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Catching up from Yesterday In the two cases shown below identical ladders are leaning against frictionless walls. In which case is the force of friction between the ladder and the ground bigger? A) Case 1 B) Case 2 Case 1 C) Same Case 2 Mechanics Lecture 18, Slide 1 CheckPoint In the two cases shown below identical ladders are leaning against frictionless walls. In which case is the force of friction between the ladder and the ground bigger? A) Case 1 B) Case 2 C) Same Case 1 A) Because the bottom of the ladder is further away from the wall. B) The angle is steeper, which means there is more normal force and thus more friction. C) Both have same mass. Mechanics Lecture 18, Slide 2 Case 2 CheckPoint Suppose you hang one end of a beam from the ceiling by a rope and the bottom of the beam rests on a frictionless sheet of ice. The center of mass of the beam is marked with an black spot. Which of the following configurations best represents the equilibrium condition of this setup? A) B) C) Mechanics Lecture 18, Slide 3 CheckPoint Which of the following configurations best represents the equilibrium condition of this setup? A) B) C) Demo If the tension has any horizontal component, the beam will accelerate in the horizontal direction. Objects tend to attempt to minimize their potential energy as much as possible. In Case C, the center of mass is lowest. Mechanics Lecture 18, Slide 4 Stability & Potential Energy CM Demos footprint footprint Mechanics Lecture 18, Slide 5 Physics 211 Lecture 19 Today’s Concepts: Angular Momentum Mechanics Lecture 19, Slide 6 Linear and Rotation Mechanics Lecture 19, Slide 7 New Cast of Characters with Old Theme Remember Mechanics Lecture 19, Slide 8 Angular Momentum We have shown that for a system of particles FEXT Momentum is conserved if dp dt FEXT 0 What is the rotational version of this? The rotational analogue of force F is torque r F Define the rotational analogue of momentum p to be Angular Momentum: Lrp For a symmetric solid object L I Mechanics Lecture 19, Slide 9 Torque & Angular Momentum EXT dL dt where Lrp L I In the absence of external torques and EXT ext r FEXT dL 0 dt Total angular momentum is conserved Mechanics Lecture 19, Slide 10 Example – Disk dropped on Disk top,initial o Demo top, final bottom, final f bottom,initial 0 Friction is an “internal force” because it is “internal” to the “system”. Friction is between two parts of the “system”. Friction may change the Kinetic Energy. But friction, being an internal force, does not change the Angular Momentum (nor the linear momentum in linear situations). Mechanics Lecture 19, Slide 11 Example – Disk dropped on Disk top,initial o top, final bottom, final f bottom,initial 0 1 Linitial MR 2o 0 2 1 MR 2o MR 2 f 2 1 1 2 L final MR f MR 2 f 2 2 1 f 0 2 Mechanics Lecture 19, Slide 12 What about Kinetic Energy? top,initial o top, final bottom, final f bottom,initial 0 2 1 2 L K I 2 2I K before L2 2 I before same K after L2 2 I after x2 Mechanics Lecture 19, Slide 13 CheckPoint In both cases shown below a solid disk of mass M, radius R, and initial angular velocity o is dropped onto an initially stationary second disk having the same radius. In Case 2 the mass of the bottom disk is twice as big as in Case 1. If there are no external torques acting on either system, in which case is the final kinetic energy of the system bigger? A) Case 1 B) Case 2 C) Same top,initial o top,initial o M M M Same initial L bottom,initial 0 bottom,initial 0 Case 1 Case 2 2M Mechanics Lecture 19, Slide 14 CheckPoint In which case is the final kinetic energy of the system bigger? A) Case 1 B) Case 2 C) Same M 2M M M Case 1 A) L is equal but bigger mass in case 2 Case 2 L2 K 2I Demo: wheel rim drop B) Case 2 has greater mass C) Conservation of angular momentum Mechanics Lecture 19, Slide 15 Your Comments Sort of . . . Work is done by friction so KE is not conserved. Okay; now what does that mean? Find the final KE in both cases. True but what about the final KE? That’s for omega but what about the final KE? True but what about the final KE? True but what about the ???? final KE? ???? This doesn’t mean anything. Mechanics Lecture 19, Slide 16 Point Particle Moving in a Straight Line L R p axis R1 1 p L1 R1 p R1 p sin( 1 ) pD R2 R3 2 p L2 R2 p R2 p sin( 2 ) pD D 3 p L3 R3 p R3 p sin( 3 ) pD Mechanics Lecture 19, Slide 17 L mvD pD L R p axis D p Demo: train on bike wheel Direction given by right hand rule (out of the page in this case) Mechanics Lecture 19, Slide 18 Playground Example Before After M Top View R m v LBefore mvR LAfter 1 I ( MR 2 mR2 ) 2 Disk LBefore LAfter Kid v 1 R 1 M 2m Mechanics Lecture 19, Slide 19 Algebra Details LBefore mvR w= LAfter mvR 1 I ( MR 2 mR2 ) 2 1 ( MR 2 + mR 2 ) 2 w= mv 1 ( MR + mR ) 2 w= LBefore LAfter mv 1 ( M + m)R 2 Disk Kid v 1 R 1 M 2m Mechanics Lecture 19, Slide 20 CheckPoint The magnitude of the angular momentum of a freely rotating disk around its center is L. You toss a heavy block onto the disk along the direction shown. Friction acts between the disk and the block so that eventually the block is at rest on the disk and rotates with it. What is the magnitude of the final angular momentum of the disk-block system: Instead of a block, imagine it’s a kid hopping onto a merry go round A) > L B) L C) < L Top View The block provides no angular momentum since its perpendicular distance from the axis of rotation is 0. No external forces act on this systems, so angular momentum is conserved. Mechanics Lecture 19, Slide 21 CheckPoint What is the magnitude of the final angular momentum of the diskblock system: A) > L B) L C) < L Top View The block provides no angular momentum since its perpendicular distance from the axis of rotation is 0. No external forces act on this systems, so angular momentum is conserved. Mechanics Lecture 19, Slide 22 Your Comments Will it be conserved in the next CheckPoint? Good! Altho’ I wouldn’t call it an external force. Good! The block’s perpendicular distance from the axis of rotation is zero. . But the block is NOT at rest. What is the final angular momentum? What does that mean for the final angular momentum? What about the Zero!>!>! How can that be? next CheckPoint? okay Tell me more. What about the block? Tell me more. !!!!! Mechanics Lecture 19, Slide 23 ACT The magnitude of the angular momentum of a freely rotating disk around its center is L. You toss a heavy block onto the disk along the direction shown. Friction acts between the disk and the block so that eventually the block is at rest on the disk and rotates with it. Is the total angular momentum of the disk-block system conserved during this? A) Yes B) No Top View Friction decreases the angular momentum of the disk But the TOTAL angular momentum of the block+disk is CONSTANT! Mechanics Lecture 19, Slide 24 CheckPoint The magnitude of the angular momentum of a freely rotating disk around its center is L. You toss a heavy block onto the disk along the direction shown. Friction acts between the disk and the block so that eventually the block is at rest on the disk and rotates with it. What is the magnitude of the final angular momentum of the disk-block system: A) > L B) L C) < L Top View Instead of a block, imagine it’s a kid hopping onto a merry go round Mechanics Lecture 19, Slide 25 CheckPoint What is the magnitude of the final angular momentum of the disk-block system: A) > L B) L C) < L Top View A) Since the block has angular momentum to begin with, that is added into L. B) L is conserved so Linitial=Lfinal C) The block adds momentum going in the opposite direction, so the total momentum is smaller than the momentum of the disk. Mechanics Lecture 19, Slide 26 Your Comments Bravo! Bravo! What about the block? What about the block? Why bigger? Angular momentum has a direction connected to it. What about the angular momentum of the block? Any better by now? No, not this time. Good. Okay but tell me more. Smaller than what? In the previous CheckPoint, the lever arm was zero. Tell me more. Mechanics Lecture 19, Slide 27 ACT A student holding a heavy ball sits on the outer edge a merry go round which is initially rotating counterclockwise. Which way should she throw the ball so that she stops the rotation? A) To her left B) To her right C) Radially outward C A B top view: initial final Mechanics Lecture 19, Slide 28 ACT A student is riding on the outside edge of a merry-go-round rotating about a frictionless pivot. She holds a heavy ball at rest in her hand. If she releases the ball, the angular velocity of the merry-go-round will: A) Increase B) Decrease C) Stay the same 1 Mechanics Lecture 19, Slide 29 Mechanics Lecture 19, Slide 30 m2 I disk 1 2 m1 R 2 m1 0 Linitial I disko Mechanics Lecture 19, Slide 31 m2 I disk 1 m1 R 2 2 m1 o 1 K initial I disko2 2 Mechanics Lecture 19, Slide 32 m2 m1 0 Linitial I disk 0 f L final ( I disk I rod ) f L final Linitial I rod 1 m2 L2 12 Solve for f Mechanics Lecture 19, Slide 33 Lfinal Linitial Mechanics Lecture 19, Slide 34 M f K final 1 ( I disk I rod ) 2f 2 Mechanics Lecture 19, Slide 35 Just like Faverage P t for linear momentum M f Lrod I rod ( f 0 ) average L t for angular momentum average I rod ( f 0 ) t Mechanics Lecture 19, Slide 36 Conservation Ideas Internal forces do not change the momentum. Internal torques do not change the angular momentum. However, internal forces or internal torques can still do work and, thus, change the Kinetic Energy. Mechanics Lecture 19, Slide 37 Heads Up! for next preLecture Mechanics Lecture 19, Slide 38