2. Chapter 11 Angular Momentum

3. Schedule 2+ Weeks left! 10- AprCh 11: Angular Mom. Ch 11: Angular Mom.+ Chapt 12.Ch 12: Statics 17- AprCh 12: StaticsCh 15: Oscillations 24- AprCh 15: OscillationsNo Class 1- MayFINAL EXAMNo Class

4. So…  This week –Angular Momentum –Statics (?) –Quiz on Friday  Next Week –Statics & Oscillations  Following Week –One Class  Then … the final examination –VERY CLOSE!

5. The Exam  You had seen most of it.  It is not graded yet.  You probably did ok! –Or not.  Let’s just make a few comments on the exam

6. As shown in the figure, a bullet of mass m and speed v passes completely through a pendulum bob of mass M. The bullet emerges with a speed of v/2. The pendulum bob is suspended by a stiff rod of length and negligible mass. What is the minimum value of v such that the pendulum bob will barely swing through a complete vertical circle? Have you seen this one before??

7. A rod of length 30.0 cm has linear mass density (mass-per- length) that varies with position. It is given by = 50.0 g/m + 20.0 x g/m2, where x is the distance from one end, measured in meters. (a) What is the mass of the rod? (b) How far from the x = 0 end is its center of mass? This one is new.

8. This one is in your notes So is the other one

9. The Vector Product (Brief Review) There are instances where the product of two vectors is another vector A X B The vector product of two vectors is also called the cross product You can learn more about this in the textbook and in the notes on the website.

10. The Vector Product and Torque The torque vector lies in a direction perpendicular to the plane formed by the position vector and the force vector  = r x F The torque is the vector (or cross) product of the position vector and the force vector

11. The Vector Product Defined Given two vectors, A and B The vector (cross) product of A and B is defined as a third vector, C C is read as “A cross B” The magnitude of C is AB sin   is the angle between A and B

12. More About the Vector Product The quantity AB sin  is equal to the area of the parallelogram formed by A and B The direction of C is perpendicular to the plane formed by A and B The best way to determine this direction is to use the right-hand rule

13. Properties of the Vector Product The vector product is not commutative. The order in which the vectors are multiplied is important To account for order, remember A x B = - B x A If A is parallel to B (  = 0 o or 180 o ), then A x B = 0 Therefore A x A = 0

14. More Properties of the Vector Product If A is perpendicular to B, then |A x B| = AB The vector product obeys the distributive law A x (B + C) = A x B + A x C

15. Final Properties of the Vector Product The derivative of the cross product with respect to some variable such as t is where it is important to preserve the multiplicative order of A and B

16. Vector Products of Unit Vectors

17. Vector Products of Unit Vectors, cont Signs are interchangeable in cross products A x (-B) = - A x B

18. Using Determinants The cross product can be expressed as Expanding the determinants gives

19. Torque Vector Example Given the force  = ?

20. Using the determinant …

21. New Definition Angular Momentum: L p=mv

22. Angular Momentum Consider a particle of mass m located at the vector position r and moving with linear momentum p This is equal to zero because v =(d r /dt) is parallel to p

23. New Rules The torque applied to a mass m is equal to the time rate of change of its angular momentum. The force applied to a mass is still equal to the time rate of change of its linear momentum. From the definition, if there is no torque applied to a mass, its angular momentum will remain constant!

24. So.. no force, no torque, no change in angular momentum! r sin(  )

25. Again ….. the definition of L The instantaneous angular momentum L of a particle relative to the origin O is defined as the cross product of the particle’s instantaneous position vector r and its instantaneous linear momentum p L = r x p

26. Torque and Angular Momentum The torque is related to the angular momentum Similar to the way force is related to linear momentum This is the rotational analog of Newton’s Second Law  and L must be measured about the same origin This is valid for any origin fixed in an inertial frame

27. New Rules … Add the following associations to our previous list

28. More About Angular Momentum The SI units of angular momentum are (kg. m 2 )/ s Both the magnitude and direction of L depend on the choice of origin The magnitude of L = mvr sin   is the angle between p and r The direction of L is perpendicular to the plane formed by r and p

29. Angular Momentum of a Particle Executing Circular Motion about a point. The vector L = r x p is pointed out of the diagram The magnitude is L = mvr sin 90 o = mvr sin 90 o is used since v is perpendicular to r A particle in uniform circular motion has a constant angular momentum about an axis through the center of its path

30. Angular Momentum of a System of Particles The total angular momentum of a system of particles is defined as the vector sum of the angular momenta of the individual particles L tot = L 1 + L 2 + …+ L n =  L i Differentiating with respect to time

31. Angular Momentum of a System of Particles, cont Any torques associated with the internal forces acting in a system of particles are zero Therefore, The net external torque acting on a system about some axis passing through an origin in an inertial frame equals the time rate of change of the total angular momentum of the system about that origin

32. Angular Momentum of a System of Particles, final The resultant torque acting on a system about an axis through the center of mass equals the time rate of change of angular momentum of the system regardless of the motion of the center of mass This applies even if the center of mass is accelerating, provided  and L are evaluated relative to the center of mass

33. Angular Momentum of a Rotating Rigid Object Each particle of the object rotates in the xy plane about the z axis with an angular speed of  The angular momentum of an individual particle is L i = m i r i 2  L and  are directed along the z axis =I i 

34. Angular Momentum of a Rotating Rigid Object, cont To find the angular momentum of the entire object, add the angular momenta of all the individual particles This also gives the rotational form of Newton’s Second Law

35. Angular Momentum of a Rotating Rigid Object, final If a symmetrical object rotates about a fixed axis passing through its center of mass, the vector form holds: L = I  where L is the total angular momentum measured with respect to the axis of rotation

36. Angular Momentum of a Bowling Ball The momentum of inertia of the ball is 2/5MR 2 The angular momentum of the ball is L z = I  The direction of the angular momentum is in the positive z direction

37. Conservation of Angular Momentum The total angular momentum of a system is constant in both magnitude and direction if the resultant external torque acting on the system is zero Net torque = 0 -> means that the system is isolated L tot = constant or L i = L f For a system of particles, L tot =  L n = constant

38. Conservation Law Summary For an isolated system - (1) Conservation of Energy: E i = E f (2) Conservation of Linear Momentum: p i = p f (3) Conservation of Angular Momentum: L i = L f

39. Conservation of Angular Momentum: The Merry-Go-Round The moment of inertia of the system is the moment of inertia of the platform plus the moment of inertia of the person Assume the person can be treated as a particle As the person moves toward the center of the rotating platform, the angular speed will increase To keep L constant

40. Motion of a Top The only external forces acting on the top are the normal force n and the gravitational force M g The direction of the angular momentum L is along the axis of symmetry The right-hand rule indicates that  = r  F = r  M g is in the xy plane

41. Motion of a Top, cont The direction of d L is parallel to that of  in part. The fact that L f = d L + L i indicates that the top precesses about the z axis. The precessional motion is the motion of the symmetry axis about the vertical The precession is usually slow relative to the spinning motion of the top

42. Gyroscope A gyroscope can be used to illustrate precessional motion The gravitational force Mg produces a torque about the pivot, and this torque is perpendicular to the axle The normal force produces no torque

43. Gyroscope, cont The torque results in a change in angular momentum d L in a direction perpendicular to the axle. The axle sweeps out an angle d  in a time interval dt. The direction, not the magnitude, of L is changing The gyroscope experiences precessional motion

44. Precessional Frequency Analyzing the previous vector triangle, the rate at which the axle rotates about the vertical axis can be found  p is the precessional frequency