# Instantaneous angular speed question

1. Jul 21, 2016

### Erenjaeger

1. The problem statement, all variables and given/known data
How do you calculate instantaneous angular speed?

2. Relevant equations
I have been told that it is when delta t approaches 0, so its just the derivative of delta theta over delta t.

3. The attempt at a solution
does it involve tangent line like normal instantaneous velocity ?

2. Jul 21, 2016

### collinsmark

You are on the right track involving derivatives. See below.
Yes, this is true.

(I'm making a few assumptions about $\theta (t)$ not having a discontinuity at the point of t in question. But for simplicity sake, I'll just say, "yes, that's true," which it is for most cases.)
Be careful here. It is not the derivative of $\frac{\Delta \theta}{\Delta t}$. Be careful of your wording there.

Rather, the instantaneous angular velocity $\omega (t)$ is the derivative of $\theta (t)$ with respect to t (not "over $\Delta t$").

Making the stipulations about smooth functions (not having discontinuities and so forth),

$$\lim_{\Delta t \rightarrow 0} \frac{\Delta \theta}{\Delta t} = \frac{d}{dt} \{ \theta (t) \} = \omega (t)$$

That question has a simple answer, but I'll let you ponder that. If you objectively know the behavior a variable or function, say $\theta(t)$ which changes as a function of time, what is the instantaneous rate of change of that function? What is the derivative of a function?