repeated with a modification. The Atwood’s machine is now set up so that the block of mass M1 is on a smooth, horizontal table and the block of mass M 2 is hanging over the side of the table, as shown in the figure above. (f) For the same values of M 1 and M 2 , is the magnitude of the tension in the string when the blocks are moving higher, lower, or equal to the magnitude of the tension in the string when the blocks are moving in the first experiment? ____ Higher. ____ Lower. ____ Equal to. Justify your answer. (g) The value determined for the acceleration due to gravity g is lower than in the first experiment. Give one physical factor that could account for this lower value and explain how this factor affected the experiment.. © 2017 The College Board. Visit the College Board on the Web: www.collegeboard.org.. -7-. GO ON TO THE NEXT PAGE..
(8) 2017 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS. 2. A block of mass m starts at rest at the top of an inclined plane of height h, as shown in the figure above. The block travels down the inclined plane and makes a smooth transition onto a horizontal surface. While traveling on the horizontal surface, the block collides with and attaches to an ideal spring of spring constant k. There is negligible friction between the block and both the inclined plane and the horizontal surface, and the spring has negligible mass. Express all algebraic answers for parts (a), (b), and (c) in terms of m, h, k, and physical constants, as appropriate. (a) i. Derive an expression for the speed of the block just before it collides with the spring.. ii. Is the speed halfway down the incline greater than, less than, or equal to one-half the speed at the bottom of the inclined plane? ____ Greater than. _____ Less than. ____ Equal to. Justify your answer. (b) Derive an expression for the maximum compression of the spring. (c) Determine an expression for the time from when the block collides with the spring to when the spring reaches its maximum compression. The block is again released from rest at the top of the incline, and when it reaches the horizontal surface it is moving with speed v0 . Now suppose the block experiences a resistive force as it slides on the horizontal surface. The magnitude of the resistive force F is given as a function of speed v by F = b v 2 , where b is a positive constant with units of kg m . (d) i. Write, but do NOT solve, a differential equation for the speed of the block on the horizontal surface as a function of time t before it reaches the spring. Express your answer in terms of m, h, k, b , v, and physical constants, as appropriate. ii. Using the differential equation from part (d)i, show that the speed of the block v(t) as a function of bt 1 1 time t can be written in the form = + , where v0 is the speed at t = 0. m v (t ) v0. © 2017 The College Board. Visit the College Board on the Web: www.collegeboard.org.. -8-. GO ON TO THE NEXT PAGE..
(9) 2017 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS (e) Sketch graphs of position x as a function of time t, velocity v as a function of time t, and acceleration a as a function of time t for the block as it is moving on the horizontal surface before it reaches the spring.. © 2017 The College Board. Visit the College Board on the Web: www.collegeboard.org.. -9-. GO ON TO THE NEXT PAGE..
(10) 2017 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS. 3. A uniform solid cylinder of mass M = 0.50 kg and radius R = 0.10 m is released from rest, rolls without slipping down a 1.0 m long inclined plane, and is launched horizontally from a horizontal table of height 0.75 m. The inclined plane makes an angle of 30° with the horizontal. The cylinder lands on the floor a distance D away from the edge of the table, as shown in the figure above. There is a smooth transition from the inclined plane to the horizontal table, and the motion occurs with no frictional energy losses. The rotational inertia of a cylinder around its center is MR 2 2 . (a) Calculate the total kinetic energy of the cylinder as it reaches the horizontal table. (b) Calculate the angular velocity of the cylinder around its axis at the moment it reaches the floor. (c) Calculate the ratio of the rotational kinetic energy to the total kinetic energy for the cylinder at the moment it reaches the floor. (d) Calculate the horizontal distance D. A sphere of the same mass and radius is now rolled down the same inclined plane. The rotational inertia of 2 a sphere around its center is MR 2 . 5 (e) i. Is the total kinetic energy of the sphere at the moment it reaches the floor greater than, less than, or equal to the total kinetic energy of the cylinder at the moment it reaches the floor? ____ Greater than. ____ Less than. ____ Equal to. Justify your answer. ii. Is the rotational kinetic energy of the sphere at the moment it reaches the floor greater than, less than, or equal to the rotational kinetic energy of the cylinder at the moment it reaches the floor? ____ Greater than. ____ Less than. ____ Equal to. Justify your answer. iii. Is the horizontal distance the sphere travels from the table to where it hits the floor greater than, less than, or equal to the horizontal distance the cylinder travels from the table to where it hits the floor? ____ Greater than. ____ Less than. ____ Equal to. Justify your answer.. STOP END OF EXAM © 2017 The College Board. Visit the College Board on the Web: www.collegeboard.org.. -10-.
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