- #1

etotheipi

- Homework Statement
- Three balls (A,B,C) are connected in series by two light rods of length ##l##, with B in the middle. All hinges are smooth. The three balls initially lie along a straight line, and an impulse delivers a speed ##v## to the ball A, perpendicular to the rods. Determine the minimum distance between the balls A and C.

- Relevant Equations
- N/A

I defined the angle ##\beta## as the angle from the right horizontal to the ball C, from B, and ##\alpha## as the angle from the left horizontal to the ball A, from B. I also work in the CM frame, which has a velocity downwards of magnitude ##\frac{v}{3}## w.r.t. the lab frame. The positions of the three balls in the CM frame are as follows $$\vec{r}_B = \begin{pmatrix}x\\y\end{pmatrix}$$ $$\vec{r}_A = \vec{r}_B - l\begin{pmatrix}\cos{\alpha}\\\sin{\alpha}\end{pmatrix}$$ $$\vec{r}_C = \vec{r}_B + l\begin{pmatrix}\cos{\beta}\\\sin{\beta}\end{pmatrix}$$From that I conserved momentum and angular momentum in the CM frame, respectively below:$$3m\vec{v}_B + ml\dot{\beta} \begin{pmatrix}-\sin{\beta}\\\cos{\beta}\end{pmatrix} + ml\dot{\alpha} \begin{pmatrix}\sin{\alpha}\\\ -\cos{\alpha}\end{pmatrix} = \vec{0}$$ $$3\vec{L}_B + ml \left[\begin{pmatrix}\cos{\beta} - \cos{\alpha}\\\sin{\beta} - \sin{\alpha} \\0\end{pmatrix} \right] \times \vec{v}_B + ml\vec{r}_B \times \left[ \dot{\beta} \begin{pmatrix}-\sin{\beta}\\\cos{\beta}\\0\end{pmatrix} + \dot{\alpha} \begin{pmatrix}-\sin{\alpha}\\\cos{\alpha}\\0\end{pmatrix} \right] + ml^2(\dot{\alpha} + \dot{\beta})\hat{z} = mlv \hat{z}$$ If we left multiply the COM equation by ##\vec{r}_B## then we can substitute the ##\vec{0}## into the COAM equation to obtain$$\begin{pmatrix}\cos{\beta} - \cos{\alpha}\\ \sin{\beta} - \sin{\alpha}\\0\end{pmatrix} \times \vec{v}_B + l(\dot{\alpha} + \dot{\beta})\hat{z} = v\hat{z}$$But I don't know what to do next. The distance between them is ##s = l\sqrt{2+2\cos{(\alpha - \beta})}##, so it is required to determine the most negative value of ##\cos{(\alpha - \beta)}##. Maybe it will also be necessary to use the centre of mass condition somewhere, which provides an extra constraint:$$3m\vec{r}_B + ml \begin{pmatrix}\cos{\beta} - \cos{\alpha}\\\sin{\beta}-\sin{\alpha}\\0\end{pmatrix} = \vec{0}$$I wondered if anyone could help out, thanks!

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