- #1

archaic

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- Homework Statement
- Four balls of equal mass lie on a horizontal, frictionless, nonconducting surface. They are connected by a lightweight, nonconducting strings of length ##a## as in the figure. Two have a charge ##+q##.

If we cut the string between the two charges, what would be the maximum speed of the uncharged balls?

- Relevant Equations
- .

The way I imagine it is that the two uncharged balls will move only vertically, while the charged balls will be coming down, and getting away from each other horizontally.

If I focus on just one side, then I notice that ##L## is decreasing from above and below by the same amount (same vertical force component).

I'll fix a reference reference frame on the uncharged balls, and so ##\theta## will be the angle between the vertical and the string between the balls.

I will directly remove second powers of differentials.$$\begin{align*}

a^2&=(L+L'd\theta)^2+(a\sin\theta+d(a\sin\theta))^2\\

a^2&=L^2+2LL'd\theta+(a\sin\theta)^2+2a^2\sin\theta\cos\theta d\theta\\

0&=2(a\cos\theta)L'd\theta+2a^2\sin\theta\cos\theta d\theta\\

L'&=-a\sin\theta

\end{align*}$$And so the maximum speed is ##v=\frac a2##, since ##L'## accounts for the vertical movement of both charges.

I feel that this is wrong, though, and that I should be using an energy approach ..

Any guidance? Thank you!