2. Physics 151: Lecture 24, Pg 1 Physics 151: Lecture 24 Today’s Agenda l Topics çAngular MomentumCh. 11.3-5 çMore fun demos !!! çGyroscopes Ch. 11.6

3. Physics 151: Lecture 24, Pg 2 Angular Momentum: Definitions & Derivations l We have shown that for a system of particles Momentum is conserved if See text: 11.3 With angular momentum L = r x p p=mv l Now we also know, Angular momentum is conserved if  EXT = 0 Animation

4. Physics 151: Lecture 24, Pg 3 What does it mean? l where and l In the absence of external torques Total angular momentum is conserved See text: 11.5 I  Also, remember that L of a rigid body about a fixed axis is: Analogue of p = mv !!

5. Physics 151: Lecture 24, Pg 4 Angular momentum of a rigid body about a fixed axis: In general, for an object rotating about a fixed (z) axis we can write L Z = I  The direction of L Z is given by the right hand rule (same as  ). We will omit the ” Z ” subscript for simplicity, and write L = I   z See text: 11.4

6. Physics 151: Lecture 24, Pg 5 Demonstration of Conservation of Angular Momentum l Figure Skating : AA z BB z Arm I A < I B  A >  B L A = L B

7. Physics 151: Lecture 24, Pg 6 Lecture 23, ACT 2 Angular momentum l Two different spinning disks have the same angular momentum, but disk 1 has more kinetic energy than disk 2. çWhich one has the biggest moment of inertia ? (a) disk 1 (b) disk 2 (c) not enough info I 1 < I 2  disk 2  disk 1 K 1 > K 2

8. Physics 151: Lecture 24, Pg 7 Example: Two Disks A disk of mass M and radius R rotates around the z axis with angular velocity  0. A second identical disk, initially not rotating, is dropped on top of the first. There is friction between the disks, and eventually they rotate together with angular velocity  F. What is  F ? 00 z FF z

9. Physics 151: Lecture 24, Pg 8 Example: Two Disks l First realize that there are no external torques acting on the two- disk system. çAngular momentum will be conserved ! l Initially, the total angular momentum is due only to the disk on the bottom: 00 z 2 1

10. Physics 151: Lecture 24, Pg 9 Example: Two Disks l First realize that there are no external torques acting on the two- disk system. çAngular momentum will be conserved ! l Finally, the total angular momentum is due to both disks spinning: FF z 2 1

11. Physics 151: Lecture 24, Pg 10 Example: Two Disks l Since L INI = L FIN 00 z FF z L INI L FIN

12. Physics 151: Lecture 24, Pg 11 Example: Two Disks l Now let’s use conservation of energy principle: 00 z FF z E INI E FIN 1/2 I  0 2 = 1/2 (I + I)  F 2  F 2 = 1/2  0 2  F =  0 / 2 1/2 E INI = E FIN

13. Physics 151: Lecture 24, Pg 12 l So we got a different answers ! Example: Two Disks  F ’ =  0 / 2 1/2 Conservation of energy ! Conservation of momentum ! Which one is correct ?  F =  0 / 2 Because this is an inelastic collision, since E is not conserved (friction) !

14. Physics 151: Lecture 24, Pg 13 Lecture 25, Act 2 A mass m=0.1kg is attached to a cord passing through a small hole in a frictionless, horizontal surface as in the Figure. The mass is initially orbiting with speed  i = 5rad/s in a circle of radius r i = 0.2m. The cord is then slowly pulled from below, and the radius decreases to r = 0.1m. How much work is done moving the mas from r i to r ? (A) 0.15 J(B) 0 J(C) - 0.15 J riri ii animation

15. Physics 151: Lecture 24, Pg 14 Lecture 25, Act 3 l A particle whose mass is 2 kg moves in the xy plane with a constant speed of 3 m/s along the direction r = i + j. What is its angular momentum (in kg · m2/s) relative to the origin? a. 0 k b. 6 (2) 1/2 k c. –6 (2) 1/2 k d. 6 k e. –6 k

16. Physics 151: Lecture 24, Pg 15 Example: Bullet hitting stick l A uniform stick of mass M and length D is pivoted at the center. A bullet of mass m is shot through the stick at a point halfway between the pivot and the end. The initial speed of the bullet is v 1, and the final speed is v 2.  What is the angular speed  F of the stick after the collision? (Ignore gravity) v1v1 v2v2 M FF beforeafter m D D/4 See text: Ex. 11.11

17. Physics 151: Lecture 24, Pg 16 Angular Momentum Conservation l A freely moving particle has a definite angular momentum about any given axis. l If no torques are acting on the particle, its angular momentum will be conserved. L l In the example below, the direction of L is along the z axis, and its magnitude is given by L Z = pd = mvd. y x v d m See text: 11.5

18. Physics 151: Lecture 24, Pg 17 Example: Bullet hitting stick... l Conserve angular momentum around the pivot (z) axis! l The total angular momentum before the collision is due only to the bullet (since the stick is not rotating yet). v1v1 D M before D/4m See text: Ex. 11.11

19. Physics 151: Lecture 24, Pg 18 Example: Bullet hitting stick... l Conserve angular momentum around the pivot (z) axis! l The total angular momentum after the collision has contributions from both the bullet and the stick.  where I is the moment of inertia of the stick about the pivot. v2v2 FF after D/4 See text: Ex. 11.11

20. Physics 151: Lecture 24, Pg 19 Example: Bullet hitting stick... l Set L BEFORE = L AFTER using v1v1 v2v2 M FF beforeafter m D D/4 See text: Ex. 11.11

21. Physics 151: Lecture 24, Pg 20 Example: Throwing ball from stool A student sits on a stool which is free to rotate. The moment of inertia of the student plus the stool is I. She throws a heavy ball of mass M with speed v such that its velocity vector passes a distance d from the axis of rotation.  What is the angular speed  F of the student-stool system after she throws the ball ? top view: before after d v M I I FF See example 13-9 See text: Ex. 11.11

22. Physics 151: Lecture 24, Pg 21 Example: Throwing ball from stool... l Conserve angular momentum (since there are no external torques acting on the student-stool system): çL BEFORE = 0  L AFTER = 0 = I  F - Mvd top view: before after d v M I I FF See example 13-9 See text: Ex. 11.11

23. Physics 151: Lecture 24, Pg 22 Gyroscopic Motion: l Suppose you have a spinning gyroscope in the configuration shown below: l If the left support is removed, what will happen ??  pivot support g See text: 11.6

24. Physics 151: Lecture 24, Pg 23 Gyroscopic Motion... l Suppose you have a spinning gyroscope in the configuration shown below: l If the left support is removed, what will happen ? çThe gyroscope does not fall down !  pivot g See text: 11.6

25. Physics 151: Lecture 24, Pg 24 Gyroscopic Motion... precesses l... instead it precesses around its pivot axis !  pivot See text: 11.6 l This rather odd phenomenon can be easily understood using the simple relation between torque and angular momentum we derived in a previous lecture.

26. Physics 151: Lecture 24, Pg 25 Gyroscopic Motion... The magnitude of the torque about the pivot is  = mgd. l The direction of this torque at the instant shown is out of the page (using the right hand rule). çThe change in angular momentum at the instant shown must also be out of the page!  L pivot d mg See text: 11.6

27. Physics 151: Lecture 24, Pg 26 Gyroscopic Motion... l Consider a view looking down on the gyroscope.  The magnitude of the change in angular momentum in a time dt is dL = Ld . çSo where  is the “precession frequency” top view L L(t) L L(t+dt) LdLLdL d  pivot See text: 11.6

28. Physics 151: Lecture 24, Pg 27 Gyroscopic Motion... l So In this example  = mgd and L = I  :  L The direction of precession is given by applying the right hand rule to find the direction of  and hence of dL/dt.  L pivot d mg  See text: 11.6

29. Physics 151: Lecture 24, Pg 28 Lecture 24, Act 1 Statics l Suppose you have a gyroscope that is supported on a gymbals such that it is free to move in all angles, but the rotation axes all go through the center of mass. As pictured, looking down from the top, which way will the gyroscope precess? (a) clockwise (b) counterclockwise (c) it won’t precess 

30. Physics 151: Lecture 24, Pg 29 Lecture 24, Act 1 Statics Remember that  /L. So what is  ?  = r x F r in this case is zero. Why? Thus  is zero. It will not precess. At All. Even if you move the base. This is how you make a direction finder for an airplane. Answer (c) it won’t precess 

31. Physics 151: Lecture 24, Pg 30 Recap of today’s lecture l Chapter 11, çConservation of Angular Momentum çSpinning Demos