Lecture 20 Chapter 12 Angular Momentum Course website:

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1 Lecture 20 Chapter 12 Angular Momentum Another Law? Am I in a Law school? 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 12.11

3 Example Pulley and mass An object of mass m is hung from a cylindrical pulley of radius R and mass M and released from rest. What is the acceleration of the object?

4 ConcepTest A puck 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

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 We will introduce angular momentum of A point mass m A rigid object L?

8 x Angular Momentum of a single particle O z r L r p m y p Suppose we have a particle with -linear momentum -positioned at r p L r p L rpsin Then, by definition: Angular momentum of a particle about point O is 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 10 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) 1000( kˆ) 1000( kˆ) 1000 iˆ kgm kgm kgm s s s z x h y r v L r p mv( rsin )( kˆ) mvh( kˆ ) 1000( 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 Thank you

17 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