Editing Centripetal force (section)

==== Calculus derivation ====
In two dimensions, the position vector <math>\textbf{r}</math>, which has magnitude (length) <math>r</math> and directed at an angle <math>\theta</math> above the x-axis, can be expressed in [[Cartesian coordinates]] using the [[unit vectors]] <math alt="x-hat">\hat\mathbf x</math> and <math alt="y-hat">\hat\mathbf y</math>:<ref>
{{cite book
| title = Vectors in physics and engineering
| author = A. V. Durrant
| publisher = CRC Press
| year = 1996
| isbn = 978-0-412-62710-1
| page = 103
| url = https://books.google.com/books?id=cuMLGAO-ii0C&pg=PA103
}}</ref>
<math display="block"> \textbf{r} = r \cos(\theta) \hat\mathbf x + r \sin(\theta) \hat\mathbf y. </math>

The assumption of [[uniform circular motion]] requires three things:
# The object moves only on a circle.
# The radius of the circle <math>r</math> does not change in time.
# The object moves with constant [[angular velocity]] <math>\omega</math> around the circle. Therefore, <math>\theta = \omega t</math> where <math>t</math> is time.

The [[velocity]] <math>\textbf{v}</math> and [[acceleration]] <math>\textbf{a}</math> of the motion are the first and second derivatives of position with respect to time:

<math display="block"> \textbf{r} = r \cos(\omega t) \hat\mathbf x + r \sin(\omega t) \hat\mathbf y, </math>
<math display="block" qid="Q11465"> \textbf{v} = \dot{\textbf{r}} = - r \omega \sin(\omega t) \hat\mathbf x + r \omega \cos(\omega t) \hat\mathbf y,  </math>
<math display="block" qid=Q11376> \textbf{a} = \ddot{\textbf{r}} = - \omega^2 (r \cos(\omega t) \hat\mathbf x + r \sin(\omega t) \hat\mathbf y). </math>

The term in parentheses is the original expression of <math>\textbf{r}</math> in [[Cartesian coordinates]]. Consequently,
<math display="block"> \textbf{a} = - \omega^2 \textbf{r}. </math>
The negative sign 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.