1 Angular Momentum Chapter 11 © 2012, 2016 A. Dzyubenko © 2004, 2012 Brooks/Cole © 2004, 2012 Brooks/Cole Phys 221

2 Vector Product Given two vectors A and B, the vector product (cross product) A×B is a vector C having a magnitude Θ is the angle between A and B

3 Vector Product, cont The quantity AB sin Θ is equal to the area of the parallelogram formed by A and B The direction of the vector C is perpendicular to the plane formed by A and B This direction is defined by the right-hand rule

4 Some Properties of the Cross Product The vector product is not commutative The order is important! Non-commutative… If A is parallel to B (Θ = 0º or 180º), then A×B = 0 A×A= 0 If A is perpendicular to B, then |A × B | = AB The vector product obeys the distributive law: Θ = 90º

5 Some Properties of the Cross Product, cont It is important to preserve the multiplicative order of A and B The derivative of the cross product with respect to some variable such as t is

6 Unit Vectors form a set of mutually perpendicular unit vectors in a right-handed coordinate system x y z

7 Cross Products of Unit Vectors Signs are interchangeable in cross product: The cross products of the rectangular unit vectors obey the following rules: x y z

8 Determinant Form of Cross Product or The cross product of any two vectors A (A x, A y, A z ) and B (B x, B y, B z ) can be expressed in the following determinant: x y z Trick to use:

9 Vector Product Example Given Find Result

10 Torque Vector Example Given the force and location Find the torque produced

11 Vector Product and Torque The torque vector τ is the cross product of the position vector r and force F The magnitude of the torque τ is Vector τ lies in a direction perpendicular to the plane formed by the position vector r and the applied force F. Along the axis of rotation! φ is the angle between r and F

12 Rotational Dynamics A particle of mass m located at position r, moves with linear momentum p The net force on the particle: Take the cross product on the left side of the equation Add the term ?

13 Angular Momentum Define the instantaneous angular momentum L of a particle relative to the origin O as the cross product of the particle’s instantaneous position vector r and its instantaneous linear momentum p looks similar in form to The torque acting on a particle is equal to the time rate of change of the particle’s angular momentum

14 Angular Momentum, cont Is the rotational analog of Newton’s second law for translational motion Torque causes the angular momentum L to change just as force causes linear momentum p to change Is valid only if Σ τ and L are measured about the same origin The expression is valid for any origin fixed in an inertial frame

15 More About Angular Momentum The SI unit of angular momentum is kg·m 2 /s L is perpendicular to the plane formed by r and p The magnitude and the direction of L depend on the choice of origin The magnitude of L is φ is the angle between r and p L is zero when r is parallel to p ( φ = 0º or 180º) L = mvr when r is perpendicular to p ( φ = 90º)

16 Angular Momentum of a System of Particles: Motivation Angular Momentum of a System of Particles: Motivation The net external force on a system of particles is equal to the time rate of change of the total linear momentum of the system The Newton’s second law for a system of particles Is there a similar statement that can be made for rotational motion?

17 Angular Momentum of a System, cont. Angular Momentum of a System, cont. Differentiate with respect to time: The total angular momentum varies in time according to the net external torque: The total angular momentum of a system of particles is the vector sum of the angular momenta of the individual particles

18 Angular Momentum of a System Relative to the System’s Center of Mass Angular Momentum of a System Relative to the System’s Center of Mass 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 theorem applies even if the center of mass is accelerating, provided τ and L are evaluated relative to the center of mass

19 Angular Momentum of a Rotating Rigid Object Angular Momentum of a Rotating Rigid Object I is the moment of inertia of the object The angular momentum of the whole object is Each particle rotates in the xy plane about the z axis with an angular speed  The magnitude of the angular momentum of a particle of mass m i about z axis is

20 Angular Momentum of a Rotating Rigid Object, cont Angular Momentum of a Rotating Rigid Object, cont If a symmetrical object rotates about a fixed axis passing through its center of mass, you can write in vector form Differentiate with respect to time, noting that I is constant for a rigid object L is the total angular momentum measured with respect to the axis of rotation  is the angular acceleration relative to the axis of rotation

21 QQ A solid sphere and a hollow sphere have the same mass and radius. They are rotating with the same angular speed. The one with the higher angular momentum is (a) the solid sphere (b) the hollow sphere (c) they both have the same angular speed (d) impossible to determine

22 Conservation of Angular Momentum Conservation of Angular Momentum For an isolated system consisting of N particles 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. That is, if the system is isolated

23 Conservation of Angular Momentum, cont If the mass of an isolated system undergoes redistribution in some way, the system’s moment of inertia I changes A change in I for an isolated system requires a change in ω

24 Conservation Laws for an Isolated System Conservation Laws for an Isolated System Energy, linear momentum, and angular momentum of an isolated system all remain constant Manifestations of some certain symmetries of space

25 Angular Momentum as a Fundamental Quantity Angular Momentum as a Fundamental Quantity The concept of angular momentum is also valid on a submicroscopic scale Angular momentum is an intrinsic property of atoms, molecules, and their constituents Fundamental unit of momentum is h is called Planck’s constant s, p, d, f, … electronic orbitals: L=0, 1, 2, 3, … in terms of

26 Reading assignment: Gyroscopes