How do you find the input force of a pulley and inclined plane?

Answer 1

A pulley and an inclined plane are two different examples of what engineers call "simple machines". Each of them can multiply the effects of an input force, so a better way of phrasing the question would be to break it into two aspects: "1. How do you calculate the ratio by which a force is increased with a pulley?", and "2. How do you calculate the ratio by which a force is increased with an inclined plane?" Answer 1. If the pulley is fixed to something solid, with the rope extended to the guy doing the pulling (call him the slave) and the other end hooked to the load, there is no multiplication of force, just a change in direction. If the load weighs 100 pounds, then it will still take 100 pounds of force to lift it. If however, you permanently affix the pulley to the movable load, with one end of the rope to a fixed anchor and the other end of the rope pulled by a slave, then you decrease the force required by the slave to lift the block by 1/2. That is, a force of 50 pounds by the slave will lift the 100-pound block. In effect, the pulley sees two ropes pulling on it, and each of the two ropes sees a load of P/2. If you install a block and tackle arrangement with multiple ropes and pulleys, you can further decrease the load pulled by the slave as a function of P(total load)/number of ropes between the block and tackle. This arrangement is better shown with a picture, but it assumes that the upper pulleys of the block and tackle are anchored to something solid, and that the lower pulleys are tied to the load. If their are six ropes between the upper and lower assemblies, then the force required to lift the block will be P/6. Note that you can't count the seventh rope segment leading down to the slave - that piece of rope merely changes the direction of the force, without any multiplying effect. In simple terms, four pulleys (2 on top and 2 on bottom) with 4 rope segments, will reduce the load pulled by the slave to P/4. Six pulleys (3 on top and 3 on bottom) with 6 rope segments will reduce load to P/6. And so forth. Note that your slave doesn't get something for nothing. He may only pull 1/4th the load, but he'll have to pull the rope 4 times as far to lift the block a specified distance. Answer 2. The ratio at which force is multiplied on an inclined plane is equal to the ratio of the vertical distance to the inclined distance along the ramp. For example, if your slave is directed to push a block up a ramp that is 20 feet long and 5 feet high, he will only have to push 5/20 or 1/4th as hard. He will have to push it 4 times farther, however. Both answers above neglect the effects of friction, so you will need additional slaves to apply plenty of grease to continuously lubricate both the pulley and the ramp to achieve maximum effifiency. Combine both questions: How much force multiplication can you get if you directed your slave to pull a 100 pound block up a ramp with a 4/20 slope using a block and tackle with 4 pulleys and 4 rope segments? Input Force = ((ramp ht/ramp length) x (1/number of rope segments)) x output force Input force = ((4ft/20ft)x(1/4 segments) x output force Input force = ((0.20 x 0.25) x output force Input force = 0.05 x output force Input force = 0.05 x 100 pounds Input force = 5 pounds Therefore, if you have plenty of lubrication and neglect friction, your slave can pull a 100 pound block up the ramp with only 5 pounds using the block and tackle. Rod Burrows, PE (professional slave)