A graphical approach to understanding the form of the centripetal acceleration.
Assumptions
We assume that we have uniform circular motion (motion with a constant radius and a constant speed centered at a fixed point in space).
The Diagram
The picture below illustrates the motion, with coordinates chosen so that the angular position at t = 0 is θ = 0.
To the right of the motion diagram is a vector diagram that shows the change in the velocity vector. The picture motivates the conclusion that if we take a very small Δt, the change in the velocity approaches:
![](/confluence/download/export/latex6278712294534889168.png)
In the infinitesimal limit, this equation becomes:
![](/confluence/download/export/latex6370563305623327768.png)
Using the fact that for uniform circular motion,
![](/confluence/download/export/latex9754724119098601601.png)
we arrive at the form of the centripetal acceleration:
![](/confluence/download/export/latex11384592413330572388.png)
Analogy with Gyroscopic Precession
Consider a gyroscope precessing. The angular momentum will trace out a circle as shown below.
The similarity to the Δv diagram implies that we can write:
![](/confluence/download/export/latex7581147794258673743.png)
where the derivative of φ with respect to time is the angular frequency of precession, usually written as Ω:
![](/confluence/download/export/latex16669302567650868840.png)
Fundamental Relationship for Gyroscopes
With that substitution, we have arrived at the fundamental relation for gyroscopes:
![](/confluence/download/export/latex11375033501299567525.png)