The magnitude of the centripetal acceleration for something with a speed v moving in a circle with radius r is given by:

That seems arbitrary
and might not be immediately obvious. Why is that the equation? Remembering that
acceleration is the change in velocity divided by the change in time, it makes sense
to try to find those two quantities.

#### Change In Velocity

You can start at an
arbitrary point on the circle with an arbitrary velocity. For simplicity, start
at the top of the circle moving clockwise with a speed of v. Noting that since motion is tangent
to the circle and tangent at the top is the positive x direction:

Now…after moving
through some small angle, the velocity will be down and to the right. Doing some basic
geometry and trig, you end up with:

Since the change in velocity
is simply the difference between those two, we find:

#### Change In Time

Noting that speed is just distance traveled divided by time, and the distance
traveled here is just the arc length for our angle, we get:

#### Combining Terms

Combining those, we
get:

That didn’t work. Can
we clean it up at all?

#### Simplifying

Since we specified a
small angle, we can approximate all terms involving our angle when it is small:

Those give us:

We have a unit vector and negative sign in there, but the
magnitude is what we wanted at the start. What’s the other stuff mean?

Remembering that we assumed the starting point was the top of the circle, this
means that the acceleration is perpendicular to the initial velocity and
pointed towards the center of the circle (negative y direction). If you think
about what it takes to keep something moving in a circle (it might be easiest
to think about putting a ball on the end of a string and spinning it), it makes
sense that in general, the force (and thus the acceleration) will point towards
the center of the circle. Thus, you have the acceleration from the formula at the beginning, and it points towards the center of the
circle.

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