Posts Tagged ‘lifting’

More Pulleys Increase Mechanical Advantage

Thursday, October 27th, 2016

    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.   Today 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.

More Pulleys Increase Mechanical Advantage

More Pulleys Increase Mechanical Advantage

   

    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 tradeoffs.   We’ll see that next time.

 Copyright 2016 – Philip J. O’Keefe, PE

Engineering Expert Witness Blog

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Mechanical Advantage of a Compound Pulley

Thursday, September 29th, 2016

    In this blog series on pulleys we’ve gone from discussing the simple pulley to the improved simple pulley to an introduction to the complex world of compound pulleys, where we began with a static representation.   We’ve used the engineering tool of a free body diagram to help us understand things along the way, and today we’ll introduce another tool to prepare us for our later analysis of dynamic compound pulleys.   The tool we’re introducing today is the engineering concept of mechanical advantage, MA, as it applies to a compound pulley scenario.

    The term mechanical advantage is used to describe the measure of force amplification achieved when humans use tools such as crowbars, pliers and the like to make the work of prying, lifting, pulling, bending, and cutting things easier.   Let’s see how it comes into play in our lifting scenario.

    During our previous analysis of the simple pulley, we discovered that in order to keep the urn suspended, Mr. Toga had to employ personal effort, or force, equal to the entire weight of the urn.

F = W                                    (1)

    By comparison, our earlier discussion on the static compound pulley revealed that our Grecian friend need only exert an amount of personal force equal to 1/2 the suspended urn’s weight to keep it in its mid-air position.   The use of a compound pulley had effectively improved his ability to suspend the urn by a factor of 2.   Mathematically, this relationship is demonstrated by,

F = W ÷ 2                              (2)

    The factor of 2 in equation (2) represents the mechanical advantage Mr. Toga realizes by making use of a compound pulley.   It’s the ratio of the urn’s weight force, W, to the employed force, F.   This is represented mathematically as,

MA = W ÷ F                            (3)

    Substituting equation (2) into equation (3) we arrive at the mechanical advantage he enjoys by making use of a compound pulley,

MA = W ÷ (W ÷ 2) = 2           (4)

Mechanical Advantage of a Compound Pulley

Mechanical Advantage of  a Compound Pulley

    Next time we’ll apply what we’ve learned about mechanical advantage to a compound pulley used in a dynamic lifting scenario.

 

 

 

                      

 Copyright 2016 – Philip J. O’Keefe, PE

Engineering Expert Witness Blog

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The Math Behind a Static Compound Pulley

Friday, September 9th, 2016

    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, F1, F2, and F3.   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 F3, we must first calculate the tension forces F1 and F2.   To do so, we’ll use a free body diagram, shown in the green box, to display the forces’ relationship to one another.

A Compound Pulley

The Math Behind a Static Compound Pulley

    The free body diagram only takes into consideration the forces inside the green box, namely F1, F2, and W.

    For the urn to remain suspended stationary in space, we know that F1 and F2 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,

F1 = F2 = W ÷ 2

    Because we know F1 and F2, we also know the value of F3, 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 F1 = F2 = F3.    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,

F1 = F2 = F3= W ÷ 2

F1 = F2 = F3 =  (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 F1 , F2 , and F3.

 

Copyright 2016 – Philip J. O’Keefe, PE

Engineering Expert Witness Blog

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The Simple Pulley Gives Us a Lift

Friday, July 8th, 2016

    Lifting heavy objects into position always presents a challenge, whether it’s a mom lifting a toddler to her hip or a construction worker lifting work materials to great heights.   During my career as an engineering expert I’ve dealt with similar challenges, some of which were handled quite nicely by incorporating a simple pulley, which we introduced last time, into my design.   But sometimes, due to certain restrictions, the addition of a simple pulley into the works isn’t enough to get the job done.   We’ll take a look at one of the restrictions working against the use of a simple pulley today.

    The simple pulley is believed to have first been used by the Greeks as far back as the 9th Century BC.   Back then it would have come in handy to lift cargo aboard ships, hoist sails on masts, and lift building materials high off the ground to supply workmen during the construction of temples and other marvels of ancient architecture.   In other words, pulleys literally saved ancient workers thousands of steps when it came to lifting things off the ground.

    Let’s return to ancient times for a moment to get an understanding of the mechanics behind the workings of the simple pulley as put to use in a basic lifting application.

 The Simple Pulley Gives Us a Lift

The Simple Pulley Gives Us a Lift

   

    With a simple pulley, the tension force F1 applied to the rope by the pull-er is equal to the tension force F2 exerted upon the object, the pull-ee.   Once lifted off the ground, these forces are also equal to the object’s weight, W, which gravity works upon to return the lifted object to its previous position on the ground.  All these forces come to bear upon whatever’s doing the pulling.   If this pull-er happens to be a human, then the simple pulley’s effectiveness to lift things is directly proportionate to that human’s strength.   In the case of the toga’d figure above, that would be about 10 pounds.   It’s this caveat that limits the usefulness of the simple pulley when relying on human power alone, particularly when it’s employed to lift extremely heavy objects like marble pillars.   A single human isn’t up to the task.

    Next time we’ll see how ancient Greeks overcame this limitation of the simple pulley by managing to cut in half the amount of brute force required to lift heavy objects.

Copyright 2016 – Philip J. O’Keefe, PE

Engineering Expert Witness Blog

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