Last time we introduced the *compound pulley* and saw how it improved upon a simple pulley, both of which I’ve engaged in my work as an **engineering expert. ** Today we’ll examine *the math behind the compound pulley*. We’ll begin with a *static* representation and follow up with an active one in our next blog.
The *compound pulley* illustrated below contains three rope sections with three representative tension forces, *F*_{1}, *F*_{2}, and *F*_{3}. Together, these three forces work to offset the weight, *W,* of a suspended urn weighing 40 lbs. Weight itself is a downward pulling force due to the effects of gravity.
To determine how our *pulley* scenario affects the man holding his section of rope and exerting force *F*_{3}, we must first calculate the tension forces *F*_{1} and *F*_{2}. To do so, we’ll use a *free body diagram*, shown in the green box, to display the forces’ relationship to one another.
**The Math Behind a Static Compound Pulley**
The free body diagram only takes into consideration the forces inside the green box, namely *F*_{1}, *F*_{2}, and *W*.
For the urn to remain suspended stationary in space, we know that *F*_{1} and *F*_{2} are each equal to one half the urn’s weight, because they’re spaced equidistant from the *pulley’s* axle, which directly supports the weight of the urn. Mathematically this looks like,
*F*_{1} = F_{2} = W ÷ 2
Because we know *F*_{1} and *F*_{2}, we also know the value of *F*_{3}, thanks to an engineering rule concerning *pulleys*. That is, when a single rope is used to support an object with pulleys, the tension force in each section of rope must be equal along the entire length of the rope, which means *F*_{1} = F_{2 }= F_{3}. This rule holds true whether the rope is threaded through one simple pulley or a complex array of fixed and moveable simple pulleys within a *compound pulley*. If it wasn’t true, then unequal tension along the rope sections would result in some sections being taut and others limp, which would result in a situation which would not make lifting the urn any easier and thereby defeat the purpose of using *pulleys*.
If the urn’s weight, *W,* is 40 pounds, then according to the aforementioned engineering rule,
*F*_{1} = F_{2} = F_{3}= W ÷ 2
*F*_{1} = F_{2} = F_{3} = (40 pounds) *÷* 2 = 20 pounds
Mr. Toga needs to exert a mere 20 pounds of personal effort to keep the immobile urn suspended above the ground. It’s the same effort he exerted when using the improved simple pulley in a previous blog, but this time he can do it from the comfort and safety of standing on the ground.
Next time we’ll examine the math and mechanics behind an active *compound pulley *and see how movement affects F_{1} , F_{2} , and F_{3.}
Copyright 2016 – Philip J. O’Keefe, PE
Engineering Expert Witness Blog
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