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## Centripetal Acceleration

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