A graphical approach to understanding the form of the centripetal acceleration. |
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 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:
\begin{large}\[ \Delta\vec{v} \rightarrow - v(\Delta \theta)\hat{r}\]\end{large} |
In the infinitesimal limit, this equation becomes:
\begin{large}\[ \frac{d\vec{v}}{dt} = - v \frac{d\theta}{dt} \hat{r}\]\end{large} |
Using the fact that for uniform circular motion,
\begin{large}\[ \frac{d\theta}{dt} = \frac{v}{r}\]\end{large} |
we arrive at the form of the centripetal acceleration:
\begin{large}\[ \vec{a} = \frac{d\vec{v}}{dt}= -\frac{v^{2}}{r} \hat{r}\]\end{large} |
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:
\begin{large}\[ \frac{d\vec{L}}{dt} = L\frac{d\phi}{dt}\hat{\phi} \]\end{large} |
where the derivative of φ with respect to time is the angular frequency of precession, usually written as Ω:
\begin{large}\[ \frac{d\phi}{dt} = \Omega\]\end{large} |
With that substitution, we have arrived at the fundamental relation for gyroscopes:
\begin{large}\[ \frac{d\vec{L}}{dt} = L\Omega\hat{\phi} \]\end{large} |