Rope Length Tradeoff in a Compound Pulley

We’re all familiar with the phrase, “too much of a good thing.”  As a professional engineer, I’ve often found this to be true.   No matter the subject involved, there inevitably comes a point when undesirable trade offs occur.   We’ll begin our look at this phenomenon in relation to compound pulleys here, and we’ll see how the pulley arrangement we’ve been working with encounters a rope length trade off.  

We’ll be working with two distance/length factors and observe what happens when the number of pulleys is increased.   Last time we saw how the compound pulley is essentially a work input-output device, which makes use of distance factors.   In our example below, the first distance/length factor, d₁, pertains to the distance the urn is lifted above the ground.   The second factor, d₂, pertains to the length of rope Mr. Toga extracts from the pulley while actively lifting.   It’s obvious that some trade off has occurred just by looking at the two lengths of rope in the image below as compared to old topic.   What we’ll see down the road is that this also affects mechanical advantage.

The compound pulley here consists of 16 pulleys, therefore it provides a mechanical advantage, MA, of 16.   For a refresher on how MA is determined, see our preceding blog.

Rope Length Trade off in a Compound Pulley

With an MA of 16 and the urn’s weight, W, at 40 pounds, we compute the force, F, Mr. Toga must exert to actively lift the urn higher must be greater than,

F > W ÷ MA

F > 40 Lbs. ÷ 16

 F > 2.5 Lbs.

Although the force required to lift the urn is a small fraction of the urn’s weight, Mr. Toga must work with a long and unwieldy length of rope.   How long?   We’ll find out next time when we’ll take a closer look at the relationship between d₁ and d₂.