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Uniform circular motion

{{See also|Uniform circular motIn two dimensions, the position vector r {\displaystyle {\textbf {r}}}, which has magnitude (length) r {\displaystyle r} and directed at an angle θ {\displaystyle \theta } above the x-axis, can be expressed in Cartesian coordinates using the unit vectors x ^ {\displaystyle {\hat {\mathbf {x} }}} and y ^ {\displaystyle {\hat {\mathbf {y} }}}: r = r cos (θ) x ^ + r sin (θ) y ^. {\displaystyle {\textbf {r}}=r\cos(\theta){\hat {\mathbf {x} }}+r\sin(\theta){\hat {\mathbf {y} }}.}

The assumption of uniform circular motion requires three things:

The velocity v {\displaystyle {\textbf {v}}} and acceleration a {\displaystyle {\textbf {a}}} of the motion are the first and second derivatives of position with respect to time:

r = r cos (ω t) x ^ + r sin (ω t) y ^, {\displaystyle {\textbf {r}}=r\cos(\omega t){\hat {\mathbf {x} }}+r\sin(\omega t){\hat {\mathbf {y} }},} v = r ˙ = − r ω sin (ω t) x ^ + r ω cos (ω t) y ^, {\displaystyle {\textbf {v}}={\dot {\textbf {r}}}=-r\omega \sin(\omega t){\hat {\mathbf {x} }}+r\omega \cos(\omega t){\hat {\mathbf {y} }},} a = r ¨ = − ω 2 (r cos (ω t) x ^ + r sin (ω t) y ^). {\displaystyle {\textbf {a}}={\ddot {\textbf {r}}}=-\omega ^{2}(r\cos(\omega t){\hat {\mathbf {x} }}+r\sin(\omega t){\hat {\mathbf {y} }}).}

The term in parentheses is the original expression of r {\displaystyle {\textbf {r}}} in Cartesian coordinates. Consequently, a = − ω 2 r. {\displaystyle {\textbf {a}}=-\omega ^{2}{\textbf {r}}.} negative shows that the acceleration is pointed towards the center of the circle (opposite the radius), hence it is called "centripetal" (i.e. "center-seeking"). While objects naturally follow a straight path (due to inertia), this centripetal acceleration describes the circular motion path caused by a centripetal force.