# Rope tangent angle over pully given position of offset load

• Wayland Bugg
In summary, the conversation discusses the challenge of symbolically resolving the angle of a rope suspending a load between two pulleys. The load is not intended to move horizontally, only vertically. The length of rope let out over the pulleys is a known factor, but the conversation explores the possibility of solving for this angle symbolically rather than using CAD software and excel. Various approaches and equations are suggested and discussed, including one involving the radius of the pulley and the vertical and horizontal distances from the center of the pulley to the attachment point. The conversation also touches on the use of dynamic geometric modeling software to solve this problem.
Wayland Bugg
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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Wayland Bugg said:
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. :)

LCKurtz said:
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.

drphysica said:
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.

Wayland Bugg said:
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 $\theta = \frac{\pi}{2} - \arctan{(\frac{V}{H})} - \arcsin{ ( \frac{R}{\sqrt{V^2 + H^2} })}$.

Stephen Tashi said:
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?

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"?

Stephen Tashi said:
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 isn't 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. that's how I divided it up into little chewable understandable pieces, but I could be crazy!

## 1. What is the formula for calculating the rope tangent angle over a pulley given the position of an offset load?

The formula for calculating the rope tangent angle over a pulley is tanθ = (F × d) / (W - F), where θ is the tangent angle, F is the force of the offset load, d is the distance from the load to the pulley, and W is the weight of the load.

## 2. How is the rope tangent angle affected by the position of the offset load?

The position of the offset load directly affects the rope tangent angle over the pulley. The closer the load is to the pulley, the larger the rope tangent angle will be. Similarly, the farther away the load is from the pulley, the smaller the rope tangent angle will be.

## 3. Can the rope tangent angle be negative?

No, the rope tangent angle cannot be negative. It is a measure of the angle formed between the rope and the horizontal line, so it will always be a positive value.

## 4. How does the weight of the load affect the rope tangent angle?

The weight of the load has a direct impact on the rope tangent angle. As the weight of the load increases, the rope tangent angle also increases. This is because a heavier load exerts a greater force on the rope, causing it to bend at a larger angle over the pulley.

## 5. Is the length of the rope a factor in calculating the rope tangent angle?

Yes, the length of the rope does play a role in calculating the rope tangent angle. However, it only affects the distance from the load to the pulley (d) in the formula. The longer the rope, the greater the distance from the load to the pulley, resulting in a smaller tangent angle.

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