ecture 20 Chapter 11 Physics I 04.14.2014 Conservation of Angular momentum Course website: http://faculty.uml.edu/andriy_danylov/teaching/physicsi ecture Capture: http://echo360.uml.edu/danylov2013/physics1spring.html
Exam III Info Exam III Friday April 25, 9:00-9:50am, OH 150. Exam III covers Chapters 9-11 Same format as Exam II Prior Examples of Exam III posted Ch. 9: inear Momentum (no section 10) Ch. 10: Rotational Motion (no section 10) Ch. 11: Angular Momentum; General Rotation (no sections 7-9) Exam Review Session TBA
Outline Chapter 11 Conservation of Angular Momentum Kepler s 2nd aw revisited
Two definitions of Angular Momentum Single particle y O r r p m r p p is perpendicular to the plane of motion created by r and p x Rigid symmetrical body I
Total angular momentum of a system Rotational N.2 nd law Torque causes angular momentum to change d dt net It is similar to translational N. 2 nd law Net Torque of a system Translational N.2 nd law F F ma dp dt Rotational N.2 nd law I d dt
Conservation of Angular Momentum Angular momentum is an important concept because, under certain conditions, it is conserved. Rotational N.2nd law If ext net 0, d dt then If the net external torque on an object is ero, then the total angular momentum is conserved. net d dt const 0
Example: Bullet strikes cylinder edge Application of Conservation of angular momentum A bullet of mass m moving with velocity v strikes and becomes embedded at theedgeofacylinderofmassmandradiusr 0. The cylinder, initially at rest, begins to rotate about its symmetry axis, which remains fixed in position. Assuming no frictional torque, what is the angular velocity of the cylinder after this collision? Is kinetic energy conserved?
Example: Bullet strikes a cylinder
Bicycle wheel/turntable as a demo of Angular Momentum Conservation
Bicycle wheel/ang. Mom. Conservation initial final w x y x Angular Momentum Conservation ( comp): fin w AD 0 AD in 0 w in fin I AD AD AD I w w w clockwise AD ( )w I AD I counterclockwise
ConcepTest 1 Spinning Bicycle Wheel You are holding a spinning bicycle wheel while standing on a stationary turntable. If you suddenly flip the wheel over so that it is spinning in the opposite direction, the turntable will: 1) remain stationary 2) start to spin in the same direction as before flipping 3) start to spin in the same direction as after flipping in fin me table me table 2 2 The total angular momentum of the system is upward, and it is conserved. So if the wheel has downward, you and the table must have +2 upward.
Angular Momentum Conservation. Examples I 1 -small 1 large I small 2 -large 2 For a rigid body 1 2 I I 1 1 I 2 2 ( I I 2 1 2) Flight: leg and hands are in to make large 1 aunch: leg and hands are out to make I large anding: leg and hands are out to dump large
Bicycle wheel precession. A bicycle wheel is spun up to high speed and is suspended from the ceiling by a wire attached to one end of its axle. Now mg try to tilt the axle downward. You expect the wheel to go down (and it would if it weren t rotating), but it unexpectedly swerves to the left and starts rotating! et s explain that. mg
wire F T x r y mg Bicycle wheel precession. fin in d et s calculate torque on the wheel r mg mg produces torque in +y direction apply Rotational N.2nd law: d dt fin in ( rmgdt) ˆj d in dt d dt Torgue changes direction of angular momentum. So rotates. Now, since I then angular momentum rotates with angular velocity It is called precession Top view of the precession rmg( ˆ) j Notice, angular momentum isn t conserved. There is an external torque produced by mg. So in +y direction
ConcepTest 2 A figure skater spins with her arms extended. When she pulls in her arms, she reduces her rotational inertia and spins faster so that her angular momentum is conserved. Compared to her initial rotational kinetic energy, her rotational kinetic energy after she pulls in her arms must be: 1KE rot = 2I 2 = 2(I ) = 2 (used = I ). Because is conserved, larger means larger KE rot. The extra energy comes from the work she does on her arms.11figure Skater A) the same B) larger because she s rotating faster C) smaller because her rotational inertia is smaller
Conservation of Angular Momentum. Kepler s second law Kepler s second law states that each planet moves so that a line from the Sun to the planet sweeps out equal areas in equal times. Use conservation of angular momentum to show this.
et s find ang. moment. of the Earth (treat the Earth as a point): r p rpsin mrvsin Kepler s second law origin Next step. et s find area swept by r in dt: da 1 2 r( vdt)sin( ) 1 2 rvdt sin da mrvsin da dt dt 2m 2m et s show that is conserved: d dt net r So if is constant, da/dt is a constant Kepler s 2 nd law is a consequence of conservation of ang. momentum! ext since net 0, then r F g ( rˆ) F g v, vdt const p 0 m dl vdt sin( )
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