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2010 F = Ma

Part of F = Ma

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2010 F = Ma Exam Problem 25

Spaceman Fred's spaceship (which has negligible mass) is in an elliptical orbit about Planet Bob. The minimum distance between the spaceship and the planet is RR; the maximum distance between the spaceship and the planet is 2R2R. At the point of maximum distance, Spaceman Fred is traveling at speed v0v_\text{0}. He then fires his thrusters so that he enters a circular orbit of radius 2R2R. What is his new speed? [asy] size(300); // Shape draw(circle((0,0),25),dashed+gray); draw(circle((0,0),3.5),linewidth(2)); draw(ellipse((5,0),20,15)); // Dashed Lines draw((25,13)--(25,-35),dotted); draw((0,-35)--(0,-3.3),dotted); draw((0,3.3)--(0,13),dotted); draw((-15,13)--(-15,-35),dotted); // Labels draw((-14,-35)--(-1,-35),Arrows(size=6,SimpleHead)); label(scale(1.2)*"RR",(-7.5,-35),N); draw((24,-35)--(1,-35),Arrows(size=6,SimpleHead)); label(scale(1.2)*"2R2R",(10,-35),N); // Blobs on Earth path A=(-1.433, 2.667)-- (-1.433, 2.573)-- (-1.360, 2.478)-- (-1.408, 2.360)-- (-1.493, 2.207)-- (-1.554, 2.160)-- (-1.614, 2.113)-- (-1.675, 2.065)-- (-1.735, 1.959)-- (-1.772, 1.877)-- (-1.723, 1.759)-- (-1.748, 1.676)-- (-1.748, 1.523)-- (-1.772, 1.369)-- (-1.760, 1.240)-- (-1.857, 1.145)-- (-1.941, 1.098)-- (-2.050, 1.122)-- (-2.111, 1.086)-- (-2.244, 1.039)-- (-2.390, 1.004)-- (-2.511, 0.909)-- (-2.486, 0.697)-- (-2.499, 0.555)-- (-2.535, 0.414)-- (-2.668, 0.308)-- (-2.765, 0.237)-- (-2.910, 0.131)-- (-3.068, 0.036)-- (-3.250, 0.024)-- (-3.310, 0.154)-- (-3.274, 0.272)-- (-3.286, 0.402)-- (-3.298, 0.532)-- (-3.250, 0.650)-- (-3.165, 0.768)-- (-3.128, 0.933)-- (-3.068, 1.074)-- (-3.032, 1.204)-- (-2.971, 1.310)-- (-2.886, 1.452)-- (-2.801, 1.558)-- (-2.729, 1.652)-- (-2.656, 1.770)-- (-2.583, 1.912)-- (-2.486, 1.995)-- (-2.365, 2.089)-- (-2.244, 2.207)-- (-2.123, 2.313)-- (-2.014, 2.419)-- (-1.905, 2.478)-- (-1.832, 2.573)-- (-1.687, 2.643)-- (-1.578, 2.714)--cycle; filldraw(A,gray); path B=(-0.397, 2.527)-- (-0.468, 2.321)-- (-0.538, 2.154)-- (-0.639, 2.065)-- (-0.760, 2.085)-- (-0.922, 2.085)-- (-0.993, 2.016)-- (-0.770, 1.918)-- (-0.649, 1.829)-- (-0.498, 1.780)-- (-0.367, 1.770)-- (-0.205, 1.751)-- (-0.084, 1.761)-- (-0.104, 1.613)-- (-0.114, 1.495)-- (-0.094, 1.358)-- (0.007, 1.220)-- (0.067, 1.131)-- (0.108, 1.013)-- (0.188, 0.905)-- (0.239, 0.787)-- (0.330, 0.650)-- (0.461, 0.620)-- (0.622, 0.620)-- (0.794, 0.591)-- (0.905, 0.610)-- (0.956, 0.689)-- (1.026, 0.591)-- (1.097, 0.483)-- (1.198, 0.374)-- (1.258, 0.276)-- (1.339, 0.188)-- (1.319, -0.009)-- (1.309, -0.166)-- (1.198, -0.343)-- (1.077, -0.432)-- (0.935, -0.520)-- (0.814, -0.589)-- (0.633, -0.677)-- (0.481, -0.727)-- (0.350, -0.776)-- (0.229, -0.894)-- (0.229, -1.041)-- (0.229, -1.228)-- (0.340, -1.346)-- (0.522, -1.415)-- (0.643, -1.513)-- (0.693, -1.651)-- (0.784, -1.798)-- (0.723, -1.936)-- (0.612, -2.044)-- (0.471, -2.123)-- (0.350, -2.201)-- (0.249, -2.270)-- (0.108, -2.339)-- (-0.013, -2.418)-- (-0.124, -2.535)-- (-0.135, -2.673)-- (-0.175, -2.811)-- (-0.084, -2.840)-- (0.067, -2.840)-- (0.209, -2.830)-- (0.350, -2.742)-- (0.522, -2.653)-- (0.582, -2.604)-- (0.713, -2.545)-- (0.845, -2.457)-- (0.935, -2.408)-- (1.057, -2.388)-- (1.228, -2.280)-- (1.329, -2.191)-- (1.460, -2.132)-- (1.581, -2.093)-- (1.692, -2.044)-- (1.793, -2.005)-- (1.844, -1.906)-- (1.844, -1.828)-- (1.904, -1.749)-- (2.005, -1.621)-- (1.955, -1.454)-- (1.894, -1.287)-- (1.773, -1.189)-- (1.632, -0.992)-- (1.592, -0.874)-- (1.491, -0.736)-- (1.410, -0.569)-- (1.460, -0.412)-- (1.561, -0.274)-- (1.592, -0.078)-- (1.622, 0.168)-- (1.551, 0.306)-- (1.440, 0.404)-- (1.420, 0.561)-- (1.551, 0.620)-- (1.703, 0.630)-- (1.824, 0.532)-- (1.955, 0.365)-- (2.046, 0.453)-- (2.116, 0.551)-- (2.167, 0.689)-- (2.096, 0.807)-- (1.965, 0.905)-- (1.834, 0.935)-- (1.743, 0.994)-- (1.622, 1.131)-- (1.531, 1.249)-- (1.430, 1.348)-- (1.359, 1.515)-- (1.420, 1.702)-- (1.511, 1.839)-- (1.571, 2.016)-- (1.672, 2.134)-- (1.592, 2.232)-- (1.440, 2.291)-- (1.289, 2.350)-- (1.178, 2.252)-- (1.127, 2.134)-- (1.067, 1.997)-- (0.986, 1.898)-- (0.845, 1.839)-- (0.693, 1.839)-- (0.522, 1.859)-- (0.471, 1.977)-- (0.380, 2.124)-- (0.289, 2.203)-- (0.188, 2.291)-- (0.047, 2.311)-- (-0.074, 2.370)-- (-0.195, 2.508)--cycle; filldraw(B,gray); [/asy] (A) 3/2v0\sqrt{3/2}v_\text{0} (B) 5v0\sqrt{5}v_\text{0} (C) 3/5v0\sqrt{3/5}v_\text{0} (D) 2v0\sqrt{2}v_\text{0} (E) 2v02v_\text{0}
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2010 F = Ma Exam Problem 24

A uniform circular disk of radius RR begins with a mass MM; about an axis through the center of the disk and perpendicular to the plane of the disk the moment of inertia is I0=12MR2I_\text{0}=\frac{1}{2}MR^2. A hole is cut in the disk as shown in the diagram. In terms of the radius RR and the mass MM of the original disk, what is the moment of inertia of the resulting object about the axis shown?
[asy] size(14cm); pair O=origin; pair A=O, B=(3,0), C=(6,0);
real r_1=1, r_2=.5;
pen my_fill_pen_1=gray(.8); pen my_fill_pen_2=white; pen my_fill_pen_3=gray(.7); pen my_circleline_draw_pen=black+1.5bp;
//fill();
filldraw(circle(A,r_1),my_fill_pen_1,my_circleline_draw_pen); filldraw(circle(B,r_1),my_fill_pen_1,my_circleline_draw_pen);
// Ellipse filldraw(yscale(.2)*circle(C,r_1),my_fill_pen_1,my_circleline_draw_pen);
draw((C.x,C.y-.75)--(C.x,C.y-.2), dashed); draw(C--(C.x,C.y+1),dashed); label("axis of rotation",(C.x,C.y-.75),3*S);
// small ellipse pair center_small_ellipse; center_small_ellipse=midpoint(C--(C.x+r_1,C.y)); //dot(center_small_ellipse); filldraw(yscale(.15)*circle(center_small_ellipse,r_1/2),white);
pair center_elliptic_arc_arrow; real gr=(sqrt(5)-1)/2; center_elliptic_arc_arrow=(C.x,C.y+gr); //dot(center_elliptic_arc_arrow);
draw(//shift((0*center_elliptic_arc_arrow.x,center_elliptic_arc_arrow.y-.2))* ( yscale(.2)* ( arc((center_elliptic_arc_arrow.x,center_elliptic_arc_arrow.y+2.4), .4,120,360+60)) ),Arrow); //dot(center_elliptic_arc_arrow);
// lower_Half-Ellipse real downshift=1;
pair C_prime=(C.x,C.y-downshift); path lower_Half_Ellipse=yscale(.2)*arc(C_prime,r_1,180,360); path upper_Half_Ellipse=yscale(.2)*arc(C,r_1,180,360); draw(lower_Half_Ellipse,my_circleline_draw_pen); //draw(upper_Half_Ellipse,red);

// Why here ".2*downshift" instead of downshift seems to be not absolutely clean. filldraw(upper_Half_Ellipse--(C.x+r_1,C.y-.2*downshift)--reverse(lower_Half_Ellipse)--cycle,gray);
//filldraw(shift(C-.1)*(circle((B+.5),.5)),my_fill_pen_2);// filldraw(circle((B+.5),.5),my_fill_pen_2);//shift(C-.1)*
/* filldraw(//shift((C.x,C.y-.45))* yscale(.2)*circle((C.x,C.y-1),r_1),my_fill_pen_3,my_circleline_draw_pen); */
draw("RR",A--dir(240),Arrow); draw("RR",B--shift(B)*dir(240),Arrow);
draw(scale(1)*"R/2\scriptstyle R/2",(B+.5)--(B+1),.5*LeftSide,Arrow); draw(scale(1)*"R/2\scriptstyle R/2",(B+.5)--(B+1),.5*LeftSide,Arrow);
[/asy] (A) (15/32)MR2\text{(15/32)}MR^2 (B) (13/32)MR2\text{(13/32)}MR^2 (C) (3/8)MR2\text{(3/8)}MR^2 (D) (9/32)MR2\text{(9/32)}MR^2 (E) (15/16)MR2\text{(15/16)}MR^2
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2010 F = Ma Exam Problem 23

Two streams of water flow through the U-shaped tubes shown. The tube on the left has cross-sectional area AA, and the speed of the water flowing through it is vv; the tube on the right has cross-sectional area A′=1/2AA'=1/2A. If the net force on the tube assembly is zero, what must be the speed v′v' of the water flowing through the tube on the right? Neglect gravity, and assume that the speed of the water in each tube is the same upon entry and exit. [asy] // Code by riben size(300); draw(arc((0,0),10,90,270)); draw(arc((0,0),7,90,270)); draw((0,10)--(25,10)); draw((0,-10)--(25,-10)); draw((0,7)--(25,7)); draw((0,-7)--(25,-7)); draw(ellipse((25,8.5),0.5,1.5)); draw(ellipse((25,-8.5),0.5,1.5)); draw((20,8.5)--(7,8.5),EndArrow(size=7)); draw((7,-8.5)--(20,-8.5),EndArrow(size=7)); draw(arc((-22,0),12,90,-90)); draw(arc((-22,0),7,90,-90)); draw((-22,12)--(-42,12)); draw((-22,-12)--(-42,-12)); draw((-22,7)--(-42,7)); draw((-22,-7)--(-42,-7)); draw(ellipse((-42,9.5),1.5,2.5)); draw(ellipse((-42,-9.5),1.5,2.5)); draw((-38,9.5)--(-23,9.5),EndArrow(size=7)); draw((-23,-9.5)--(-38,-9.5),EndArrow(size=7)); [/asy] (A) 1/2v1/2v (B) vv (C) 2v\sqrt{2}v (D) 2v2v (E) 4v4v
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2010 F = Ma Exam Problem 20

Consider the following graph of position vs. time, which represents the motion of a certain particle in the given potential. [asy] import roundedpath; size(300); picture pic; // Rectangle draw(pic,(0,0)--(20,0)--(20,15)--(0,15)--cycle); label(pic,"0",(0,0),S); label(pic,"2",(4,0),S); label(pic,"4",(8,0),S); label(pic,"6",(12,0),S); label(pic,"8",(16,0),S); label(pic,"10",(20,0),S); label(pic,"-15",(0,2),W); label(pic,"-10",(0,4),W); label(pic,"-5",(0,6),W); label(pic,"0",(0,8),W); label(pic,"5",(0,10),W); label(pic,"10",(0,12),W); label(pic,"15",(0,14),W); label(pic,rotate(90)*"x (m)",(-2,7),W); label(pic,"t (s)",(11,-2),S); // Tick Marks draw(pic,(4,0)--(4,0.3)); draw(pic,(8,0)--(8,0.3)); draw(pic,(12,0)--(12,0.3)); draw(pic,(16,0)--(16,0.3)); draw(pic,(20,0)--(20,0.3)); draw(pic,(4,15)--(4,14.7)); draw(pic,(8,15)--(8,14.7)); draw(pic,(12,15)--(12,14.7)); draw(pic,(16,15)--(16,14.7)); draw(pic,(20,15)--(20,14.7)); draw(pic,(0,2)--(0.3,2)); draw(pic,(0,4)--(0.3,4)); draw(pic,(0,6)--(0.3,6)); draw(pic,(0,8)--(0.3,8)); draw(pic,(0,10)--(0.3,10)); draw(pic,(0,12)--(0.3,12)); draw(pic,(0,14)--(0.3,14)); draw(pic,(20,2)--(19.7,2)); draw(pic,(20,4)--(19.7,4)); draw(pic,(20,6)--(19.7,6)); draw(pic,(20,8)--(19.7,8)); draw(pic,(20,10)--(19.7,10)); draw(pic,(20,12)--(19.7,12)); draw(pic,(20,14)--(19.7,14)); // Path add(pic); path A=(0.102, 6.163)-- (0.192, 6.358)-- (0.369, 6.500)-- (0.526, 6.642)-- (0.643, 6.712)-- (0.820, 6.830)-- (0.938, 6.901)-- (1.075, 7.043)-- (1.193, 7.185)-- (1.369, 7.256)-- (1.506, 7.374)-- (1.644, 7.445)-- (1.840, 7.515)-- (1.958, 7.586)-- (2.134, 7.657)-- (2.291, 7.752)-- (2.468, 7.846)-- (2.625, 7.846)-- (2.899, 7.893)-- (3.095, 8.035)-- (3.350, 8.035)-- (3.586, 8.106)-- (3.860, 8.106)-- (4.135, 8.106)-- (4.371, 8.035)-- (4.606, 8.035)-- (4.881, 8.012)-- (5.155, 7.917)-- (5.391, 7.823)-- (5.665, 7.728)-- (5.960, 7.563)-- (6.175, 7.468)-- (6.332, 7.374)-- (6.528, 7.232)-- (6.725, 7.161)-- (6.882, 6.996)-- (7.117, 6.854)-- (7.333, 6.712)-- (7.509, 6.523)-- (7.666, 6.358)-- (7.902, 6.146)-- (8.098, 5.980)-- (8.274, 5.791)-- (8.451, 5.649)-- (8.647, 5.484)-- (8.882, 5.248)-- (9.196, 5.059)-- (9.392, 4.894)-- (9.628, 4.752)-- (9.824, 4.634)-- (10.118, 4.516)-- (10.452, 4.350)-- (10.785, 4.232)-- (11.001, 4.185)-- (11.315, 4.138)-- (11.648, 4.114)-- (12.002, 4.114)-- (12.257, 4.091)-- (12.610, 4.067)-- (12.825, 4.161)-- (13.081, 4.185)-- (13.316, 4.279)-- (13.492, 4.327)-- (13.689, 4.445)-- (13.826, 4.516)-- (14.022, 4.587)-- (14.159, 4.705)-- (14.316, 4.823)-- (14.532, 4.964)-- (14.669, 5.059)-- (14.866, 5.177)-- (15.062, 5.248)-- (15.278, 5.461)-- (15.474, 5.697)-- (15.650, 5.838)-- (15.847, 6.004)-- (16.043, 6.169)-- (16.258, 6.334)-- (16.415, 6.523)-- (16.592, 6.736)-- (16.788, 6.830)-- (17.063, 7.067)-- (17.357, 7.232)-- (17.573, 7.397)-- (17.808, 7.515)-- (18.063, 7.634)-- (18.358, 7.704)-- (18.573, 7.870)-- (18.887, 7.941)-- (19.142, 8.012)-- (19.358, 8.035)-- (19.574, 8.082)-- (19.770, 8.130); draw(shift(1.8*up)*roundedpath(A,0.09),linewidth(1.5)); [/asy] What is the total energy of the particle?
(A) -5 J\text{-5 J} (B) 0 J\text{0 J} (C) 5 J\text{5 J} (D) 10 J\text{10 J} (E) 15 J\text{15 J}
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2010 F = Ma Exam Problem 19

Consider the following graphs of position vs. time. [asy] size(500); picture pic; // Rectangle draw(pic,(0,0)--(20,0)--(20,15)--(0,15)--cycle); label(pic,"0",(0,0),S); label(pic,"2",(4,0),S); label(pic,"4",(8,0),S); label(pic,"6",(12,0),S); label(pic,"8",(16,0),S); label(pic,"10",(20,0),S); label(pic,"-15",(0,2),W); label(pic,"-10",(0,4),W); label(pic,"-5",(0,6),W); label(pic,"0",(0,8),W); label(pic,"5",(0,10),W); label(pic,"10",(0,12),W); label(pic,"15",(0,14),W); label(pic,rotate(90)*"x (m)",(-2,7),W); label(pic,"t (s)",(11,-2),S); // Tick Marks draw(pic,(4,0)--(4,0.3)); draw(pic,(8,0)--(8,0.3)); draw(pic,(12,0)--(12,0.3)); draw(pic,(16,0)--(16,0.3)); draw(pic,(20,0)--(20,0.3)); draw(pic,(4,15)--(4,14.7)); draw(pic,(8,15)--(8,14.7)); draw(pic,(12,15)--(12,14.7)); draw(pic,(16,15)--(16,14.7)); draw(pic,(20,15)--(20,14.7)); draw(pic,(0,2)--(0.3,2)); draw(pic,(0,4)--(0.3,4)); draw(pic,(0,6)--(0.3,6)); draw(pic,(0,8)--(0.3,8)); draw(pic,(0,10)--(0.3,10)); draw(pic,(0,12)--(0.3,12)); draw(pic,(0,14)--(0.3,14)); draw(pic,(20,2)--(19.7,2)); draw(pic,(20,4)--(19.7,4)); draw(pic,(20,6)--(19.7,6)); draw(pic,(20,8)--(19.7,8)); draw(pic,(20,10)--(19.7,10)); draw(pic,(20,12)--(19.7,12)); draw(pic,(20,14)--(19.7,14)); // Path add(pic); path A=(0,14)--(20,14); draw(A); label("I.",(8,-4),3*S); path B=(0,6)--(20,6); picture pic2=shift(30*right)*pic; draw(shift(30*right)*B); label("II.",(38,-4),3*S); add(pic2); path C=(0,12)--(20,14); picture pic3=shift(60*right)*pic; draw(shift(60*right)*C); label("III.",(68,-4),3*S); add(pic3); [/asy] Which of the graphs could be the motion of a particle in the given potential?
(A) I\text{I} (B) III\text{III} (C) I and II\text{I and II} (D) I and III\text{I and III} (E) I, II, and III\text{I, II, and III}
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2010 F = Ma Exam Problem 18

Which of the following represents the force corresponding to the given potential? [asy] // Code by riben size(400); picture pic; // Rectangle draw(pic,(0,0)--(22,0)--(22,12)--(0,12)--cycle); label(pic,"-15",(2,0),S); label(pic,"-10",(5,0),S); label(pic,"-5",(8,0),S); label(pic,"0",(11,0),S); label(pic,"5",(14,0),S); label(pic,"10",(17,0),S); label(pic,"15",(20,0),S); label(pic,"-2",(0,2),W); label(pic,"-1",(0,4),W); label(pic,"0",(0,6),W); label(pic,"1",(0,8),W); label(pic,"2",(0,10),W); label(pic,rotate(90)*"F (N)",(-2,6),W); label(pic,"x (m)",(11,-2),S); // Tick Marks draw(pic,(2,0)--(2,0.3)); draw(pic,(5,0)--(5,0.3)); draw(pic,(8,0)--(8,0.3)); draw(pic,(11,0)--(11,0.3)); draw(pic,(14,0)--(14,0.3)); draw(pic,(17,0)--(17,0.3)); draw(pic,(20,0)--(20,0.3)); draw(pic,(0,2)--(0.3,2)); draw(pic,(0,4)--(0.3,4)); draw(pic,(0,6)--(0.3,6)); draw(pic,(0,8)--(0.3,8)); draw(pic,(0,10)--(0.3,10)); draw(pic,(2,12)--(2,11.7)); draw(pic,(5,12)--(5,11.7)); draw(pic,(8,12)--(8,11.7)); draw(pic,(11,12)--(11,11.7)); draw(pic,(14,12)--(14,11.7)); draw(pic,(17,12)--(17,11.7)); draw(pic,(20,12)--(20,11.7)); draw(pic,(22,2)--(21.7,2)); draw(pic,(22,4)--(21.7,4)); draw(pic,(22,6)--(21.7,6)); draw(pic,(22,8)--(21.7,8)); draw(pic,(22,10)--(21.7,10)); // Paths path A=(0,6)--(5,6)--(5,4)--(11,4)--(11,8)--(17,8)--(17,6)--(22,6); path B=(0,6)--(5,6)--(5,2)--(11,2)--(11,10)--(17,10)--(17,6)--(22,6); path C=(0,6)--(5,6)--(5,5)--(11,5)--(11,7)--(17,7)--(17,6)--(22,6); path D=(0,6)--(5,6)--(5,7)--(11,7)--(11,5)--(17,5)--(17,6)--(22,6); path E=(0,6)--(5,6)--(5,8)--(11,8)--(11,4)--(17,4)--(17,6)--(22,6); draw(A); label("(A)",(9.5,-3),4*S); draw(shift(35*right)*B); label("(B)",(45.5,-3),4*S); draw(shift(20*down)*C); label("(C)",(9.5,-23),4*S); draw(shift(35*right)*shift(20*down)*D); label("(D)",(45.5,-23),4*S); draw(shift(40*down)*E); label("(E)",(9.5,-43),4*S); add(pic); picture pic2=shift(35*right)*pic; picture pic3=shift(20*down)*pic; picture pic4=shift(35*right)*shift(20*down)*pic; picture pic5=shift(40*down)*pic; add(pic2); add(pic3); add(pic4); add(pic5); [/asy]
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2010 F = Ma Exam Problem 13

A ball of mass MM and radius RR has a moment of inertia of I=25MR2I=\frac{2}{5}MR^2. The ball is released from rest and rolls down the ramp with no frictional loss of energy. The ball is projected vertically upward off a ramp as shown in the diagram, reaching a maximum height ymaxy_{max} above the point where it leaves the ramp. Determine the maximum height of the projectile ymaxy_{max} in terms of hh. [asy] size(250); import roundedpath; path A=(0,0)--(5,-12)--(20,-12)--(20,-10); draw(roundedpath(A,1),linewidth(1.5)); draw((25,-10)--(12,-10),dashed+linewidth(0.5)); filldraw(circle((1.7,-1),1),lightgray); draw((25,-1)--(-1.5,-1),dashed+linewidth(0.5)); draw((23,-9.5)--(23,-1.5),Arrows(size=5)); label(scale(1.1)*"hh",(23,-6.5),2*E); [/asy] (A) hh
(B) 2549h\frac{25}{49}h
(C) 25h\frac{2}{5}h
(D) 57h\frac{5}{7}h
(E) 75h\frac{7}{5}h
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2010 F = Ma Exam Problem 11

The three masses shown in the accompanying diagram are equal. The pulleys are small, the string is lightweight, and friction is negligible. Assuming the system is in equilibrium, what is the ratio a/ba/b? The figure is not drawn to scale! [asy] size(250); dotfactor=10; dot((0,0)); dot((15,0)); draw((-3,0)--(25,0),dashed); draw((0,0)--(0,3),dashed); draw((15,0)--(15,3),dashed); draw((0,0)--(0,-15)); draw((15,0)--(15,-10)); filldraw(circle((0,-16),1),lightgray); filldraw(circle((15,-11),1),lightgray); draw((0,0)--(4,-4)); filldraw(circle((4.707,-4.707),1),lightgray); draw((15,0)--(5.62,-4.29)); draw((0.5,3)--(14.5,3),Arrows(size=5)); label(scale(1.2)*"aa",(7.5,3),1.5*N); draw((2.707,-4.707)--(25,-4.707),dashed); draw((25,-0.5)--(25,-4.2),Arrows(size=5)); label(scale(1.2)*"bb",(25,-2.35),1.5*E); [/asy] (A) 1/21/2 (B) 11 (C) 3\sqrt{3} (D) 22 (E) 232\sqrt{3}
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2010 F = Ma Exam Problem 10

A block of mass m1m_\text{1} is on top of a block of mass m2m_\text{2}. The lower block is on a horizontal surface, and a rope can pull horizontally on the lower block. The coefficient of kinetic friction for all surfaces is μ\mu. What is the resulting acceleration of the lower block if a force FF is applied to the rope? Assume that FF is sufficiently large so that the top block slips on the lower block. [asy] size(200); import roundedpath; draw((0,0)--(30,0),linewidth(3)); path A=(7,0.5)--(17,0.5)--(17,5.5)--(7,5.5)--cycle; filldraw(roundedpath(A,1),lightgray); path B=(10,6)--(15,6)--(15,9)--(10,9)--cycle; filldraw(roundedpath(B,1),lightgray); label("1",(12.5,6),1.5*N); label("2",(12,0.5),3*N); draw((17,3)--(27,3),EndArrow(size=13)); label(scale(1.2)*"FF",(22,3),2*N); [/asy] (A) a2=(F−μg(2m1+m2))/m2a_\text{2}=(F-\mu g(2m_\text{1}+m_\text{2}))/m_\text{2} (B) a2=(F−μg(m1+m2))/m2a_\text{2}=(F-\mu g(m_\text{1}+m_\text{2}))/m_\text{2} (C) a2=(F−μg(m1+2m2))/m2a_\text{2}=(F-\mu g(m_\text{1}+2m_\text{2}))/m_\text{2} (D) a2=(F+μg(m1+m2))/m2a_\text{2}=(F+\mu g(m_\text{1}+m_\text{2}))/m_\text{2} (E) a2=(F−μg(m2−m1))/m2a_\text{2}=(F-\mu g(m_\text{2}-m_\text{1}))/m_\text{2}
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2010 F = Ma Exam Problem 9

A point object of mass MM hangs from the ceiling of a car from a massless string of length LL. It is observed to make an angle θ\theta from the vertical as the car accelerates uniformly from rest. Find the acceleration of the car in terms of θ\theta, MM, LL, and gg. [asy] size(250); import graph; // Left draw((-3,0)--(-23,0),linewidth(1.5)); draw((-13,0)--(-13,-14)); filldraw(circle((-13,-15),2),gray); draw((-13,-15)--(-21,-15),dashed); draw((-21,-14)--(-21,-1),EndArrow(size=5)); draw((-21,-1)--(-21,-14),EndArrow(size=5)); label(scale(1.5)*"LL",(-21,-7.5),2*E); // Right draw((3,0)--(23,0),linewidth(1.5)); draw((13,0)--(13,-19),dashed); draw((13,0)--(5,-12)); filldraw(circle((3.89,-13.66),2),gray); label(scale(1.5)*"θ\theta",(12,-9),1.5*W); real f(real x){ return 5x^2/12-95x/12+25; } draw(graph(f,12,7),Arrows); [/asy] (A) Mgsin⁡θMg \sin \theta (B) MgLtan⁡θMgL \tan \theta (C) gtan⁡θg \tan \theta (D) gcot⁡θg \cot \theta (E) Mgtan⁡θMg \tan \theta
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2010 F = Ma Exam Problem 7

Harry Potter is sitting 2.02.0 meters from the center of a merry-go-round when Draco Malfoy casts a spell that glues Harry in place and then makes the merry-go-round start spinning on its axis. Harry has a mass of 50.0 kg\text{50.0 kg} and can withstand 5.0 g’s\text{5.0} \ g\text{'s} of acceleration before passing out. What is the magnitude of Harry's angular momentum when he passes out?
(A) 200 kg⋅m2/s\text{200 kg} \cdot \text{m}^2\text{/s} (A) 330 kg⋅m2/s\text{330 kg} \cdot \text{m}^2\text{/s} (A) 660 kg⋅m2/s\text{660 kg} \cdot \text{m}^2\text{/s} (A) 1000 kg⋅m2/s\text{1000 kg} \cdot \text{m}^2\text{/s} (A) 2200 kg⋅m2/s\text{2200 kg} \cdot \text{m}^2\text{/s}
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2010 F = Ma Exam Problem 5

Two projectiles are launched from a 3535 meter ledge as shown in the diagram. One is launched from a 3737 degree angle above the horizontal and the other is launched from 3737 degrees below the horizontal. Both of the launches are given the same initial speed of v0=50 m/sv_\text{0} = \text{50 m/s}. [asy] size(300); import graph; draw((-8,0)--(0,0)--(0,-11)--(30,-11)); draw((0,-11)--(-4.5,-11),dashdotted); draw((0,0)--(12,0),dashdotted); label(scale(0.75)*"35 m",(0,-5.5),5*W); draw((-4,-4.5)--(-4,-0.5),EndArrow(size=5)); draw((-4,-6)--(-4,-10.5),EndArrow(size=5)); // Projectiles real f(real x){ return -11x^2/49; } draw(graph(f,0,7),dashed+linewidth(1.5)); real g(real x){ return -6x^2/145+119x/145; } draw(graph(g,0,29),dashed+linewidth(1.5)); // Labels label(scale(0.75)*"Projectile 1",(20,2),E); label(scale(0.75)*"Projectile 2",(6,-7),E); [/asy] The difference in the times of flight for these two projectiles, t1−t2t_1-t_2, is closest to
(A) 3 s\text{3 s} (B) 5 s\text{5 s} (C) 6 s\text{6 s} (D) 8 s\text{8 s} (E) 10 s\text{10 s}
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2010 F = Ma Exam Information (1, 2, and 3)

Questions 1 to 3 refer to the figure below which shows a representation of the motion of a squirrel as it runs in a straight-line along a telephone wire. The letters A through E refer to the indicated times. [asy] size(250); // Dashes draw((0,-4)--(10,-4),dashdotted+linewidth(0.35)); draw((6,-4)--(6,4),dashdotted+linewidth(0.35)); draw((0,4)--(6,4),dashdotted+linewidth(0.35)); draw((8,0)--(8,-4),dashdotted+linewidth(0.35)); draw((10,0)--(10,-4),dashdotted+linewidth(0.35)); draw((18,0)--(18,-4),dashdotted+linewidth(0.35)); // Grid draw((0,0)--(20,0)); draw((0,-4)--(0,4)); draw((2,0.25)--(2,-0.25)); draw((4,0.25)--(4,-0.25)); draw((6,0.25)--(6,-0.25)); draw((8,0.25)--(8,-0.25)); draw((10,0.25)--(10,-0.25)); draw((12,0.25)--(12,-0.25)); draw((14,0.25)--(14,-0.25)); draw((16,0.25)--(16,-0.25)); draw((18,0.25)--(18,-0.25)); draw((18,0)--(20,0),EndArrow(size=10)); draw((-0.25,1)--(0.25,1)); draw((-0.25,2)--(0.25,2)); draw((-0.25,3)--(0.25,3)); draw((-0.25,4)--(0.25,4)); draw((-0.25,-1)--(0.25,-1)); draw((-0.25,-2)--(0.25,-2)); draw((-0.25,-3)--(0.25,-3)); draw((-0.25,-4)--(0.25,-4)); // Labels label("A",(0,-4),S*2); label("B",(6,-4),S*2); label("C",(8,-4),S*2); label("D",(10,-4),S*2); label("E",(18,-4),S*2); label(scale(0.75)*"time",(19.5,0),3*S); // Path path A=(0,0)--(6,4)--(10,-4)--(18,0); draw(A,linewidth(1.5)); [/asy]

2010 F = Ma Exam Information (15 and 16)

The following figure is used for questions 15 and 16. [asy] size(250); draw((0,0)--(30,0),linewidth(2)); filldraw((1,0.5)--(3,0.5)--(3,2)--(1,2)--cycle,gray); label(scale(1.2)*"mm",(2,2),N); draw((3.5,1.25)--(6,1.25),EndArrow(size=9)); label(scale(1.2)*"v0v_0",(5.5,1.25),2*N); filldraw((9,0.5)--(25,0.5)--(25,9)--cycle,lightgray); label(scale(1.2)*"MM",(21,2),N); [/asy] A small block of mass mm is moving on a horizontal table surface at initial speed v0v_\text{0}. It then moves smoothly onto a sloped big block of mass MM. The big block can also move on the table surface. Assume that everything moves without friction.

2010 F = Ma Exam Information (18, 19, and 20)

The following graph of potential energy is used for questions 18 through 20. [asy] size(300); // Rectangle draw((0,0)--(22,0)--(22,14)--(0,14)--cycle); label("-15",(2,0),S); label("-10",(5,0),S); label("-5",(8,0),S); label("0",(11,0),S); label("5",(14,0),S); label("10",(17,0),S); label("15",(20,0),S); label("-10",(0,3),W); label("-5",(0,7),W); label("0",(0,11),W); label(rotate(90)*"U (J)",(-2,7),W); label("x (m)",(11,-2),S); // Tick Marks draw((2,0)--(2,0.3)); draw((5,0)--(5,0.3)); draw((8,0)--(8,0.3)); draw((11,0)--(11,0.3)); draw((14,0)--(14,0.3)); draw((17,0)--(17,0.3)); draw((20,0)--(20,0.3)); draw((0,3)--(0.3,3)); draw((0,7)--(0.3,7)); draw((0,11)--(0.3,11)); draw((2,14)--(2,13.7)); draw((5,14)--(5,13.7)); draw((8,14)--(8,13.7)); draw((11,14)--(11,13.7)); draw((14,14)--(14,13.7)); draw((17,14)--(17,13.7)); draw((20,14)--(20,13.7)); draw((22,3)--(21.7,3)); draw((22,7)--(21.7,7)); draw((22,11)--(21.7,11)); // Path path A=(0,11)--(5,11)--(11,3)--(17,11)--(22,11); draw(A,linewidth(1.2)); [/asy]