# Rope tangent angle over pully given position of offset load

## Main Question or Discussion Point

I am trying to symbolically resolve the angle of rope that suspends a load (rectangle) between two pulleys given the length of rope that is let out over the pulleys. See attached image.

the load is not intended to move horizontally, only vertically. so you can imagine each pulley must have an equal length of line pulled or let out between them.

I can solve for this for any number of points for known lengths of rope with the assistance of CAD software and excel to generate a polynomial curve, but I would like to solve this only in terms of rope let out over the pulleys symbolically to achieve modularity and increased accuracy.

I am not sure this can be done, so I am seeking some insight and advice here.

Thank you

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LCKurtz
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I am trying to symbolically resolve the angle of rope that suspends a load (rectangle) between two pulleys given the length of rope that is let out over the pulleys. See attached image.

the load is not intended to move horizontally, only vertically. so you can imagine each pulley must have an equal length of line pulled or let out between them.

I can solve for this for any number of points for known lengths of rope with the assistance of CAD software and excel to generate a polynomial curve, but I would like to solve this only in terms of rope let out over the pulleys symbolically to achieve modularity and increased accuracy.

I am not sure this can be done, so I am seeking some insight and advice here.

Thank you
I don't see why you would assume each pulley has an equal length of line to the rectangle when your two pulleys don't seem to be at the same position vertically.

Can you be more specific on what you trying to solve because it sounds more complicated than it might actually be. :)

I don't see why you would assume each pulley has an equal length of line to the rectangle when your two pulleys don't seem to be at the same position vertically.
You are right they are not at the same height or distance from the load. I misspoke when trying to articulate a visual. But a visual is all intended with that statement. If what I am trying to do were possible, one would only have to enter the vertical and horizontal distances from the center of the pulley to the load attachment point (for example) and could find the angle for any pulley/rope combo.

Can you be more specific on what you trying to solve because it sounds more complicated than it might actually be. :)
I'd like the angle so I can calculate the vertical and horizontal force components for any given position.

If it helps, you could imagine a winch on the other side of the pulley.

Stephen Tashi
one would only have to enter the vertical and horizontal distances from the center of the pulley to the load attachment point (for example) and could find the angle for any pulley/rope combo.
You'd also need to enter the radius R of the pulley.
V = vertical height center of pulley above level attachment point
H = horizontal distance of center of pulley to vertical line through attachment point

I think the angle in radians is $\theta = \frac{\pi}{2} - \arctan{(\frac{V}{H})} - \arcsin{ ( \frac{R}{\sqrt{V^2 + H^2} })}$.

You'd also need to enter the radius R of the pulley.
V = vertical height center of pulley above level attachment point
H = horizontal distance of center of pulley to vertical line through attachment point

I think the angle in radians is $\theta = \frac{\pi}{2} - \arctan{(\frac{V}{H})} - \arcsin{ ( \frac{R}{\sqrt{V^2 + H^2} })}$.
Thank you for you response! I think that is closer than what I have gotten so far. At least it is another approach. I will work on seeing if this can be adapted to define the angle based only on the amount of line let out at the winch.

I attached another approach I have thought about earlier

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im not sure this can be solved for in terms of line let out at the winch. is this a problem for dynamic geometric modeling software? does anyone have any experience using such tools that could suggest one with a shorter learning curve?

Stephen Tashi
Is the circle in the upper left of the picture a winch or is it just a pulley?
Which distance in the picture is "line let out of the winch"?

Is the circle in the upper left of the picture a winch or is it just a pulley?
Which distance in the picture is "line let out of the winch"?
thats just a pulley.

the line let out of the winch isnt directly represented in that picture, but if you could think of the line payed out at the winch, it would pass over the pulley.
in the last image I attached, you see the red line. with the load positioned all the way up, this length is known, so we can say this 'base length' + whatever the winch paid out, is the length of the red line. the blue segment is wrapped around the pulley as the load descends. thats how I divided it up into little chewable understandable pieces, but I could be crazy!