Torque the vector cross product, p is the particle's linear momentum, and r is the displacement vector from the origin (the origin is assumed to be a fixed location anywhere in space). The time-derivative of this is: This result can easily be proven by splitting the vectors into components and applying the product rule. Now using the definition of force formula_17 (whether or not mass is constant) and the definition of velocity formula_18 The cross product of momentum formula_20 with its associated velocity formula_21 is zero because velocity and momentum are parallel, so the second term vanishes. By definition, torque τ
Angular velocity is formula_5, and the linear velocity is formula_6, so that formula_7. In the general case of a particle moving in the plane, the angular velocity is measured relative to a chosen center point, called the origin. The diagram shows the radius vector r from the origin "O" to the particle "P", with its polar coordinates formula_8. (All variables are functions of time "t".) The particle has linear velocity splitting as formula_9, with the radial component formula_10 parallel to the radius, and the cross-radial (or tangential or circular) component formula_11 perpendicular to the radius. When there is no radial component, the