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{excerpt}The acceleration directed toward the center of rotation that results from the change in direction (not magnitude) of the velocity when an object is in circular motion.{excerpt}

h3. Form of the Acceleration

We can derive the form of the acceleration required to produce [uniform circular motion|Uniform Circular Motion].  We begin by writing the position of an object undergoing [uniform circular motion|Uniform Circular Motion] as a function of time.  We can, with complete generality, choose a coordinate system that has its origin at the center of the circle and which aligns the x-y plane with the plane of the circle.  Further, we can choose to look at the motion from the side which makes it clockwise.  Finally, we can set our clocks such that _t_ = 0 occurs when the object intersects the positive x-axis.  With these choices, the form of the position as a function of time will be:
{latex}\begin{large}\[ \vec{r} = R\cos\left(\frac{vt}{R}\right)\hat{x} - R \sin\left(\frac{vt}{R}\right)\hat{y}\]\end{large}{latex}
where _v_ is the speed of the motion and _R_ is the radius, both of which are constants for [uniform circular motion|Uniform Circular Motion].

We can now obtain the acceleration by performing two derivatives with respect to time.  The first
derivative gives the velocity:
{latex}\begin{large}\[ \vec{v} =\frac{d\vec{r}}{dt} = -v\sin\left(\frac{vt}{R}\right) \hat{x}
- v \cos\left(\frac{vt}{R}\right) \hat{y} \]\end{large}{latex}
and the second gives the acceleration:
{latex}\begin{large}\[ \vec{a} = \frac{d^{2}\vec{r}}{dt^{2}}
= -\frac{v^{2}}{R}\cos\left(\frac{vt}{R}\right) + \frac{v^{2}}{R} \sin\left(\frac{vt}{R}\right)\]
\end{large}{latex}
Comparing the expression for the acceleration with our original expression for the position, we see that:
{latex}\begin{large}\[ \vec{a} = -\frac{v^{2}}{R} \hat{r}\]\end{large}{latex}
Comparing the expressions found for _r_, _v_ and _a_ leads to the following picture:

!uniformcirc.png!

Note that the acceleration points _at all times_ toward the center of the circle (which is the reason for the name _centripetal_) and is perpendicular to the velocity.  The acceleration has magnitude:
{panel:title=magnitude of the centripetal acceleration}
{latex}\begin{large}\[ a = \frac{v^{2}}{r} \] \end{large}{latex}
{panel}
{note}It is important to note that the equation relates the magnitude of the velocity to the magnitude of the acceleration. Both the acceleration and the velocity will be constantly changing directions during the motion.
{note}

h3. Centripetal Acceleration in Angular Variables

Using a regular _x_ and _y_ coordinate system is awkward when describing circular motion. It is simpler to instead describe the motion in terms of an *[angular position]*. Suppose that we describe the position of the object executing circular motion by giving its angle θ measured counterclockwise from the x-axis. It will turn out to be most useful to measure the angle in [radians]. This angular position will be constantly changing as the object moves around the circle. Thus, we can define its time derivative, which will be called the *[angular velocity]* ω:
{panel:title=angular velocity}
{latex}\begin{large}\[ \omega = \frac{d\theta}{dt} \] \end{large}{latex}
{panel}
For the case of uniform circular motion, this angluar velocity will have a constant magnitude equal to *2* *π* radians per period *T*. Thus:
{panel:title=uniform circular motion}
{latex}\begin{large}\[ |\omega| = \frac{2\pi}{T} \] \end{large}{latex}
{panel}
We can now find a relationship between the magnitude of the linear velocity as the particle goes around the circle and the angular velocity. The _x_ and _y_ components of the linear velocity were given above. By using the Pythagorean theorem and the fact that sin ^2{^}θ + cos ^2{^}θ = 1, you can show from those expressions that
{latex}\begin{large}\[ v = \frac{2\pi r}{T} \]\end{large}{latex}
{note}This expression is also clearly true since 2 π _r_ is the distance of the circle's circumference and _T_ is the time taken to move around 1 circumference of the circle.
{note}
Thus, we have the relationship:
{latex}\begin{large}\[ v = r |\omega| \] \end{large}{latex}
This relationship means that we can write the centripetal acceleration in another way:
{panel:title =magnitude of the centripetal acceleration}
{latex}\begin{large}\[ a = \omega^{2} r \] \end{large}{latex}
{panel}