Sunday, April 17, 2016

Published 8:05 PM by with 0 comment

Where Does The Formula For Centripetal Acceleration Come From?

I tried deriving the formula

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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