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Circular motion

In uniform circular motion, that is moving with constant speed along a circular path, a particle experiences an acceleration resulting from the change of the direction of the velocity vector, while its magnitude remains constant. The derivative of the location of a point on a curve with respect to time, i.e. its velocity, turns out to be always exactly tangential to the curve, respectively orthogonal to the radius in this point. Since in uniform motion the velocity in the tangential direction does not change, the acceleration must be in radial direction, pointing to the center of the circle. This acceleration constantly changes the direction of the velocity to be tangent in the neighboring point, thereby rotating the velocity vector along the circle.

Expressing centripetal acceleration vector in polar components, where r {\displaystyle \mathbf {r} } 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 a c = − v 2 | r | ⋅ r | r |. {\displaystyle \mathbf {a_{c}} =-{\frac {v^{2}}{|\mathbf {r} |}}\cdot {\frac {\mathbf {r} }{|\mathbf {r} |}}\,.}

As usual in rotations, the speed v {\displaystyle v} of a particle may be expressed as an angular speed with respect to a point at the distance r {\displaystyle r} as ω = v r. {\displaystyle \omega ={\frac {v}{r}}.}

Thus a c = − ω 2 r. {\displaystyle \mathbf {a_{c}} =-\omega ^{2}\mathbf {r} \,.}

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 r {\displaystyle r} for the centripetal acceleration. The tangential component is given by the angular acceleration α {\displaystyle \alpha }, i.e., the rate of change α = ω ˙ {\displaystyle \alpha ={\dot {\omega }}} of the angular speed ω {\displaystyle \omega } times the radius r {\displaystyle r}. That is, a t = r α. {\displaystyle a_{t}=r\alpha.}

The sign of the tangential component of the acceleration is determined by the sign of the angular acceleration (α {\displaystyle \alpha }), and the tangent is always directed at right angles to the radius vector.