BELIEVE ME NOT!   A SKEPTIC's GUIDE
Next: Kepler
Up: Circular Motion
Previous: Rate of Change of a Vector
Figure:
Differences between vectors at slightly different times
for a body in uniform circular motion.

From Fig. 10.3 we can see the relationship between
the change in position
and the change in velocity
in a short time interval .
As all three get smaller and smaller,
gets to be more and more exactly in the centripetal direction
(along
)
and its scalar magnitude
will always (from similar triangles) be given by
where I have been careful to write
rather than
since the magnitude of the
radius vector, r, does not change!
Now is a good time to note that, for a tiny sliver of a circle,
there is a vanishingly small difference between
and the actual distance
travelled along the arc, which is given exactly by
.
Thus
If we divide both sides by
and then take the limit
as
,
the approximation becomes arbitrarily good
and we get
We can now combine this with the definitions of acceleration
(
)
and angular velocity
(
)
to give
(after multiplying both sides by v)
.
We need only divide the equation
by
and let
to realize that
.
If we substitute this result into our equation
for the acceleration, it becomes

(10.2) 
which is our familiar result for the centripetal acceleration
in explicitly vectorial form.
Next: Kepler
Up: Circular Motion
Previous: Rate of Change of a Vector
Jess H. Brewer 
Last modified: Sat Nov 14 12:24:50 PST 2015