Advanced Placement PHYSICS 1 Angular Momentum Student 2014-2015 Angular Momentum and its Conservation What I Absolutely Have to Know to Survive the AP* Exam If you have a problem involving a collision or an explosion, the first approach you should consider is conservation of either momentum or angular momentum. Linear momentum is not conserved if an object that is part of the system is fixed in place by a hinge. Angular momentum can be conserved if the point about which angular momentum is taken is the hinge where external forces are applied. Whenever, you have a collision or an explosion, never assume that mechanical energy is conserved. In a system in which there is both rotation and translation, you must include both rotational and translational kinetic energy in the same conservation of energy expressions. If you are looking for a velocity or angular velocity, always think first of Newton’s Second Law. If you are looking for a velocity or angular velocity, and if there is motion in which potential energy is changing, always first think about energy considerations. Newton’s Second Law is possible if the body is moving in a circular path, but energy is usually the best approach. The temptation is to memorize formulas, but know the relationships and understand the concepts. Key Formulas and Relationships Linear and Angular analogs - variables Linear x Linear Distance (m) Δx v a m F p Angular Rotational distance (radians) Linear Displacement (m) Rotational displacement (radians) Linear Velocity (m/s) Rotational velocity (radians/s) 2 Linear Acceleration (m/s ) Rotational acceleration (radians/s2) Mass (kg) Rotational inertia (kg.m2) Force (N) Torque (N.m or mN) Linear momentum Angular momentum . (kg m/s) (kg.m2/s) θ Δθ ω α Ι τ L AP* is a trademark of the College Entrance Examination Board. The College Entrance Examination Board was not involved in the production of this material. ® Copyright © 2013 National Math + Science Initiative , Inc., Dallas, TX. All rights reserved. Angular Momentum and its Conservation Linear and Angular analogs – kinematics and energy equations Linear Angular Constant Motion x = xo + vt θ = θo + ω t Motion with Constant Acceleration v = vo + at ω = ω o + αt 1 x = (vo + v)t 2 1 x = xo + vot + at 2 2 2 2 vo = v + 2ax 1 θ = (ω o + ω )t 2 1 θ = θ o + ω ot + α t 2 2 2 2 ω o = ω + 2αθ ΣF Fnet a= = m m 2nd Law of Motion Στ τ net α= = I I Linear and Angular Momentum Linear momentum p = mv Angular momentum for a particle L = rmv sin θ for a system rotating about an axis L = Iω ΔL = τΔt Conservation of Linear and Angular Momentum psystem = constant Lsystem = constant ® Copyright © 2013 National Math + Science Initiative , Inc., Dallas, TX. All rights reserved. Angular Momentum and its Conservation Multiple Choice 1. A stunt man is jumping several cars with a motorcycle. While the rider and the motorcycle are in the air, he uses the throttle to cause the rear wheel to spin faster. What happens to the rotation of the rider-motorcycle system when the rear wheel spins faster? a. b. c. d. Front of motorcycle rotates upward Rear of motorcycle rotates upward System does not rotate System rotates to the left 2. The angular velocity of an object increases, ω initial < ω final . Which of the following would explain why the angular momentum would not increase? a. b. c. d. The object’s rotational inertia remains constant. The object had mass added to it. The object’s rotational inertia decreased. The object began to roll down hill. 3. A child on a spinning merry-go-round walks from the center to the outside edge of the merry-go-round. What happens to the angular momentum, angular speed and kinetic energy of the child-merry-go-round system? Angular Momentum a. constant b. constant c. decrease d. increase Angular Speed constant decrease constant increase Kinetic Energy constant decrease decrease increase 4. A 50-kg student runs with an initial speed of 5 m/s along a path tangential to the rim of a 2-m radius merry-go-round. The merry-go-round has a rotational inertia of I = 1000 kg.m2, and is initially at rest. The student jumps on to the merry-goround at its rim. What is the final angular speed of the student - merry-go-round system? a. b. c. d. 1/100 radians/sec 1/20 radians/sec 2/5 radians/sec 5/12 radians/sec ® Copyright © 2013 National Math + Science Initiative , Inc., Dallas, TX. All rights reserved. Angular Momentum and its Conservation 5. A spherical spinning star has a uniform density, mass M, radius R and an angular speed ω1. It begins to collapse and shrinks to ½ its original radius. What will be the ratio of the initial angular speed ω1 to the final angular speed ω2? The rotational inertia for a 2 uniform sphere is MR 2 . 5 ω1 ω2 ω1 b. ω2 ω1 c. ω2 ω1 d. ω2 a. = 0.25 = 0.50 = 1.0 = 2.0 6. A hockey puck of mass M is moving in a straight line at a constant speed v across the ice rink. The puck is a distance x from the center of the rink at its nearest approach to the center. What is the puck’s angular momentum about the center of the rink? a. Iω cos θ b. Mvx c. Mv d. Mvx sin 60 0 7. A pendulum bob is released and swings downward. The pendulum string strikes a peg and begins to wrap around the peg, with the pendulum bob spiraling inward. Neglecting air resistance, what happens to the mechanical energy and angular momentum as the string with the pendulum bob wraps around the pole? Mechanical Energy a. remains constant b. increases c. decreases d. remains constant Angular Momentum conserved conserved conserved decreases 8. Two identical spoked-wheels have a common axis and are initially not touching one another. Wheel # 1 is initially spinning and wheel # 2 is not. When the two wheels are brought into contact, they stick together. Which combination of the total kinetic energy and total angular momentum is true for the two-wheel system? a. b. c. d. Total Kinetic Energy ¼ original value ½ original value value is unchanged value is unchanged Total Angular Momentum value is unchanged ½ original value value is unchanged ¼ original value ® Copyright © 2013 National Math + Science Initiative , Inc., Dallas, TX. All rights reserved. Angular Momentum and its Conservation 1 MR 2 is rotating about a radial 2 axis as shown in the diagram. Its angular momentum is L. What is its kinetic energy? 9. A solid disk of mass M, radius R and rotational inertia L2 MR 2 1 b. MR 2 2 1 c. MR 2ω 2 d. 2MRL a. 10. A student holding two textbooks sits on a spinning lab stool with her arms extended outward. What happens to her rotational kinetic energy as she pulls her arms and books inward? a. b. c. d. remains constant increases decrease doubles 11. A student holding two textbooks sits on a spinning lab stool with her arms extended outward. What happens to her angular momentum if she drops the books? a. remains constant b. increases c. decrease d. doubles ® Copyright © 2013 National Math + Science Initiative , Inc., Dallas, TX. All rights reserved. Angular Momentum and its Conservation Free Response Question 1 An ice skater starts her performance-ending spin with her arms outstretched, rotating at 2.5 rev/second. As she pulls her arms inward, her rotational inertia decreases to 0.70 of its original value. A. What is her final angular speed? B. What is the ratio of her final kinetic energy to initial kinetic energy? C. If the ratio between the final kinetic energy and the initial kinetic energy is less than 1, where did the energy go? If the ratio is greater than 1, where did the extra energy come from? Justify your answer. ® Copyright © 2013 National Math + Science Initiative , Inc., Dallas, TX. All rights reserved. Angular Momentum and its Conservation Question 2 The two wheels shown above are connected by a belt. The belt does not slip as the wheels rotate. Wheel #2 has a radius that is 2.5 times larger than the radius of wheel #1. I A. What would the ratio of 1 be equal to if the both wheels had the same angular I2 momentum? B. What would the ratio of I1 be equal to if the both wheels had the same rotational I2 kinetic energy? ® Copyright © 2013 National Math + Science Initiative , Inc., Dallas, TX. All rights reserved. Angular Momentum and its Conservation Question 3 A metal rod of mass 5.0 kg and length of 2.0 meters is attached to a pivot at the top of 1 the rod. The rotational inertia about this pivot is ML2 . A 0.35-kg rubber ball hits 3 the rod with a speed of 15.0 m/s at 1.5 meters from the pivot and bounces off with a speed of 10.0 m/s in the opposite direction. A. What is the initial angular momentum of the ball-rod system? ® Copyright © 2013 National Math + Science Initiative , Inc., Dallas, TX. All rights reserved. Angular Momentum and its Conservation B. What is the angular speed of the rod immediately after the collision? C. Is this an elastic collision? Justify your answer. ® Copyright © 2013 National Math + Science Initiative , Inc., Dallas, TX. All rights reserved.