Angular Momentum Conservation of Angular Momentum

Transcription

1 Lecture 22 Chapter 12 Physics I Angular Momentum Conservation of Angular Momentum Course website:

2 IN THIS CHAPTER, you will continue discussing rotational dynamics Today we are going to discuss: Chapter 12: Angular Momentum: Section Rotational Newton s 2 nd Law (general form): Section Conservation of Angular Momentum: Section 12.11

3 Now we can write Torque as a vector product F r rf sin Torque is a turning force (the rotational equivalent of force). Axis of rotation Now, with the vector product r notation F we can rewrite torque as Torque direction out of page (right hand rule)

4 ConcepTest Figure Skater A student gives a quick push to a puck that can rotate in a horizontal circle on a frictionless table. After the push has ended, the puck s angular speed A) Steadily increases B) Increases for awhile, then holds steady C) Holds steady D) Decreases for awhile, then holds steady A torque causes angular acceleration which leads to changes of the angular velocity. With no torque, the angular velocity stays the same. K rot I

5 Angular velocity as a vector A more general description of rotational motion requires us to replace the scalars ω and τ with the vector quantities and The magnitude of the angular velocity vector is ω. The angular velocity vector points along the axis of rotation in the direction given by the right-hand rule as illustrated.

6 Angular Momentum We will introduce angular momentum of A point mass m A rigid object

7 For translational motion we needed the concepts of force, F linear momentum, p mass, m For rotational motion we needed the concepts of torque, angular momentum, L moment of inertia, I Angular momentum is the rotational equivalent of linear momentum p mv L?

8 x O O Angular Momentum of a single particle z r r L r p m y p Suppose we have a particle with -linear momentum -positioned at r Then, by definition: Angular momentum of a particle about point O is L r p L rpsin p Carefull: Let s calculate angular momentum of m about point O L r p since r p, rpsin Thus, angular momentum of m 0 0 so 0, sin 0 L but 0 O L O Angular Momentum is not an intrinsic property of a particle. It depends on a choice of origin So, never forget to indicate which origin is being used

9 Example Angular momentum (about the origin) of an object of mass m dropped from rest. (The shortest distance between the origin and the line of motion)

10 ConcepTest A car of mass 1000 kg drives away from a traffic light h=10 m high, as shown below, at a constant speed of v=10 m/s. What is the angular momentum of the car with respect to the light? Traffic light/car A) B) C) 100,000( kˆ) 10,000( kˆ) 100,000 iˆ kgm kgm kgm 2 2 s 2 s s z x h y r v L r p mv( rsin )( kˆ) mvh( kˆ ) 100,000( k ˆ )

11 Angular Momentum of a rigid body For the rotation of a symmetrical object about the symmetry axis, the angular momentum and the angular velocity are related by (without a proof) L I L I I moment of inertia of a body L points towards L I L I

12 Two definitions of Angular Momentum Summary Single particle L r p L r p Rigid symmetrical body L I L

13 Rotational N. 2 nd law Let s rewrite our rotational Newton s 2 nd Law in terms of angular momentum: I I d d ( I ) dl dl (We use the angular momentum expression for a rigid body but it can also be shown for a point mass. See the end of the presentation) dl Rotational N. 2nd law written in terms of L. Torque causes the particle s angular momentum to change

14 Translational vs- Rotational N. 2 nd law Translational N.2 nd law F F ma dp Rotational N.2 nd law I dl End of the class

15 Example Angular momentum (about the origin) of an object of mass m dropped from rest (cont.). (cont.)

16 dl Conservation of Angular Momentum Angular momentum is an important concept because, under certain conditions, it is conserved. In Ch11 we derived the linear momentum conservation, where we showed that internal forces cancel each other out and cannot change the total linear momentum. Similar, torques produced by the internal forces cancel each other out and cannot change the total angular momentum. So, only external torques are left to play the game. int ernal 0 dl ext ext 0, If the net external torque on an object is zero, then the total angular momentum is conserved. If L const then dl 0, so

17 Example Figure Skater s Jump Angular Momentum Conservation helps to solve many problems I1 large -small 1 I 2 2 small -large Angular Momentum stays constant throughout the whole jump Flight: leg and hands are in to make large 1 2 L 1 L 2 L I I 1 1 I 2 2 I I For a rigid body ( 2 1 2) 1 Launch: leg and hands are out to make I large Landing: leg and hands are out to dump large

18 ConcepTest 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: Figure Skater A) the same B) larger because she s rotating faster C) smaller because her rotational inertia is smaller 11KE rot = 2I 2 = 2(I ) = L (used L = I ). Because L is conserved, larger means larger KE rot. The extra energy comes from the work she does on her arms.122 K rot 1 2 I

19 Example Bullet strikes cylinder edge A bullet of mass m moving with velocity v strikes and becomes embedded at the edge of a cylinder of mass M and radius R. 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?

20 Thank you See you on Monday

21 L r p Rotational N. 2 nd law Let s find relationship between angular momentum and torque for a point particle: Read if only if you want dl dl dr p dp r mv r F v dp N. 2 nd law F p mv dl Rotational N. 2nd law written in terms of L. Torque causes the particle s angular momentum to change