Shade Sails and the 4:1 Mechanical Advantage of Pulleys

When installing a sail, there are several options for corner tensioning hardware. One option is to use turnbuckles; the other option is to use pulley corners. What are pulley corners? Great question. A pulley corner is a system of two double pulleys. A rope is attached to one pulley and then travels back and forth 4 times before exiting the pulleys. When the loose end of the rope is pulled, it draws the two pulleys together.

Pulley corners are great because they make tensioning so easy! Our pulleys have a 4:1 mechanical advantage, which means that you only have to pull with 1/4 of the force you need to tension the sail. For example, to apply 400 lbs to the sail, you'll only need to pull with a force of 100 lbs.

Additionally, fewer tools are required to put up or take down a sail. So, pulley corners are a great option if your sail will be going up or coming down often.

If you’re feeling adventurous or are curious about how a pulley corner’s 4:1 mechanical advantage works, read on.

Warning: This article contains algebra and trace amounts of trigonometry. Proceed with caution.

Newton's First Law of Motion

In order for a body to be stationary, the net, or total forces in each direction must be zero. For the sake of simplicity, we'll stick with the forces in the x direction.

Looking at the pulley as a whole, we find that the force of the sail is the same as the force of the anchor, as shown below:

Looking at one side of the pulley at a time, we find that the force, or tension, in each rope is 1/4 of the force of the sail, as shown below:

The results are the same for the other side, which is to be expected as the force of the sail and the force of the anchor are the same:

4:1 Mechanical Advantage

We can assume that the tension in the rope is distributed evenly, thus the downward force in the extra rope is the same as the tension in the rest of the rope. Therefore, in order to apply the necessary tensioning force to your sail, you need only pull the rope with 1/4 of that force. This is called a 4:1 mechanical advantage. 

If you've had enough math and you're satisfied with this proof, feel free to stop here!

...but if you're still curious for more...

Something for Nothing?

Pulleys are great! It almost seems like you're getting something for nothing, right?

Wrong.

It is a fundamental law that energy is conserved in an isolated system. It is impossible to put in 1x energy and get 4x energy out! It may seem like that's what happens with pulleys, but that's not the case.

On a straight line, energy, or work, can be calculated as a force over a distance, or:

where Δx is the distance traveled. The energy you put into the system by pulling the rope must be the same as the work done tensioning the sail. Therefore, the distance the rope is pulled is 4x the distance the sail moves, as shown below:

We see that with a 4:1 pulley, you're pulling the rope 4x as far, with 1/4 the force in order to tension the sail with 4x the force, moving it 1/4 as far.

Balancing the forces in the Y direction 

This is your last chance to turn back before the trigonometry hits you. You really don't need this information, but if you're curious what happened to the forces in the Y direction, here you go:

To counteract the downward force of the rope heading down the pole to the cleat tie off, there must be an equal force pulling up. It's not obvious where this force comes from, but it's actually supplied by the rope and the anchor point. In order to supply that reactant force, we'll see that the pulley won't be perfectly horizontal. In fact, there will be an angle of about 7°, as shown below:

Still reading? Congratulations on trudging through all that math. Hopefully, you understand how pulleys work a bit better, now. If not, it doesn't really matter! All you need to know is that they do work, and that they make for an incredibly easy install and take down! Don't believe us? Try it yourself!