Angular Momentum

Moments  The moment of a vector at a point is the wedge product.  This is applied to physical variables in rotating systems. Applied to momentum for angular momentum Applied to force for torque  Moments are summed for systems of particles. x1x1 x2x2 x3x3 r A

Shifted Moments Theorem:  If the total momentum is zero, the angular momentum is independent of the point. Select an arbitrary point aSelect an arbitrary point a The angular momentum is unchangedThe angular momentum is unchanged  The equivalent theorem is true for torque. however so

Law of Angular Inertia  The time derivative of angular momentum vector is the net torque vector.  By the law of reaction, all internal torques come in canceling pairs. Only need external torquesOnly need external torques

Rigid Body rr rr  v CM  If the positions of separate masses are fixed compared to the center of mass it is a rigid body.  Rigid body motion can be expressed in terms of the center of mass. Translational motion Rotational motion

Angular Velocity  In a rigid body each point is a fixed distance from the origin. Velocity must be perpendicular to the radius Use wedge product Any two points are fixed x1x1 x2x2 x3x3

Angular Momentum  The angular momentum can be defined in terms of the inertia and angular velocity. Accounts for non-collinear vectors J r p

Inertia Tensor  Rotational inertia is represented by a tensor. Symmetric tensorSymmetric tensor Diagonal elements are moments of inertiaDiagonal elements are moments of inertia Off-diagonal are products of inertiaOff-diagonal are products of inertia

Rotational Energy  The kinetic energy can be expressed for a rotating system. Inertia tensor and angular velocityInertia tensor and angular velocity next rr 