Recent content by Ludwig

I'm trying to derive a general work function (provided force and displacement vector-valued functions). Below are my best guesses. Can someone let me know whether these are valid? Rigid-System: ## \sum W = \int \left ( \sum \vec{F}(t)\cdot \vec{r}\,'(t) \right ) dt ## Deformable-system...

So, does that mean I would evaluate this like so? ## \int (\sum \vec{F} \cdot \frac{d\vec{r}}{dt})dt ## I.e., evaluate the dot product of the the force and derivative of curve vectors, then integrate with respect to t.

My physics text gives the following as a general definition of work done by a varying force on a system: ## \sum W = \int (\sum \vec{F}) \cdot d \vec{r} ## Unfortunately, I haven't the faintest idea how to evaluate this. I know how to evaluate an integral with respect to some parameter, but not...

Chain rule. Right. Well that clarifies a lot! So the magnitudes end up being: ##v=\omega r## ##a=\omega^{2}r,## right? Excellent. This is a useful result! Thanks for indicating the way out of my confusion.

Yes.

Even with the correct arguments, won't the magnitudes of the vectors still be equal? I.e., ##\sqrt{(\pm rCos(\omega t))^{2}+(\pm rSin(\omega t))^{2}}=\sqrt{(- rSin(\omega t))^{2}+(\pm rCos(\omega t))^{2}} = r##, right? I'm sure that there is something I'm missing, but I don't know what.

The position function for a particle moving on a circle (with constant speed) is: ## \vec{r}(t)=\left \langle r\,sin(t), \, r\,cos(t) \right \rangle ## Taking the first and second derivatives, ## \vec{v}(t)=\left \langle -r\,cos(t), \, r\,sin(t) \right \rangle ## ## \vec{a}(t)=\left \langle...