FINAL EXAM, PHYSICS 1403-001, December 11, 2009, Dr. Charles W. Myles INSTRUCTIONS: Please read ALL of these before doing anything else!!! 1. 2. 3. 4. PLEASE put your name on every sheet of paper you use and write on ONE SIDE of the paper only!! PLEASE DO NOT write on the exam sheets, there will not be room! PLEASE show all work, writing the essential steps in the problem solution. Write appropriate formulas first, then put in numbers. Partial credit will be LIBERAL, provided that essential work is shown. Organized, logical, easy to follow work will receive more credit than disorganized work. The setup (PHYSICS) of a problem will count more heavily than the math of working it out. PLEASE write neatly. Before handing in solutions, PLEASE: a) number the pages & put the pages in numerical order, b) put the problem solutions in numerical order, & c) clearly mark your final answers. If I can’t read or find your answer, you can't expect me to give it the credit it deserves. I HAVE 145 EXAMS TO GRADE!!! PLEASE HELP ME GRADE THEM EFFICIENTLY BY FOLLOWING THESE SIMPLE INSTRUCTIONS!!! FAILURE TO FOLLOW THEM MAY RESULT IN A LOWER GRADE!! Three 8.5’’ x 11’’ pieces of paper with anything on them & a calculator are allowed. Problem 1 (Conceptual) is REQUIRED! NOTE: The following is DIFFERENT for this Exam!! You MUST] ALSO WORK either Problem 2 or Problem 3 (both on rotational dynamics)!! Also work any three (3) of the remaining problems for five (5) problems total on the exam. Each problem is equally weighted & worth 20 points, for a total of 100 points on this exam. Answer all parts of the ones you choose! 1. MANDATORY QUESTIONS (Physical Principles)!! Use WORDS to explain the Physical Principles asked for. Answer in complete English sentences. Keep symbols/formulas to a minimum. Use WORDS! If you use a formula, DEFINE ALL SYMBOLS. I prefer that you try to answer using NO symbols/formulas. (Note: Answers with ONLY symbols, without WORD explanations get ZERO credit!). a. State Newton’s 1st Law. How many objects at a time does it apply to? State Newton’s 3rd Law. How many objects at a time does it apply to? b. State the Work-Energy Principle. c. State the Principle of Conservation of Mechanical Energy. What types of forces must be the only ones present in order for this principle to hold? d. State Newton’s 2nd Law in terms of Momentum (The form that Newton first used! ∑F = ma will get e. f. g. ZERO credit!! Despite my saying this on Exam III, many of you used ∑F = ma anyway! WHY?) State the Law of Conservation of Linear Momentum. Under what conditions is momentum conserved? State Newton’s 2nd Law for Rotational Motion. (The answer ∑F = ma will get ZERO credit!). A box of mass m sits statically (not moving!) on a flat horizontal table. The figure is the box’s free body diagram when the only forces acting on it are the normal force FN upward & it’s weight mg downward. A calculation of the normal force FN in this case will obviously give a result that FN is equal in size & opposite in direction to the box’s weight mg. Which Newton’s Law of Motion is used to find this result? 2. NOTE!!! Work EITHER Problem 2 OR Problem 3! The figure is a compound wheel. The inner wheel radius is r1 = 0.41 m. The outer wheel radius is r2 = 0.76 m. (They are bolted together, & so rotate rigidly together). Two equal forces, F1 = F2 = 70 N, are applied as shown. F1 is tangent to the inner wheel in the direction shown (down in figure). F2 is applied at the top of the outer wheel, at angle 60° with respect to the dashed vertical line in the figure (30° with respect to the tangent to the large wheel there.). The moment of inertia of this compound wheel is I = 3.22 kg m2. (Hints: This means that I = (½)MR2 should NOT be used! The wheel mass is NOT needed! The gravitational acceleration g, is irrelevant to this problem!! ) a. Calculate the torque, τ1 due to F1 & the torque, τ2 due to F2. Be sure to include the signs of these in your answers. (Hint: Sign convention: A torque which, if it acted alone, would cause a counterclockwise rotation is positive & a torque which, if it acted alone, would cause a clockwise rotation is negative.) Calculate the net torque τnet on the compound wheel. Calculate the wheel’s angular acceleration α. What physical principle did you use to do this? Calculate the tangential acceleration atan of a point on the outer rim (at r2). The wheel starts from rest at time t = 0. At time t = 8 s later, calculate: c. It’s angular velocity ω. d. The translational velocity v of a point on the rim (at radius r2). e. It’s rotational kinetic energy & it’s angular momentum. b. F1 Ff F2 NOTE!!! Work EITHER Problem 2 OR Problem 3! 3. See figure. A solid SPHERE is on a track. It has radius R = 0.25 m & mass M = 18 kg. It’s moment of inertia is I = (2/5)MR2. It starts from rest at point A. It rolls without slipping to the right & down the track, past point B & then moves to point C. It continues to the right past C, but that isn’t part of this problem! The height difference between A & B, y1 in the figure, is unknown. The height difference between B & C is y2 = 8 m. When it reaches B, it’s angular velocity (about an axis through it’s center of mass) is ω = 20 rad/s. A a. Calculate the sphere’s rotational kinetic energy at B. b. Calculate the translational velocity V of the sphere’s center of | mass at B. Use this to calculate it’s translational kinetic | energy there. Use these results plus the kinetic energy from y1 part a to calculate the sphere’s total kinetic energy at B. | B c. Calculate the sphere’s initial gravitational potential energy | at A. Also calculate it’s initial height y1 at A. d. Calculate the sphere’s potential energy when it reaches C. Calculate it’s total kinetic energy there. e. What Physical Principle did you use to find the sphere’s potential energy at point A and it’s total kinetic energy at C? See figures. A helicopter, mass M = 7.8 103 kg, moves UP with acceleration a = 2.1 m/s2. It lifts an object of mass m = 2.4 103 kg, which is connected to it’s bottom by a massless cable. The upward force, FP, on the helicopter exerted by the air on the rotors is unknown. So is the tension in the cable, FT, which, of course, acts down on the helicopter & up on the object. The free body diagrams for the helicopter & object are in the figure. (Hints: The motion is vertical, but the acceleration is obviously NOT g down, but a up! If it were g downward, the object would be in free fall & FT would be zero! Also, FT can’t possibly be = mg, or a would be zero!). a. Apply Newton’s 2nd Law to the helicopter & to the object to find the two equations needed to calculate FP & FT. Writing these without substituting in numbers, will receive more credit than writing them with numbers substituted in! b. Using the equations from part a, calculate FT & FP. Assume that all forces are constant, so that the upward acceleration a = 2.1 m/s2 is constant. Once the cable is tight (time t = 0), the upward velocity of the system is v0 = 7 m/s. Consider the system after it has moved up a distance y = 14 m. c. Calculate the upward velocity v of the system at that point. (Hints: Use a kinematic equation e. y2 | a Mg FT FT from Ch. 2! Remember that the acceleration is a NOT g!) d. | FP NOTE: Work any three (3) of Problems 4, 5, 6 and 7! 5. C Calculate the initial kinetic energy of the system (when v0 = 7 m/s) & it’s final kinetic energy (when it is moving with the velocity v found in part c). Calculate the Net Work done in this process (Hint: Rather than calculate the work done due to each force separately & adding them, it’s easier to use the Work-Kinetic Energy Principle! Use the results of part d!). mg NOTE: Work any three (3) of Problems 4, 5, 6 and 7! 5. See figure. A mass m = 7.0 kg slides with initial velocity v = 3.8 m/s across a horizontal, frictionless surface until it encounters a spring with constant k = 280 N/m. It comes instantaneously to rest after compressing the spring a distance x. (Hint: In the following, PLEASE try to take square roots properly!) k = 280 N/m Initially, v = 3.8 m/s a. b. c. Finally, x = ?, v = 0 Calculate the initial kinetic energy of the mass (left hand figure). Calculate the elastic potential energy of the spring-mass system at the point where the mass is instantaneously at rest (right hand figure). Also, calculate the distance x the spring is compressed there. What physical principle did you use to find these? Calculate the elastic potential energy of the spring-mass system & the distance the spring has been compressed when the mass’s speed has slowed down to 2.1 m/s. (This is not shown in the figures! This occurs after the mass touches the spring, but before it has come to rest in as in the right hand figure!) d. e. The mass in the left hand figure was given its initial velocity by sliding it from rest down a frictionless inclined plane from a height h. (This is not shown in the figures! This happened sometime before the situation shown in the left hand figure!) Calculate the potential energy the mass had at the top of the inclined plane & the height h from which the mass started. What physical principle did you use to find these results? Calculate the FORCE (magnitude & direction) the spring exerts on the mass at the final position (right hand figure) where the mass has stopped moving. NOTE: Work any three (3) of Problems 4, 5, 6 and 7! 6. See figures. A mass m1 = 8 kg, is on a horizontal, frictionless surface. In Fig. 1, it moves to the right at velocity v1 = 9 m/s. It collides with a mass m2 = 15 kg which is initially at rest. The collision is inelastic, so they stick together, as in Fig. 2 & move to the right with velocity v´. Figure 1 Figure 2 m1 = 8 kg v1 = 9 m/s m2 = 15 kg v2 = 0 stick together! v´ = ? a. Calculate the momentum & kinetic energy of mass m1 before the collision. b. Calculate the momentum of the two stuck together masses & their velocity v2 after the collision. What Physical Principle did you use to find these? c. Calculate the impulse given to m2 by m1 due to this collision. d. If the collision time is Δt = 4.6 × 10-3 s, calculate the average collision force that m1 exerts on m2 in this collision. What Physical Principle did you use to calculate this? e. Calculate the kinetic energy of the two stuck together masses in Fig. 2. (Note: It is physically impossible for kinetic energy to increase in a collision!) Is kinetic energy conserved in this collision? (Is this an elastic collision?) Why or why not?? NOTE: Work any three (3) of Problems 4, 5, 6 and 7! 7. See figure. A stock clerk pushes on a carton on a dolly. He pushes with a force F = 71 N which makes an angle θ = 37° below the horizontal. The mass of the carton + dolly is m = 12 kg. There is a frictional force Ff between the dolly & a the floor, acting in the opposite direction of the motion. The coefficient of kinetic friction between the box & the floor is μk = 0.19. To solve this problem, θ F use the x & y axes shown. a. Sketch the free body diagram of the carton + dolly, properly labeling all forces. Don’t forget the weight & the normal force, not shown in the figure. Ff Calculate the x & y components of the pushing force F. b. Calculate the weight of the carton plus the dolly & the normal force FN the floor exerts on them. Is FN equal (& oppositely directed) to the weight? If so, why? If not, why not? Which Newton’s Law of Motion is used answer this question? Calculate the frictional force Ff that the dolly experiences as it moves to the right. c. Use Newton’s 2nd Law to find the acceleration a experienced by the carton plus the dolly. What forces cause this acceleration? Assume that all forces are constant, so that the acceleration a from part c is constant. The dolly starts from rest (time t = 0) & moves to the right a distance x = 20 m. d. Calculate the velocity v of the cart at that point. (Hint: Use a kinematic equation from Ch. 2). Use this velocity to calculate the cart’s kinetic energy at that point. e. Calculate the Net Work done on the cart after it starts until it has moved the distance x = 20 m. (Hint: Rather than calculate the work done due to each force separately & adding them, it’s easier to use the Work-Kinetic Energy Principle! Use the results of part d!). 8. BONUS!! This semester, I did some demonstrations. Sometimes, I also briefly shared with you some interesting (to me!) facts about the history of physics (which are NOT in the book!). For example, I told you a lot about Newton’s life & his dealings with fellow scientists. If you were in class on demonstration days and/or on days when I mentioned some history, you should be able to answer the following questions, which are worth five (5) Bonus points each added to your Final Exam grade. If you “cut” class on such days you will (probably) not be able to answer them. a. If you were present for any of the demonstrations, write a few short, complete sentences telling about ONE of the them. BE SPECIFIC! Tell me what demonstration I did AND what physical principle I was trying to illustrate. b. If you were present for any of the history discussions, write a few short, complete, sentences telling about ONE of the historical facts I shared with you. BE SPECIFIC! Tell me which historical person in physics I was discussing AND what I told you about him. Have a good holiday season and good luck in the future!