Chapter 11 Angular Momentum; General Rotation

Introduction Recap from Chapter 10 –Used torque with axis fixed in an inertial frame –Used equivalent of Newton’s Second Law –Rotational kinetic energy as mechanical energy What’s ahead –General rotation can be quite complicated –Our goal is understanding several important new aspects and our limitations –Awareness of limits to what is known; e.g. spin and its technological uses

Our approach Angular momentum provides insight New precision and math: the meaning and use of the cross (vector) product Torque and angular momentum for a particle, a system of particles, rigid bodies Conservation of angular momentum (lab) Combining translation and rotation Comments on more complex systems and rotating frames of reference

Angular Momentum of Objects Rotating about a Fixed Axis Angular momentum as analogue of linear momentum (What could we conclude?) Scalar expressions for angular momentum, the relation of torque and angular momentum, and conservation of angular momentum Examples (demo, sports, weather) Question, QuestionQuestion

Angular Momentum of Objects Rotating about a Fixed Axis… Direction of angular momentum –Right-hand rule –Vectors and pseudovectors (axial vectors) –Directions of angular momentum and angular velocity for a symmetrical object rotating about a symmetry axis –What remains true otherwise? Symmetry of nature corresponding to conservation of angular momentum

Vector Cross Product Math for more precision, generality Torque as a vector The cross product –Definition –Examples –Question, QuestionQuestion The torque vector –QuestionQuestion Examples

Angular momentum of a particle Considering p and L in a simple system of particles and their constancy if Force and Torque are zero A more precise definition of angular momentum Analogue of Newton’s 2 nd Law –Derivation questionquestion –Limitations –QuestionQuestion

Angular momentum & torque for a system of particles Relation between angular momentum and torque –Definitions of total angular momentum, net torque, and external torque –Relationship –Picture –Limitations?

Angular momentum & torque for a rigid object Relation between component of angular momentum along axis of rotation and angular velocity about that axis –Derivation in book Relation between angular momentum and angular velocity when rotation axis is a symmetry axis through CM –Discussion in book

Angular momentum & torque for a rigid object (cont’d) General relation between angular momentum and torque when calculation is done about either (1) the origin of an inertial frame, or (2) CM of system –Previous result since a rigid body is a special case of a system of particles

Angular momentum & torque for a rigid object (cont’d…) From the previous, the relations among the external torque along the rotation axis, the angular momentum along the rotation axis, the moment of inertia about the rotation axis, the angular velocity about the rotation axis, and the angular acceleration about the rotation axis Examples of rotation only (Precession) Examples with translation (Pulleys, Physlet 10.12, more)

More… More complicated examples –Kepler’s Second Law (in text) –Zero torque analogue (in Physlets) Rotating frames of reference –Resulting fictitious forces

the end

What three general conditions could make A x B = 0? (Write examples out of actual vectors as specifically as possible on the whiteboard provided.) back

What is r x F, where r = (1m,.5m, -2m) F = (4N, 2N, -1N) (Use the whiteboard provided.) back

A particle moves with a constant speed in a straight line. How does its angular momentum, calculated about any point not in its path, change in time? Hint: A physics argument will show that the net force on the particle is zero, so the net torque must be zero about any point. Make a mathematical argument (on the whiteboard) Make a diagrammatic argument (on the whiteboard) back

Describe how astronauts floating in space can move their arms to turn upside down? Or turn around to face the opposite direction? (Demonstrate this.) back

Start with L = r x p How is the net torque related to dL/dt? Assume L is about a point at rest in an inertial frame. use whiteboards provided back

Conservation of angular momentum is useful for understanding the motion of one object, whereas conservation of linear momentum is not and is useful for a system of more than one object. Why (in general)? Give an example of its use for one object. back

We learned previously that v = rω, which is true for the diagram given. Which of the following is/are true: 1.v = rω 2.v = ωr 3.v = r x ω 4.v = ω x r back