Instantaneous angular speed question

[Erenjaeger]

Homework Statement

How do you calculate instantaneous angular speed?[/B]

Homework Equations

I have been told that it is when delta t approaches 0, so its just the derivative of delta theta over delta t.[/B]

The Attempt at a Solution

does it involve tangent line like normal instantaneous velocity ?
[/B]

 

[collinsmark]

Homework Statement

How do you calculate instantaneous angular speed?[/B]

You are on the right track involving derivatives. See below.

Homework Equations

I have been told that it is when delta t approaches 0, [/B]

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.)

so its just the derivative of delta theta over delta t.

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),

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

The Attempt at a Solution

does it involve tangent line like normal instantaneous velocity ?[/B]

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?