Instructions Write your name, student ID and name of your TA instructor clearly on all sheets and fill your name and student ID on the bubble sheet. Solve all multiple choice questions. No penalty is given for wrong answers. Solve each problem on a different sheet of paper. Solutions to the problems should begin from the following basic physical principles: If r(t) is the position of the object as a function of time than velocity is v(t) = d r dt and acceleration is a(t) = d2 x dt 2. When the acceleration is a constant a then r(t) = r 0 + v 0 t + 1 2 at2. For motion in a circle of radius R, v = Rω, s = Rφ, and the centripetal acceleration is a c = ω 2 R Newton s Laws: F = m a and F 12 = F 21 Common forces include static friction (F µ s F N ), kinetic friction (F = µ k F N ), gravitational force (F = mg), drag (F = 1 2 ρac Dv 2 ) and the spring force (F = kx). Kinetic energy is 1 2 mv2, work is W = R F d x, gravitational potential energy is U g = mgh, and spring potential energy is U s = 1 2 kx2. Show all steps in the derivation of the answers. Make sure you write neatly and orderly. It is YOUR RESPONSIBILITY to make sure that the grader understands your solution. S/he will not give full points if they can not follow the solution, even if the final answer is correct. The acceleration due to gravity on Earth is 9.8 m/s 2. The solutions to the quadratic equation 0 = ax 2 + bx + c are given by x = b± b 2 4ac 2a. The following derivatives and integrals may be useful: If y(x) = Ax m, where A and m are constants, then Z y dx = dy dx = Amxm 1 A m + 1 xm+1, for m 1 You can use a calculator. Page 1
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Last name : First name : Student id : TA : Problem 1 (25 points) At the light rail terminus station, you notice a large horizontal spring at the end of the track where the train comes in. This is a safety device to stop the train so that it will not go plowing through the station if the engineer misjudges the stopping distance. While waiting, you wonder what would be the fastest train that the spring could stop by being fully compressed, by 1.2 m. To keep the passengers as safe as possible when the spring stops the train, you assume that the maximum stopping acceleration of the train, caused by the spring, is g/2. You make a guess that a train might have a mass of 0.35 million kilograms. For the purpose of getting your answer, you assume that all frictional forces are negligible. The main issue is to determine the spring constant. We can do this by using the maximum stopping acceleration, which will occur when the spring reaches maximum compression. F max = mg/2 = kx max k = mg 2x max E i = E f 1 2 mv2 = 1 2 kx2 k v max = m x2 max mg = 2x max m x2 max g = 2 x max = 2.4 m/s Page 3
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Last name : First name : Student id : TA : Problem 2 (25 points) Finally you are leaving Minneapolis to get a few days of Spring Break, but your car breaks down in the middle of nowhere. A pickup truck weighing 5000 lbs comes along and agrees to tow your car, which weighs 2500 lbs, to the nearest town. The driver of the truck attaches his rope to your car at an angle of 20 o to the horizontal. He tells you that his rope has a strength of 500 lbs. He plans to take 15 seconds to tow your car at a constant acceleration from rest in a straight line along the flat road until he reaches the minimum speed limit of 40 miles/hour (18 m/s). Can the driver carry out his plan? You assume that rolling friction behaves like kinetic friction, and the coefficient of rolling friction between your tires and the road is at most 0.10 and that you can neglect drag. FR FN F f Write the forces. Fg F r = F r cosθî + F r sinθ ĵ F g = mg ĵ F f = µ k F N î F N = F N ĵ a = v max î t Balance the forces in the vertical to determine the normal force. ma y = 0 = F r sinθ + F N mg F N = mg F r sinθ Evaluate the forces in the horizontal to determine the necessary rope force. m v max t No. The rope will break. m v max t = F r cosθ µ k [mg F r sinθ] + µ k mg = F r [cosθ + µ k sinθ] F r = v max t g mg k cosθ + µ k sinθ = 0.228mg = 571 lb Page 5
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Multiple Choice Questions (5 points each) Question 1 : A toy rocket with a mass of 0.15 kg is propelled upward by burning a fuel pellet. The fuel is totally burned out when the rocket reaches a height of 2.0 m, at which point the rocket is moving at 12 m/s. How much work was done by the rocket s engine? A) 2.94 J B) 3.60 J C) 7.86 J D) 10.8 J E) 13.7 J The correct answer is E. Question 2 : If a car s engine can produce 400 HP (300 kw), what is the maximum speed it can attain given the force of drag? Assume that the density of air is ρ = 1.3 kg/m 3, the drag cofficient is C D = 0.4, and the cross-sectional area of the car is A = 2.4 m 2. A) 86 m/s B) 65 m/s C) 78 m/s D) 690 m/s E) 99 m/s The correct answer is C. Question 3 : A simple pendulum of length 2.00 m is made with a mass of 2.00 kg. The mass has a speed of 3.00 m/s when the pendulum is 30.0 o above its lowest position. What is the maximum angle away from the lowest position the pendulum will reach? A) 74.2 o B) 63.2 o C) 50.5 o D) 83.1 o E) 67.4 o The correct answer is C. Page 7 (OVER)
Question 4 : A car drives over a hilltop that has a radius of curvature 120 m at the top of the hill. At what speed would the car be traveling when it tires just barely lose contact with the road when the car is at the top of the hill? A) 45.5 m/s B) 34.3 m/s C) 41.8 m/s D) 22.2 m/s E) 27.6 m/s The correct answer is B. Question 5 : Which of the potential energy versus position plots best represents the situation of an inclined plane with a spring at the bottom? (A) (B) (C) (D) The correct answer is A. (E) Page 8