Physics 50 Winter 2016 Final Exam

Transcription

1 Physics 50 Winter 2016 Final Exam Name: Mar 24, 2016 Please show your work! Answers are not complete without clear reasoning. When asked for an expression, you must give your answer in terms of the variables given in the question and/or fundamental constants, like g. Answer as many questions as you can. Do not forget to include appropriate units when giving a number as an answer. Calculators are allowed. If you detach any pages from the test, please write your name on every loose page. g = 9.81 m s 2 Equations v f = v 0 + at v avg = v 0+v f 2 x = v 0 t at2 x = v f t 1 2 at2 v 2 f = v a x x = v avg t ω = 2π T a c = v2 r f k = µ k n f s,max = µ s n F = kx K = 1 2 mv2 U g = mgy U s = 1 2 kx2 W = F d P = W = F v t Trigonometric Identities sin 2 θ + cos 2 θ = 1 sin(2θ) = 2 sin(θ) cos(θ). cos(2θ) = cos 2 θ sin 2 θ sin(α ± β) = sin α cos β ± cos α sin β cos(α ± β) = cos α cos β sin α sin β cos α cos β = 1 [cos(α β) + cos(α + β)] 2 sin α sin β = 1 [cos(α β) cos(α + β)] 2 sin α cos β = 1 [sin(α + β) + sin(α β)] 2 sin ( θ + π 2 ) = cos θ cos ( θ + π 2 ) = sin θ sec θ := 1 cos θ csc θ := 1 sin θ cot θ := 1 tan θ 1

2 speed of 5.8 m/s when he decides to hit the dirt. The chicken slides for 1.1 s, just reaching the plate as he stops (safe, of course). (a) What is the magnitude and direction of the chicken s acceleration? (b) How far did the chicken slide? 1. A car in stop-and-go traffic starts at rest, moves forward 15 m in 8.0 s, then comes 60. A bicyclist is finishing his repair of a flat tire when a friend to rest again. The rides velocity-versus-time by with a constant speed plot of 3.5 form/s. thistwo carseconds is shown later in the figure. What Section 2 7 Freely Falling Object distance does the the car bicyclist coverhops in on his bike and accelerates at 2.4 m/s 2 until 64. Legend has it that Isaac Newton he catches his friend. (a) How much time does it take until he falling apple, thus triggering his thou (a) the first 4.0 catches seconds his of friend? its motion (b) How and far has he traveled in this time? the story to be true, estimate the s (c) What is his speed when he catches up? struck Newton. (b) the last 2.0 seconds of its motion? 61. A car in stop-and-go traffic starts at rest, moves forward (c) What is the 13 constant m in 8.0 s, speed then comes V that to rest characterizes again. The velocity-versus-time the middle portion of its motion? plot for this car is given in Figure What distance does the car cover in (a) the first 4.0 seconds of its motion and (b) the last 2.0 seconds of its motion? (c) What is the constant speed V that characterizes the middle portion of its motion? (d) Sketch a position-versus-time graph for the car. (e) Sketch an acceleration-versus-time graph for the car. (a) Find the magnitude of the cart s (b) Show that your results for part with the formula, a = g sin u. (We Chapter 5.) 65. The cartoon below shows a car in made in the cartoon accurate? Justify V Velocity (m/s) O 2 4 Time (s) FIGURE 2 33 Problem YOU DO NOT NEED TO DO A HYPOTHESIS FOR THIS QUESTION. 62. Acar and a truck are heading directly toward one another on a straight and narrow street, but they avoid a head-on collision by simultaneously applying their brakes at t = 0. The resulting velocity versus time graphs are shown in Figure What is the separation between the car and the truck when they have come to rest, given that at t = 0 the car is at x = 15 m and the truck is at x = -35 m. (Note that this information determines which line in the graph corresponds to which vehicle.) SKETCH (of car s journey): [2pts] SOLUTION Parts (a), (b), and (c): [8pts] (Problems 65 and 2

3 2.0 seconds of its motion? (c) What is the constant speed V that characterizes the middle portion of its motion? V Velocity (m/s) O 2 4 Time (s) FIGURE 2 33 Problem Parts (d) and (e): [12pts] (Make sure to show the correct shapes of the functions in your drawings and label the axes with numerical values.) 62. Acar and a truck are heading directly toward one another on a straight and narrow street, but they avoid a head-on collision by simultaneously applying their brakes at t = 0. The resulting velocity versus time graphs are shown in Figure What is the separation between the car and the truck when they have come to rest, given that at t = 0 the car is at x = 15 m and the truck is at x = -35 m. (Note that this information determines which line in the graph corresponds to which vehicle.) (Problems 65 and 3

4 2. A swimmer wants to swim across a river that runs from east to west (see diagram) with a current of 0.5 m/s and is 30 m wide. He plans to swim from the south bank (south side) of the river to the north bank and can swim at a steady speed of 2 m/s. North bank current θ W N S E South bank (a) If he wants to arrive on the north bank directly north of his starting point (the point directly on the opposite side of the river), at what angle relative to due North (θ in the diagram) should he choose to swim? (b) At what angle should he swim if he instead wants to minimize his time to cross the stream, and doesn t care where on the north bank he arrives? Give some justification for your answer, either in words or equations. HYPOTHESIS (just for part (a)): [1pt] VECTOR DIAGRAM and SOLUTION: Part (a): [4pts] ANSWER REASONABLE / AGREES WITH HYPOTHESIS? (just for part (a)): 4

5 Part (b): [4pts] (You don t actually need this much space.) 5

6 3. A shot-putter throws the shot with an initial speed of 11.2 m/s from a height of 2.00 m above the ground. What is the range of the shot if the launch angle is 30.0? HYPOTHESIS: [1pt] SKETCH: [2pts] SOLUTION: [6pts] ANSWER REASONABLE / AGREES WITH HYPOTHESIS?: [1pt] 6

7 Contact force 1 mg 20 FIGURE 6 37 Problem 65 Center line 4. A kg toy airplane fast is enough tiedto make to the you feel ceiling weightless withat the a string. top. (a) HowWhen the airplane s motor many seconds does it take to complete one revolution in this is started, it moves with case? (b) a constant How does your speed answer to ofpart 1.21 (a) depend m/son your a horizontal circle of radius mass? Explain. (c) What is the direction and magnitude of your 0.44 m, as illustrated in the diagram. Find Contact force 2 (a) the angle the string makes with the vertical and [pts] (b) the tension in the string. 2.0-kg box rests on a plank that is inclined at an angle of above the horizontal. The upper end of the box is attached spring with a force constant of 360 N/m, as shown in re If the coefficient of static friction between the box the plank is 0.22, what is the maximum amount the spring be stretched and the box remain at rest? as shown in Figure The tension in string 2 is 1.7 N. (a) Is the tension in string 1 greater than, less than or equal to 1.7 N? Explain. (b) Verify your answer to part (a) by calculating the tension in string 1. (c) What is the mass of the picture? 70. IP Referring to Problem 49, suppose the Ferris wheel rotates acceleration when you are at the bottom of the wheel? Assume that its rotational speed has remained constant. 71. A Conical Pendulum A kg toy airplane is tied to the ceiling with a string. When the airplane s motor is started, it moves with a constant speed of 1.21 m/s in a horizontal circle of radius 0.44 m, as illustrated in Figure Find (a) the angle the string makes with the vertical and (b) the tension in the string. FIGURE 6 40 Problem kg A tugboat tows a barge at constant speed with a 3500-kg cable, as shown in Figure If the angle the cable makes with the horizontal where it attaches to the barge and the tugboat is 22, find the force the cable exerts on the barge in the forward direction. HYPOTHESIS (for angle and tension): [2pts] FIGURE 6 38 Problem 66 P The blocks shown in Figure 6 39 are at rest. (a) Find the ional force exerted on block A given that the mass of block A 50 kg, the mass of block B is 2.25 kg, and the coefficient of ic friction between block A and the surface on which it rests 320. (b) If the mass of block A is doubled, does the frictional e exerted on it increase, decrease, or stay the same? Explain. A SKETCH (a free body diagram for the airplane): [3pts] FIGURE 6 41 Problem 72 SOLUTION (for (a) and (b)): [6pts] 73. IP Two blocks, stacked one on top of the other, can move without friction on the horizontal surface shown in Figure The surface between the two blocks is rough, however, with a coefficient of static friction equal to (a) If a horizontal force F is applied to the 5.0-kg bottom block, what is the maximum value F can have before the 2.0-kg top block begins to slip? (b) If the mass of the top block is increased, does the maximum value of F increase, decrease, or stay the same? Explain. 2.0 kg 22 B F 5.0 kg FIGURE 6 42 Problem 73 IGURE 6 39 Problems 67 and Find the coefficient of kinetic friction between a 4.5-kg block and the horizontal surface on which it rests if an 85 N/m ANSWERS REASONABLE / AGREE WITH HYPOTHESIS?: [2pts] 7

8 5. A driver on a test track accelerates in a straight line at 7.0 m/s 2 without spinning the tires. (a) Determine the minimum coefficient of static friction between the tires and the road needed to make this possible. [4pts] (b) If the car reaches a speed of 25 m/s in 3 s, what distance did it cover during its acceleration? HYPOTHESIS (for (a) and (b)): [2pts] SKETCH (a free body diagram for car): [4pts] SOLUTION Part (a): [5pts] Part (b): [4pts] ANSWERS REASONABLE / AGREE WITH HYPOTHESIS?: [2pts] 8

9 6. A block of mass m 1 on a frictionless table top is attached by a string to a hanging block of mass m 2, as shown. The blocks are released from rest and allowed to move freely. (a) Is the tension in the string greater than, less than, or equal to the weight of the hanging mass? Why? (b) Find an expression for the acceleration of the blocks and (c) find an expression for the tension in the string. (d) Find an expression for the speed of the blocks once the hanging block has descended a distance d. (Give your answers in terms of m 1, m 2, g, and d only.) YOU DO NOT NEED TO DO A HYPOTHESIS FOR THIS QUESTION. Part (a) answer: [2pts] SKETCH: [7pts] SOLUTION for parts (b)-(d): [9pts] 9

10 7. At a playground, a 21-kg child plays on a slide that drops through a height of 2.3 m. The child starts at rest at the top of the slide. On the way down, the slide does a nonconservative work of 370 J on the child. What is the child s speed at the bottom of the slide? HYPOTHESIS: [1pt] SKETCH (the child s path down the slide): [2pts] SOLUTION: [5pts] ANSWER REASONABLE / AGREES WITH HYPOTHESIS?: [1pt] 10

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