Expressing start of a vector using a point on the vector

[yotama9]

Hi guys. I hope this is the right thread for my question. The question is for my project and I can assure you it's not my homework. I know the solution to my problem isn't that complicated but I just lost my self with the problem and I don't have anyone to talk with about my problem.

Anyhow, here is my problem. I need to find the forces between two rods. Each rod is constructed from 3 particles (O--O--O) and the potential between the rod is given by:
$$U = f(\bar r) \cdot g(\bar u_1 - \bar u_2)$$.


Where

$$\bar r = \bar r_{22} - r_{21}$$

That is, $$r$$ is the distance between the two middle particles. $$\bar u_i$$ is give by

$$\bar u_i = \bar r_{i3} - \bar r_{i1}$$

which means it is the vector that shows the direction of the rod. I need to find forces between the particles:

$$F = \frac{dU}{dr}$$.

My problem is with the derivative of $$\bar u_1 - \bar u_2$$ with respect to $$\bar r$$. I tried several ways but I alway end up with $$\frac{du_1}{d \bar r} - \frac{du_2}{d \bar r}$$.

What have I missed?

Thanks.

 

[HallsofIvy]

Yes, that is what you should end up with. Now what is the real difficulty?

 

[yotama9]

The problem is to express the derivative as a function of the distance between the two vectors.
Thanks

 

[DrRocket]

The problem is to express the derivative as a function of the distance between the two vectors.
Thanks

Which you did. So what is the difficulty ?

 

[yotama9]

I don't know how to compute the derivative. I feel like I'm missing something here

 

[yotama9]

Hi, I'm bumping the problem up again.
I'll try to clarify my problem. I need to find the forces on ball number 1 and ball number 3 of each chain. I know that it should have something with the distance between the balls ($$\vec{r}_1 - \vec{r}_2$$) and the angle ([tex]\theta[\tex]) but I'm not sure what it is.

Thanks.