velocity vector along the circle. For a given speed , the magnitude of this geometrically caused acceleration (centripetal acceleration) is inversely proportional to the radius of the circle, and increases as the square of this speed: Note that, for a given angular velocity , the centripetal acceleration is directly proportional to radius . This is due to the dependence of velocity on the radius . Expressing centripetal acceleration vector in polar components, where is a vector from the centre of the circle to the particle with magnitude equal to this distance, and considering the orientation of the acceleration towards the center, yields As usual in rotations, the speed of a particle may be expressed as an angular speed with respect to a point at the distance as Thus This acceleration and the mass of the particle determine the necessary centripetal force, directed toward the centre of the circle, as the net force acting on this particle to keep it in this uniform circular motion. The so-called 'centrifugal force', appearing to act outward on the body, is a so-called pseudo force experienced in the frame of reference of the body in circular motion, due to the body's linear momentum, a vector tangent to the circle of motion. In a nonuniform circular motion, i.e., the speed along the curved path is changing, the acceleration has a non-zero component tangential to the curve, and is not confined to the principal normal, which directs to the center of the osculating circle, that determines the radius for the centripetal acceleration. The tangential component is given by the angular acceleration , i.e., the rate of change of the angular speed times the radius . That is, The sign of the tangential component of the acceleration is determined by the sign of the angular acceleration (), and the tangent is always directed at right angles to the radius vector. Relation to relativity Special relativity The special theory of relativity describes the behavior of objects traveling relative to other objects at speeds approaching that of light in vacuum. Newtonian mechanics is exactly revealed to be an approximation to reality, valid to great accuracy at lower speeds. As the relevant speeds increase toward the speed of light, acceleration no longer follows classical equations. As speeds approach that of light, the acceleration produced by a given force decreases, becoming infinitesimally small as light speed is approached; an object with mass