When calculating the mechanical advantage of a lever what two pieces of information are needed?

How do you calculate the ideal mechanical advantage of a pulley system?

A pulley system creates mechanical advantage by dividing force over a length of rope or its equivalent, that is greater in length than the maximum distance the load can travel by using the pulley system. Through the use of movable pulleys or their equivalent, a system creates a mechanical advantage through the even division of force over multiple rope strands of a continuous rope. As rope, or its equivalent, is removed from the system, pulleys, or their equivalent, allow the side of the rope to apply force to the load. As the the system contracts, the load is lifted or moved (depending on the direction of the pull). The more strands created by the configuration, the greater the mechanical advantage. This is because every strand of rope or its equivalent created by the configuration of the system will take an equal amount of length of rope removed as the system contracts. Thus if there are three strands of rope created by the system, and three units of rope are removed from the system, each strand will contract by one unit. As the strands are parallel, or function in as parallel the overall contraction of the system is one unit, moving the load only one unit for every three units of rope removed. By distributing the force needed to move the load one unit over three units of the rope, this decreases the force needed on the pulling end by 1/3. This would be a mechanical advantage of 3:1. One of the most common systems of mechanical advantage is a shoe lace system. The grommets of the system are the equivalent of movable pulleys. As lace is removed from the system, force is applied to grommet, contracting the system. The laces are much longer than the space that they are contracting, and to fully contract the space nearly all the lace must be removed, so we can clearly see that many more units of lace must be removed for every one unit of contraction in the system, thus mechanical advantage is created. Of course in a lace system friction quickly overcomes and limits the advantage created. But on the other hand the friction helps to hold the force exerted allowing you to cinch up you shoes more easily. Now with this example in mind, let's look at a more traditional pulley system. The easiest way to understand how mechanical advantage is achieved may be to focus on the geometry of the system. Specifically by focusing on how force is applied to the load and why the configuration of movable pulleys distributes force and creates mechanical advantage. Imagine a weight to which a rope is directly attached. The rope is fed though a pulley mounted on the ceiling (fixed pulley). If you were to pull the rope the weight would move up a distance equal to the length of rope pulled. This is because the rope is directly attached to the load. There is no mechanical advantage. If we want to create a mechanical advantage we must attach a pulley to the load/weight so that force is applied via the rope's contact with the movable pulley . So in the next scenario imagine the rope is directly attached to the ceiling, and is fed through a pulley attached to the load (movable pulley as the load can move). The distance from the movable pulley to the ceiling is 10 feet. Now imagine you were to grab the rope exiting the pulley (imagine the system has no slack), and raise it to the ceiling. Now you have 10 foot section of rope with both ends on the ceiling. Where does that leave the load? Since the load is connected to the system by a wheel that can travel over the rope it has not followed the end of the rope the 10 feet to the ceiling, instead it has stayed in the center of the rope, constantly dividing the distance of the remaining section of rope. The load will now be 5 feet from the ceiling (10 feet / 2 section of rope). It has move only 1 unit of distance for every 2 units the rope has moved. Therefore only 1/2 the force is needed to move the rope 1 unit. This movable pulley system therefore has a 2:1 mechanical advantage. Now we will add another pulley to the ceiling. This is a fixed pulley and will not add any mechanical advantage, but will only redirect the force applied to the system. If we add another pulley to the load we will then have added mechanical advantage. When calculating the advantage added, you must observe the movable pulleys and their relationship to the load. Imagine a system with a rope directly connected to a load. The rope travels through a fixed pulley on the ceiling to another pulley on the load and back up to a fixed pulley on the ceiling. Drawn on paper this system will have four rope strands. For calculating mechanical advantage you must not count the strand exiting the final fixed pulley as the fixed pulley does not add mechanical advantage. (if the system was to end with a pulley attached to the load you would want to count the final strand). In this scenario we have three strands of rope contributing to the mechanical advantage of the system so the advantage should be 3:1. But how can you prove this. Imagine each section is ten feet long. Thus we have 30 total feel in the system. We pull out 10 feet of rope, how far has the load traveled? Well, we know we now have 20 feet of rope in the system distributed over 3 equal strands of rope. That would make each strand approximately 6.66 feet long. The load would therefore be approximately 6.66 feet from the ceiling or 3.33 feet from the ground (10 - 6.66). By traveling only 3.33 feet for 10 feet of rope removed from the system we have 3:1 mechanical advantage ratio (10:3.33). A final thought exercise to intuitively understand what can be a very unintuitive process. Imagine a 10 ft tall pulley system. Now focus on the amount of rope in the system. If you have three strands going back and forth you will have 20 to 30 feet of rope in the system (depending on if the final pulley is attached to the load or a fixed point). If you have four strand you'll have 30 to 40 feet. The particular amount is not important. What is important is to see that the only way the load can travel the 10 feet to the top of the pulley system is for nearly all the rope in the system to be removed be it 20, 30, 40, 50... ect. The more rope that must be remove/the more strands that divide the amount removed, the greater the division of the force over the rope and the less force is required on the pulling end of the system. Of course this is a basic pulley system. If you attach pulley systems to pulley systems (piggy back systems) you can begin doubling forces quickly, and strands need not be equal in length for their dividing power to function. Z rigs, trucker's hitches, and others create mechanical force through attaching or creating a movable pulley to/on the rope. The overall geometry of the systems and the relationships of elements stay the same as does the reason for the mechanical advantage. It is also important to note that there are configurations where a pulley or its equivalent may not be "movable", but mechanical advantage is created. Imagine multiple pulleys fixed to a ceiling and floor of a room. If one end of a cable was fixed to either the floor, ceiling or one of the pulleys and the system was threaded, it certainly would be creating a mechanical advantage. Though all pulleys are technically "fixed" the opposition force is magnified just as in any other system, and depending on the strength of the cable, ceiling, or anchors, one element may eventually fail because of the tension in the system. The amount of tension in the system is created though the mechanical advantage of the configuration, and though nothing may move but the cable, magnified force is applied to the elements of the system. In summary, it may be helpful to focus on the geometric relationships in pulley systems to better and more intuitively understand the way in which they create mechanical advantage.

What benefits do pulley simple machines have to man?

A pulley is a mechanism with a wheel and a simple frame that can be connected to something, either a fixed object or a movable object. The purpose of the pulley is to decrease friction when redirecting the pull/force of a rope, chain, cable or its equivalent. A pulley creates mechanical advantage only when configured in a particular way (see below). A pulley system creates mechanical advantage by dividing force over a length of rope or its equivalent, that is greater in length than the maximum distance the load can travel by using the pulley system. Through the use of movable pulleys or their equivalent, a system creates a mechanical advantage through the even division of force over multiple rope strands of a continuous rope (in a continuous system). As rope, or its equivalent, is removed from the system, pulleys, or their equivalent, allow the side of the rope to apply force to the load. As the the system contracts, the load is lifted or moved (depending on the direction of the pull). The more strands created by the configuration, the greater the mechanical advantage. This is because every strand of rope or its equivalent created by the configuration of the system will equally distribute the loss of rope as rope is removed from the system. Thus if there are three strands of rope created by the system, and three units of rope are removed from the system, each strand will contract by one unit. As the strands are parallel, or function in parallel, the overall contraction of the system is one unit, moving the load only one unit for every three units of rope removed. By distributing the work needed to move the load one unit over three units of the rope, the work needed to move the rope one unit decreases to 1/3 of what it would be if it was directly connected to the load. The force needed to move the load also decreases by 1/3, and thus this example system makes someone's work 3 times "easier" (though doesn't reduce the total work done, it just stretches it out over 3 times the rope). This would be a mechanical advantage of 3:1. One of the most common systems of mechanical advantage is a shoe lace system. The grommets of the system are the equivalent of movable pulleys. As lace is removed from the system, force is applied to grommet, contracting the system. The laces are much longer than the space that they are contracting, and to fully contract the space nearly all the lace must be removed, so we can clearly see that many more units of lace must be removed for every one unit of contraction in the system, thus mechanical advantage is created. Of course in a lace system friction quickly overcomes and limits the advantage created. But on the other hand the friction helps to hold the force exerted allowing you to cinch up you shoes more easily. Now with this example in mind, let's look at a more traditional pulley system. The easiest way to understand how mechanical advantage is achieved may be to focus on the geometry of the system. Specifically by focusing on how force is applied to the load and why the configuration of movable pulleys distributes force and creates mechanical advantage. Imagine a weight to which a rope is directly attached. The rope is fed though a pulley mounted on the ceiling (fixed pulley). If you were to pull the rope the weight would move up a distance equal to the length of rope pulled. This is because the rope is directly attached to the load. There is no mechanical advantage. If we want to create a mechanical advantage we must attach a pulley to the load/weight so that force is applied via the rope's contact with the movable pulley . So in the next scenario imagine the rope is directly attached to the ceiling, and is fed through a pulley attached to the load (movable pulley as the load can move). The distance from the movable pulley to the ceiling is 10 feet. Now imagine you were to grab the rope exiting the pulley (imagine the system has no slack), and raise it to the ceiling. Now you have 10 foot section of rope with both ends on the ceiling. Where does that leave the load? Since the load is connected to the system by a wheel that can travel over the rope it has not followed the end of the rope the 10 feet to the ceiling, instead it has stayed in the center of the rope, constantly dividing the distance of the remaining section of rope. The load will now be 5 feet from the ceiling (10 feet / 2 section of rope). It has move only 1 unit of distance for every 2 units the rope has moved. Therefore only 1/2 the force is needed to move the rope 1 unit. This movable pulley system therefore has a 2:1 mechanical advantage. Now we will add another pulley to the ceiling. This is a fixed pulley and will not add any mechanical advantage, but will only redirect the force applied to the system. But, if we add another pulley to the load we will have added mechanical advantage. It is important to note, when calculating the advantage added, you must observe the movable pulleys and their relationship to the load. Now imagine a system with a rope directly connected to a load. The rope travels through a fixed pulley on the ceiling to another pulley on the load and back up to a fixed pulley on the ceiling, and back down to the ground where it can be pulled. Drawn on paper this system will have four rope strands. For calculating mechanical advantage you must not count the strand exiting the final fixed pulley as the final fixed pulley only redirects force and does not add mechanical advantage. (if the system was to end with a pulley attached to the load you would want to count the final strand). In this scenario we have three strands of rope contributing to the mechanical advantage of the system so the advantage should be 3:1. But how can you prove this? Imagine each section is ten feet long. Thus we have 30 total feel in the system. We pull out 10 feet of rope, how far has the load traveled? Well, we know we now have 20 feet of rope in the system distributed over 3 equal strands of rope. That would make each strand approximately 6.66 feet long. The load would therefore be approximately 6.66 feet from the ceiling or 3.33 feet from the ground (10 - 6.66). By traveling only 3.33 feet for 10 feet of rope removed from the system we have 3:1 mechanical advantage ratio (10:3.33). A final thought exercise to intuitively understand what can be a very unintuitive process. Imagine a 10 ft tall pulley system. Now focus on the amount of rope in the system. If you have three strands going back and forth you will have 20 to 30 feet of rope in the system (depending on if the final pulley is attached to the load or a fixed point). If you have four strand you'll have 30 to 40 feet. The particular amount is not important. What is important is to see that the only way the load can travel the 10 feet to the top of the pulley system is for nearly all the rope in the system to be removed be it 20, 30, 40, 50... ect. The more rope that must be removed and the more strands that divide the amount removed, the greater the division of the force over the rope and the less force is required on the pulling end of the system. Of course this is a basic pulley system. If you attach pulley systems to pulley systems (piggy back systems) you can begin doubling forces quickly, and strands need not be equal in length for their dividing power to function. Z rigs, trucker's hitches, and others create mechanical force through attaching or creating a movable pulley to/on the rope. The overall geometry of the systems and the relationships of elements stay the same as does the reason for the mechanical advantage. It is also important to note that there are configurations where a pulley or its equivalent may not be "movable", but mechanical advantage is created. Imagine multiple pulleys fixed to a ceiling and floor of a room. If one end of a cable was fixed to either the floor, ceiling or one of the pulleys and the system was threaded and the end of the system was pulled, there would be a mechanical advantage. Though all pulleys are technically "fixed" the opposition force is magnified just as in any other system, and depending on the strength of the cable, ceiling, or anchors, one element may eventually fail(move/break) because of the tension in the system. The amount of tension in the system is created through the mechanical advantage of the configuration, and though nothing may move but the cable (until failure of an element), magnified force is applied to the elements of the system. In summary, it may be helpful to focus on the geometric relationships in pulley systems to better and more intuitively understand the way in which they create mechanical advantage. I hope this approach to explaining the how pulleys work has been useful. Now get out there and move something!

When calculating the mechanical advantage of a lever what two pieces of information are needed?

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