Optimal Length for Triple-Segment Metal Rod Hanging | Torque Equations

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To determine the optimal length of the third segment of a metal rod bent into three perpendicular segments, the goal is to achieve equilibrium with two segments horizontal when supported by a hook. The torque equations are set up as T1 = L, T2 = L, and T3 = X, where X is the unknown length of the third segment. The challenge lies in visualizing the configuration and solving the torque balance equation. The solution requires ensuring that the sum of the torques around the pivot point equals zero. Properly calculating the length X in terms of L will allow the unit to hang correctly.
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Homework Statement



thin uniform metal rod is bent into three perpendicular segments, two of which have length L . You want to determine what the length of the third segment should be so that the unit will hang with two segments horizontal when it is supported by a hook

u <--hook
L
---------
| L
|
----------------
x

Homework Equations



Torque = 0

The Attempt at a Solution



I tried to do three separate torques, T1 = L T2= L2 and T3 = X but i don't know how to solve
 
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I'm having trouble visualizing this.
 
its a backward C

the top and perpendicular part is length L and the bottom length X which is what we are looking for in terms of L

It looks like ______ <--- L
......|
......| <--- L
......|
________________|
...x...
 

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