Rigid body dynamics r and velocities v. Select a reference point R and compute the relative position and velocity vectors, The total linear and angular momentum vectors relative to the reference point R are and If R is chosen as the center of mass these equations simplify to To specialize these formulas to a rigid body, assume the particles are rigidly connected to each other so P, i=1...,n are located by the coordinates r and velocities v. Select a reference point R and compute the relative position and velocity vectors, where ω is the angular velocity of the system. The linear momentum and
Angular velocity particle moves around the origin in a circle; but when there is no cross-radial component, it moves in a straight line from the origin. Since radial motion leaves the angle unchanged, only the cross-radial component of linear velocity contributes to angular velocity. The angular velocity "ω" is the rate of change of angle with respect to time, which can be computed from the cross-radial velocity as: Here the cross-radial speed formula_13 is the signed magnitude of formula_11, positive for counter-clockwise motion, negative for clockwise. Taking polar coordinates for the linear velocity v gives magnitude "v" (linear speed) and angle "θ"