MathDB

2009 F = Ma

Part of F = Ma

Subcontests

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2009 F = Ma Exam Problem 25

Two discs are mounted on thin, lightweight rods oriented through their centers and normal to the discs. These axles are constrained to be vertical at all times, and the discs can pivot frictionlessly on the rods. The discs have identical thickness and are made of the same material, but have differing radii r1r_\text{1} and r2r_\text{2}. The discs are given angular velocities of magnitudes ω1\omega_\text{1} and ω2\omega_\text{2}, respectively, and brought into contact at their edges. After the discs interact via friction it is found that both discs come exactly to a halt. Which of the following must hold? Ignore effects associated with the vertical rods. [asy] //Code by riben, Improved by CalTech_2023 // Solids import solids;
//bigger cylinder draw(shift(0,0,-1)*scale(0.1,0.1,0.59)*unitcylinder,surfacepen=white,black); draw(shift(0,0,-0.1)*unitdisk, surfacepen=black); draw(unitdisk, surfacepen=white,black); draw(scale(0.1,0.1,1)*unitcylinder,surfacepen=white,black);
//smaller cylinder draw(rotate(5,X)*shift(-2,3.2,-1)*scale(0.1,0.1,0.6)*unitcylinder,surfacepen=white,black); draw(rotate(4,X)*scale(0.5,0.5,1)*shift(1,8,0.55)*unitdisk, surfacepen=black); draw(rotate(4,X)*scale(0.5,0.5,1)*shift(1,8,0.6)*unitdisk, surfacepen=white,black); draw(rotate(5,X)*shift(-2,3.2,-0.2)*scale(0.1,0.1,1)*unitcylinder,surfacepen=white,black);

// Lines draw((0,-2)--(1,-2),Arrows(size=5)); draw((4,-2)--(4.7,-2),Arrows(size=5)); // Labels label("r1",(0.5,-2),S); label("r2",(4.35,-2),S); // Curved Lines path A=(-0.694, 0.897)-- (-0.711, 0.890)-- (-0.742, 0.886)-- (-0.764, 0.882)-- (-0.790, 0.873)-- (-0.815, 0.869)-- (-0.849, 0.867)-- (-0.852, 0.851)-- (-0.884, 0.844)-- (-0.895, 0.837)-- (-0.904, 0.824)-- (-0.879, 0.800)-- (-0.841, 0.784)-- (-0.805, 0.772)-- (-0.762, 0.762)-- (-0.720, 0.747)-- (-0.671, 0.737)-- (-0.626, 0.728)-- (-0.591, 0.720)-- (-0.556, 0.715)-- (-0.504, 0.705)-- (-0.464, 0.700)-- (-0.433, 0.688)-- (-0.407, 0.683)-- (-0.371, 0.685)-- (-0.316, 0.673)-- (-0.271, 0.672)-- (-0.234, 0.667)-- (-0.192, 0.664)-- (-0.156, 0.663)-- (-0.114, 0.663)-- (-0.070, 0.660)-- (-0.033, 0.662)-- (0.000, 0.663)-- (0.036, 0.663)-- (0.067, 0.665)-- (0.095, 0.667)-- (0.125, 0.666)-- (0.150, 0.673)-- (0.187, 0.675)-- (0.223, 0.676)-- (0.245, 0.681)-- (0.274, 0.687)-- (0.300, 0.696)-- (0.327, 0.707)-- (0.357, 0.709)-- (0.381, 0.718)-- (0.408, 0.731)-- (0.443, 0.740)-- (0.455, 0.754)-- (0.458, 0.765)-- (0.453, 0.781)-- (0.438, 0.795)-- (0.411, 0.809)-- (0.383, 0.817)-- (0.344, 0.829)-- (0.292, 0.839)-- (0.254, 0.846)-- (0.216, 0.851)-- (0.182, 0.857)-- (0.153, 0.862)-- (0.124, 0.867); draw(shift(0.2,0)*A,EndArrow(size=5)); path B=(2.804, 0.844)-- (2.790, 0.838)-- (2.775, 0.838)-- (2.758, 0.831)-- (2.740, 0.831)-- (2.709, 0.827)-- (2.688, 0.825)-- (2.680, 0.818)-- (2.660, 0.810)-- (2.639, 0.810)-- (2.628, 0.803)-- (2.618, 0.799)-- (2.604, 0.790)-- (2.598, 0.778)-- (2.596, 0.769)-- (2.606, 0.757)-- (2.630, 0.748)-- (2.666, 0.733)-- (2.696, 0.721)-- (2.744, 0.707)-- (2.773, 0.702)-- (2.808, 0.697)-- (2.841, 0.683)-- (2.867, 0.680)-- (2.912, 0.668)-- (2.945, 0.665)-- (2.973, 0.655)-- (3.010, 0.648)-- (3.040, 0.647)-- (3.069, 0.642)-- (3.102, 0.640)-- (3.136, 0.632)-- (3.168, 0.629)-- (3.189, 0.627)-- (3.232, 0.619)-- (3.254, 0.624)-- (3.281, 0.621)-- (3.328, 0.618)-- (3.355, 0.618)-- (3.397, 0.617)-- (3.442, 0.616)-- (3.468, 0.611)-- (3.528, 0.611)-- (3.575, 0.617)-- (3.611, 0.619)-- (3.634, 0.625)-- (3.666, 0.622)-- (3.706, 0.626)-- (3.742, 0.635)-- (3.772, 0.635)-- (3.794, 0.641)-- (3.813, 0.646)-- (3.837, 0.654)-- (3.868, 0.659)-- (3.886, 0.672)-- (3.903, 0.681)-- (3.917, 0.688)-- (3.931, 0.697)-- (3.943, 0.711)-- (3.951, 0.720)-- (3.948, 0.731)-- (3.924, 0.745)-- (3.900, 0.757)-- (3.874, 0.774)-- (3.851, 0.779)-- (3.821, 0.779)-- (3.786, 0.786)-- (3.754, 0.792)-- (3.726, 0.797)-- (3.677, 0.806)-- (3.642, 0.812); draw(shift(0.7,0)*B,EndArrow(size=5)); [/asy]
(A) ω12r1=ω22r2\omega_\text{1}^2r_\text{1}=\omega_\text{2}^2r_\text{2} (B) ω1r1=ω2r2\omega_\text{1}r_\text{1}=\omega_\text{2}r_\text{2} (C) ω1r12=ω2r22\omega_\text{1}r_\text{1}^2=\omega_\text{2}r_\text{2}^2 (D) ω1r13=ω2r23\omega_\text{1}r_\text{1}^3=\omega_\text{2}r_\text{2}^3 (E) ω1r14=ω2r24\omega_\text{1}r_\text{1}^4=\omega_\text{2}r_\text{2}^4
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2009 F = Ma Exam Problem 24

A uniform rectangular wood block of mass MM, with length bb and height aa, rests on an incline as shown. The incline and the wood block have a coefficient of static friction, μs\mu_s. The incline is moved upwards from an angle of zero through an angle θ\theta. At some critical angle the block will either tip over or slip down the plane. Determine the relationship between aa, bb, and μs\mu_s such that the block will tip over (and not slip) at the critical angle. The box is rectangular, and a≠ba \neq b. [asy] draw((-10,0)--(0,0)--20/sqrt(3)*dir(150)); label("θ\theta",(0,0),dir(165)*6); real x = 3; fill((0,0)*dir(60)+(-x*sqrt(3),x)--(3,0)*dir(60)+(-x*sqrt(3),x)--(3,3)*dir(60)+(-x*sqrt(3),x)--(0,3)*dir(60)+(-x*sqrt(3),x)--cycle,grey); draw((0,0)*dir(60)+(-x*sqrt(3),x)--(3,0)*dir(60)+(-x*sqrt(3),x)--(3,3)*dir(60)+(-x*sqrt(3),x)--(0,3)*dir(60)+(-x*sqrt(3),x)--cycle); label("aa",(0,0)*dir(60)+(-x*sqrt(3),x)--(3,0)*dir(60)+(-x*sqrt(3),x)); label("bb",(3,3)*dir(60)+(-x*sqrt(3),x)--(3,0)*dir(60)+(-x*sqrt(3),x),dir(60)); [/asy]
(A) μs>a/b\mu_s > a/b (B) μs>1−a/b\mu_s > 1-a/b (C) μs>b/a\mu_s >b/a (D) μs<a/b\mu_s < a/b (E) μs<b/a−1\mu_s < b/a-1
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2009 F = Ma Exam Problem 18

A simple pendulum of length LL is constructed from a point object of mass mm suspended by a massless string attached to a fixed pivot point. A small peg is placed a distance 2L/32L/3 directly below the fixed pivot point so that the pendulum would swing as shown in the figure below. The mass is displaced 55 degrees from the vertical and released. How long does it take to return to its starting position? [asy] // Code by riben size(275); draw(circle((0,0),1),linewidth(2)); filldraw(circle((0,0),1),gray); draw((0,0)--(0,-70.8)); draw(circle((0,-71.8),3)); filldraw(circle((0,-71.8),3),gray); draw(circle((0,-45),1)); filldraw(circle((0,-45),1),gray); filldraw(circle((15,-70),3),gray,linewidth(0.2)); filldraw(circle((-15,-67),3),gray,linewidth(0.2)); draw((0,0)--(14.5,-66.5),dashed); draw((0,-45)--(-13,-65),dashed); // Labels label("Fixed Pivot Point",(0,0),4*E); label("Small Peg",(0,-45),12*E); label("Point Object of mass m",(0,-70),17*E); draw((-40,1)--(-40,-76.8),EndArrow(size=5)); draw((-40,-76.8)--(-40,1),EndArrow(size=5)); label("L",(-40,-37.9),E*2); [/asy]
(A) πLg(1+23)\pi \sqrt{\frac{L}{g}} \left(1+\sqrt{\frac{2}{3}}\right) (B) πLg(2+23)\pi \sqrt{\frac{L}{g}} \left(2+\frac{2}{\sqrt{3}}\right) (C) πLg(1+13)\pi \sqrt{\frac{L}{g}} \left(1+\frac{1}{3}\right) (D) πLg(1+3)\pi \sqrt{\frac{L}{g}} \left(1+\sqrt{3}\right) (E) πLg(1+13)\pi \sqrt{\frac{L}{g}} \left(1+\frac{1}{\sqrt{3}}\right)
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2009 F = Ma Exam Problem 14

A wooden block (mass MM) is hung from a peg by a massless rope. A speeding bullet (with mass mm and initial speed v0v_\text{0}) collides with the block at time t=0t = \text{0} and embeds in it. Let SS be the system consisting of the block and bullet. Which quantities are conserved between t=−10 st = -\text{10 s} and t=+10 s t = \text{+10 s}? [asy] // Code by riben draw(circle((0,0),0.3),linewidth(2)); filldraw(circle((0,0),0.3),gray); draw((0,-0.8)--(0,-15.5),linewidth(2)); draw((5,-15.5)--(-5,-15.5)--(-5,-20.5)--(5,-20.5)--cycle,linewidth(2)); filldraw((5,-15.5)--(-5,-15.5)--(-5,-20.5)--(5,-20.5)--cycle,gray); draw((-15,-18)--(-16,-17)--(-18,-17)--(-18,-19)--(-16,-19)--cycle,linewidth(2)); filldraw((-15,-18)--(-16,-17)--(-18,-17)--(-18,-19)--(-16,-19)--cycle,gray); [/asy]
(A) The total linear momentum of SS. (B) The horizontal component of the linear momentum of SS. (C) The mechanical energy of SS. (D) The angular momentum of SS as measured about a perpendicular axis through the peg. (E) None of the above are conserved.
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2009 F = Ma Exam Problem 11

A 2.25 kg\text{2.25 kg} mass undergoes an acceleration as shown below. How much work is done on the mass? [asy] // Code by riben size(350); // Axes draw((0,0)--(12,0),lightgray); draw((0,-3)--(0,5)); // Tick Marks draw((2,0)--(2,-0.2)); label("2",(2,-0.2),S*2); draw((4,0)--(4,-0.2)); label("4",(4,-0.2),S*2); draw((6,0)--(6,-0.2)); label("6",(6,-0.2),S*2); draw((8,0)--(8,-0.2)); label("8",(8,-0.2),S*2); draw((10,0)--(10,-0.2)); label("10",(10,-0.2),S*2); draw((12,0)--(12,-0.2)); label("12",(12,-0.2),S*2); draw((0,-2)--(-0.2,-2)); label("-2",(-0.2,-2),W); draw((0,0)--(-0.2,0),lightgray); label("0",(-0.2,0),W); draw((0,2)--(-0.2,2),lightgray); label("2",(-0.2,2),W); draw((0,4)--(-0.2,4)); label("4",(-0.2,4),W); // Dashed Lines draw((0,-2)--(12,-2),dashed); draw((0,2)--(12,2),dashed+lightgray); draw((0,4)--(12,4),dashed); draw((2,5)--(2,0.2),dashed); draw((4,5)--(4,0.2),dashed); draw((6,5)--(6,0.2),dashed); draw((8,5)--(8,0.2),dashed); draw((10,5)--(10,0.2),dashed); draw((12,5)--(12,0.2),dashed); draw((2,-1)--(2,-3),dashed); draw((4,-1)--(4,-3),dashed); draw((6,-1)--(6,-3),dashed); draw((8,-1)--(8,-3),dashed); draw((10,-1)--(10,-3),dashed); draw((12,-1)--(12,-3),dashed); // Path path A=(0,0)--(2,4)--(4,4)--(6,2)--(8,0)--(10,-2)--(12,0); draw(A,linewidth(1.5)); // Labels label(scale(0.85)*rotate(90)*"Acceleration (m/s/s)",(0,1),W*7); label(scale(0.75)*"Position (m)",(11,0),N); [/asy]
(A) 36 J\text{36 J} (B) 22 J\text{22 J} (C) 5 J\text{5 J} (D)-17 J\text{-17 J} (E) -36 J\text{-36 J}
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2009 F = Ma Exam Problem 5

Three equal mass satellites AA, BB, and CC are in coplanar orbits around a planet as shown in the figure. The magnitudes of the angular momenta of the satellites as measured about the planet are LAL_A, LBL_B, and LCL_C. Which of the following statements is correct? [asy] // Code created by riben size(250); dotfactor=12; draw(circle((0,0),1.5),linewidth(2)); draw(circle((0,0),6),dashdotted); draw(circle((0,0),14),dashed); draw(ellipse((4,0),10,8),linewidth(1)); pair A,B,C; A=(-7,12.12); B=(5,7.9); C=(5.7,-1.87); dot(A); dot(B); dot(C); label("A",A,NW*1.5); label("B",B,NW*1.5); label("C",C,E*1.5); filldraw((-1.500, 0.078)-- (-1.428, 0.080)-- (-1.337, 0.094)-- (-1.295, 0.157)-- (-1.246, 0.209)-- (-1.186, 0.227)-- (-1.143, 0.290)-- (-1.148, 0.357)-- (-1.135, 0.469)-- (-1.057, 0.505)-- (-0.996, 0.563)-- (-0.936, 0.526)-- (-0.852, 0.557)-- (-0.773, 0.587)-- (-0.772, 0.716)-- (-0.765, 0.828)-- (-0.781, 0.955)-- (-0.732, 1.035)-- (-0.648, 1.083)-- (-0.605, 1.162)-- (-0.604, 1.246)-- (-0.645, 1.295)-- (-0.736, 1.270)-- (-0.796, 1.229)-- (-0.851, 1.193)-- (-0.941, 1.135)-- (-1.014, 1.076)-- (-1.105, 0.995)-- (-1.154, 0.921)-- (-1.227, 0.841)-- (-1.288, 0.760)-- (-1.349, 0.669)-- (-1.398, 0.556)-- (-1.453, 0.465)-- (-1.485, 0.357)-- (-1.510, 0.239)--cycle,gray); filldraw((-0.119, 1.245)-- (-0.130, 1.193)-- (-0.146, 1.095)-- (-0.202, 1.056)-- (-0.327, 1.033)-- (-0.262, 1.031)-- (-0.278, 0.979)-- (-0.193, 0.949)-- (-0.108, 0.943)-- (-0.013, 0.941)-- (0.032, 0.915)-- (0.026, 0.840)-- (0.015, 0.779)-- (0.019, 0.705)-- (0.074, 0.646)-- (0.113, 0.582)-- (0.162, 0.533)-- (0.167, 0.463)-- (0.241, 0.400)-- (0.311, 0.412)-- (0.416, 0.410)-- (0.465, 0.342)-- (0.541, 0.410)-- (0.611, 0.347)-- (0.679, 0.242)-- (0.728, 0.132)-- (0.732, 0.048)-- (0.671, -0.037)-- (0.615, -0.104)-- (0.540, -0.172)-- (0.409, -0.209)-- (0.324, -0.244)-- (0.253, -0.293)-- (0.188, -0.314)-- (0.162, -0.389)-- (0.181, -0.486)-- (0.270, -0.534)-- (0.340, -0.537)-- (0.380, -0.596)-- (0.424, -0.688)-- (0.418, -0.772)-- (0.352, -0.825)-- (0.281, -0.883)-- (0.241, -0.926)-- (0.145, -0.981)-- (0.044, -1.044)-- (-0.006, -1.107)-- (-0.007, -1.190)-- (0.077, -1.216)-- (0.162, -1.213)-- (0.253, -1.163)-- (0.323, -1.128)-- (0.404, -1.075)-- (0.510, -1.015)-- (0.605, -0.980)-- (0.671, -0.931)-- (0.731, -0.920)-- (0.817, -0.852)-- (0.898, -0.798)-- (0.963, -0.777)-- (0.964, -0.708)-- (1.024, -0.645)-- (1.025, -0.571)-- (0.976, -0.488)-- (0.912, -0.425)-- (0.878, -0.347)-- (0.823, -0.289)-- (0.779, -0.225)-- (0.744, -0.193)-- (0.756, -0.100)-- (0.816, -0.033)-- (0.837, 0.047)-- (0.838, 0.122)-- (0.824, 0.200)-- (0.800, 0.307)-- (0.796, 0.381)-- (0.872, 0.416)-- (0.967, 0.414)-- (1.016, 0.360)-- (1.096, 0.381)-- (1.117, 0.428)-- (1.058, 0.506)-- (0.998, 0.564)-- (0.954, 0.591)-- (0.914, 0.617)-- (0.860, 0.676)-- (0.800, 0.716)-- (0.751, 0.775)-- (0.757, 0.859)-- (0.797, 0.921)-- (0.823, 0.987)-- (0.889, 1.096)-- (0.850, 1.160)-- (0.780, 1.176)-- (0.700, 1.183)-- (0.645, 1.125)-- (0.579, 1.039)-- (0.518, 0.986)-- (0.438, 0.956)-- (0.343, 0.967)-- (0.289, 1.049)-- (0.249, 1.117)-- (0.195, 1.176)-- (0.125, 1.192)-- (0.030, 1.208)-- (-0.040, 1.220)--cycle,gray); [/asy]
(A) LA>LB>LCL_\text{A} > L_\text{B} > L_\text{C} (B) LC>LB>LAL_\text{C} > L_\text{B} > L_\text{A} (C) LB>LC>LAL_\text{B} > L_\text{C} > L_\text{A} (D) LB>LA>LCL_\text{B} > L_\text{A} > L_\text{C} (E) The relationship between the magnitudes is different at various instants in time.
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2009 F = Ma Exam Information (2 and 3)

The following information is used for questions 2 and 3. Three blocks of identical mass are placed on a frictionless table as shown. The center block is at rest, whereas the other two blocks are moving directly towards it at identical speeds vv. The center block is initially closer to the left block than the right one. All motion takes place along a single horizontal line. [asy] // Code by riben import roundedpath; // Platform filldraw((-75,0)--(75,0)--(75,4)--(-75,4)--cycle,gray); // Blocks path A=(-65,6)--(-52,6)--(-52,13)--(-65,13)--cycle; path B=(65,6)--(52,6)--(52,13)--(65,13)--cycle; path C=(-20,6)--(-7,6)--(-7,13)--(-20,13)--cycle; draw(roundedpath(A,3),linewidth(2)); filldraw(roundedpath(A,3),gray); draw(roundedpath(B,3),linewidth(2)); filldraw(roundedpath(B,3),gray); draw(roundedpath(C,3),linewidth(2)); filldraw(roundedpath(C,3),gray); // Arrows draw((-51,9.5)--(-41,9.5),EndArrow(size=7)); draw((51,9.5)--(41,9.5),EndArrow(size=7)); [/asy]

2009 F = Ma Exam Information (8 and 9)

The following information is used for questions 8 and 9. A flat disk rotates about an axis perpendicular to the plane of the disk and through the center of the disk with an angular velocity as shown in the graph below. [asy] size(400); // x-axis draw((0,0)--(12,0)); // x-axis tick marks draw((4,0)--(4,-0.2)); draw((8,0)--(8,-0.2)); draw((12,0)--(12,-0.2)); //y-axis draw((0,-3)--(0,5)); //y-axis tick marks draw((-0.2,-2)--(0,-2)); draw((-0.2,0)--(0,0)); draw((-0.2,2)--(0,2)); draw((-0.2,4)--(0,4)); // y-axis labels label("-2",(0,-2),2*W); label("0",(0,0),2*W); label("2",(0,2),2*W); label("4",(0,4),2*W); // Axis Labels label("Time (s)",(10,0),N); label(rotate(90)*"Angular Velocity (rad/s)",(0,1),6*W); //Dashes draw((0,4)--(12,4),dashed); draw((0,2)--(12,2),dashed); draw((0,-2)--(12,-2),dashed); draw((4,-3)--(4,-1),dashed); draw((4,0)--(4,5),dashed); draw((8,-3)--(8,-1),dashed); draw((8,0)--(8,5),dashed); draw((12,-3)--(12,-1),dashed); draw((12,0)--(12,5),dashed); //Line draw((0,4)--(12,-2)); // x-axis labels label("1",(4,0),4*S); label("2",(8,0),4*S); label("3",(12,0),4*S); [/asy]

2009 F = Ma Exam Information (21 and 22)

The following information is used for questions 21 and 22. Two stars orbit their common center of mass as shown in the diagram below. The masses of the two stars are 3M3M and MM. The distance between the stars is dd. [asy] size(250); draw((0,0)--(sqrt(2)/4,sqrt(2)/4),linewidth(2)); draw((0,0)--(sqrt(2)/4,-sqrt(2)/4),linewidth(2)); draw((0,0)--(-sqrt(2)/4,sqrt(2)/4),linewidth(2)); draw((0,0)--(-sqrt(2)/4,-sqrt(2)/4),linewidth(2)); draw((4,0)--(-8,0),lightgray); draw((4,0)--(-8,0),black+dashed); filldraw(circle((7,0),3),lightgray,linewidth(2)); filldraw(circle((-9.5,0),1.5),lightgray,linewidth(2)); path A=(-8.786, 2.387)-- (-8.779, 2.585)-- (-8.780, 2.765)-- (-8.781, 2.873)-- (-8.782, 2.999)-- (-8.783, 3.179)-- (-8.775, 3.323)-- (-8.794, 3.503)-- (-8.752, 3.719)-- (-8.762, 3.935)-- (-8.738, 4.223)-- (-8.721, 4.385)-- (-8.714, 4.583)-- (-8.689, 4.763)-- (-8.664, 5.015)-- (-8.657, 5.302)-- (-8.624, 5.536)-- (-8.590, 5.752)-- (-8.563, 5.976)-- (-8.528, 6.164)-- (-8.511, 6.376)-- (-8.482, 6.614)-- (-8.452, 6.788)-- (-8.411, 6.963)-- (-8.375, 7.163)-- (-8.346, 7.338)-- (-8.317, 7.538)-- (-8.269, 7.701)-- (-8.245, 7.851)-- (-8.198, 8.026)-- (-8.169, 8.263)-- (-8.127, 8.425)-- (-8.079, 8.575)-- (-8.038, 8.800)-- (-7.984, 8.938)-- (-7.930, 9.138)-- (-7.876, 9.300)-- (-7.840, 9.463)-- (-7.786, 9.638)-- (-7.726, 9.850)-- (-7.654, 10.013)-- (-7.606, 10.212)-- (-7.546, 10.387)-- (-7.486, 10.487)-- (-7.432, 10.662)-- (-7.378, 10.812)-- (-7.305, 10.987)-- (-7.239, 11.150)-- (-7.179, 11.287)-- (-7.125, 11.412)-- (-7.058, 11.562)-- (-6.992, 11.712); path B=(4.931, -4.882)-- (4.933, -5.080)-- (4.925, -5.260)-- (4.900, -5.422)-- (4.901, -5.638)-- (4.885, -5.836)-- (4.825, -6.052)-- (4.783, -6.412)-- (4.732, -6.664)-- (4.681, -6.880)-- (4.638, -7.078)-- (4.587, -7.276)-- (4.544, -7.438)-- (4.457, -7.636)-- (4.405, -7.816)-- (4.345, -8.013)-- (4.258, -8.265)-- (4.180, -8.463)-- (4.076, -8.661)-- (3.997, -8.823)-- (3.910, -8.967)-- (3.823, -9.093)-- (3.726, -9.219)-- (3.674, -9.345)-- (3.508, -9.543)-- (3.386, -9.741)-- (3.263, -9.867)-- (3.175, -9.939)-- (3.061, -10.011); draw(shift(-0.7,-0.5)*A,EndArrow(size=7)); draw(shift(2.3,1.5)*B,EndArrow(size=7)); [/asy]
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