More Pulleys Increase Mechanical Advantage

Last time we saw how compound pulleys within a dynamic lifting scenario result in increased mechanical advantage to the lifter, mechanical advantage being an engineering phenomenon that makes lifting weights easier. here we’ll see how the mechanical advantage increases when more fixed and movable pulleys are added to the compound pulley arrangement we’ve been working with.

The image shows a more complex compound pulley than the one we previously worked with. To determine the mechanical advantage of this pulley, we need to determine the force, F₅, Mr. Toga exerts to hold up the urn.

The urn is directly supported by four equally spaced rope sections with tension forces F₁, F₂, F₃, and F₄. The weight of the urn, W, is distributed equally along the rope, and each section bears one quarter of the load. Mathematically this is represented by,

F₁ = F₂ = F₃ = F₄= W ÷ 4

If the urn’s weight wasn’t distributed equally, the bar directly above it would tilt. This tilting would continue until equilibrium was eventually established, at which point all rope sections would equally support the urn’s weight.

Because the urn’s weight is equally distributed along a single rope that’s threaded through the entire pulley arrangement, the rope rule, as I call it, applies. The rule posits that if we know the tension in one section of rope, we know the tension in all rope sections, including the one Mr. Toga is holding onto. Therefore,

F₁ = F₂ = F₃ = F₄= F₅= W ÷ 4

Stated another way, the force, F₅ , Mr. Toga must exert to keep the urn suspended is equal to the weight force supported by each section of rope, or one quarter the total weight of the urn, represented by,

F₅= W ÷ 4

If the urn weighs 40 pounds, Mr. Toga need only exert 10 pounds of bicep force to keep it suspended, and today’s compound pulley provides him with a mechanical advantage, MA, of,

MA = W ÷ F₅

MA = W ÷ (W ÷ 4)

MA = 4

It’s clear that adding the two extra pulleys results in a greater benefit to the man doing the lifting, decreasing his former weight bearing load by 50%. If we added even more pulleys, we’d continue to increase his mechanical advantage, and he’d be able to lift far heavier loads with a minimal of effort. Is there any end to this mechanical advantage? No, but there are undesirable trade offs.