Crossed Spring Equilibrium

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117.Crossed Spring Equilibrium

+10

Two identical rigid bars, each of length

2a

, are joined by a frictionless pivot through their midpoints, free to rotate independently in a horizontal plane. The bars rest in equilibrium with the angle between them equal to

60°

(so the supplementary angle on the other side of the pivot is

120°

). Spring 1 connects the two bar endpoints whose chord subtends the

60°

angle at the pivot; spring 2 connects the two bar endpoints whose chord subtends the

120°

angle. The springs lie in the same horizontal plane as the bars. What is the ratio

T_1/T_2

of the tension in spring 1 to the tension in spring 2?

$Two thin rigid bars crossing at a single pivot $O$ at their common midpoint, drawn in the plane of the page. Bar 1 is horizontal with endpoints labeled $A$ on the right and $C$ on the left. Bar 2 is rotated $60°$ counterclockwise from bar 1, with endpoint $B$ in the upper right (on the same side of the pivot as $A$) and endpoint $D$ in the lower left (on the same side as $C$). The pivot is marked with a small filled dot and labeled $O$. The angle $\angle AOB = 60°$ is indicated with a small arc near the pivot. Spring 1 is drawn as a zigzag connecting $A$ to $B$ across the $60°$ gap; spring 2 is a zigzag connecting $A$ to $D$, spanning the $120°$ gap on the opposite side of bar 2. Label the springs as "spring 1" and "spring 2".$

\sqrt{3}/3

1/2

1

\sqrt{3}

EThe ratio cannot be determined without the spring constants.

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