(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

[[F(tang) meansthe tangential force]]

Consider a bead constrained to move on a tract curved in three-dimensional space, with the bead's position specified by its distance s, measured along the wire, from the origin.

One force on the bead is the normal force N of the wire (which constrains the bead to stay on the wire). If we assume that all other forces (gravity, etc.) are conservative, then their resultant can be derived from a potential energy U. Prove that F(tang) = -dU/ds. This shows that one-dimensional systems of this type can be treated just like linear systems, with x replaced by s and Fx by F(tang).

2. Relevant equations

3. The attempt at a solution

I have before me a solution someone wrote that looks like it could be right, but I can't follow one of the steps. Here is what they have written:

F = N - [tex]\nabla[/tex]U

Where N is the normal force, and F is the net force.

The tangental force will be F(tang) = [tex]\hat{}v[/tex](-[tex]\nabla[/tex]U)

Now consider a small displacement along the wire. Then we should have

ds= ds[tex]\hat{}v[/tex]

***

Then dU = ds[tex]\hat{}v[/tex][tex]\nabla[/tex]U

***

= -F(tang)ds

Therefore you can write

-dU/ds = F(tang)

The part I don't follow is enclosed with asterisks.

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# Homework Help: Proving F(tang) = -dU/ds

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