Your sketch is difficult to understand. You say there is a "fixed point" at the left (which would be a pivot) plus a pivot at the circle. How can a lever have two pivot points? If the lever rotates about the fixed point at the left, then the dimension Z will compress as the lever rotates, yet you call Z a "fixed dimension" - so clearly I don't understand the configuration. So I'll give two answers - maybe one of them captures your intent:
1. If the fixed pivot is at the left end (where the "X" is), so that Z compresses as the lever rotates, then if you calculate the torque about the left end of the lever and set it to zero you get:
$\displaystyle F_d (x+y) = F_uy$
where F_u is the upward force and F_d is the downward force applied at the right end of the lever.
Rearranging:
$\displaystyle F_d = F_u \frac y {x+y} $
This is independent of the angle, as long as both forces act purely in the vertical direction. Rearrange to find x:
$\displaystyle x = \frac {(F_u-F_d)y}{F_d} $
If x is greater than or equal to this value the lever will rotate clockwise. If less than this value it will rotate counterclockwise.
2. If the fixed pivot is at the point where the upward force acts, and the left end has no forces acting on it and is free to rise, then zero force is required at the right end of the lever in order to rotate it clockwise, and it doesn't matter what the length of "X" is, as long as it's greater than 0.
Perhaps you can clarify what the point "X" is meant to signify, what Z is, and where the upward force comes from as well.
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Last edited by ChipB; Feb 27th 2017 at 01:19 PM.
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