You come into contact with force/distance tradeoffs in all sorts of simple machines. For example, a lever is an example of this phenomenon:

In this diagram a force F is being applied to the left end of the lever. The left end of the lever is twice as long (2X) as the right end (X). Therefore on the right end of the lever a force of 2F is available, but it acts through half of the distance (Y) that the left end moves (2Y). Changing the relative lengths of the left and right end of the lever changes the multipliers.

Gears can do the same thing:

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In this diagram the left-hand gear has twice the diameter of the right-hand gear. For every turn of the left-hand gear, the right-hand gear turns twice. If you apply a certain amount of torque to the left-hand gear through one rotation, the right-hand gear will exert half as much torque but will turn two revolutions.

Another good example is a simple hydraulic system, as shown below:

Assume that you have two cylinders full of water with a pipe connecting the two cylinders together as shown. If you apply a force F to the left-hand plunger, it creates a pressure in the left-hand cylinder. Let's say you apply a 10 pound downward force to the left-hand cylinder. Let's also say that the radius of the left-hand cylinder is 0.57 inches. Therefore, the area of the left-hand piston is Pi * 0.57 * 0.57 = 1 square inch. If the radius of the right-hand cylinder is 4 times greater, or 2.28 inches, then the area of the right-hand piston is 16 square inches, or 16 times greater. If you push the left-hand piston down through 16 inches with a force of 10 pounds, then the right-hand piston will rise 1 inch with a force of 160 pounds. Hydraulic cylinders of all sorts take advantage of this simple force-multiplying effect every day.

You can see that a block and tackle, a lever, a gear train and a hydraulic system all do the same thing: they let you magnify a force by proportionally diminishing the distance through which the magnified force can act. It turns out that this sort of force multiplication is an extremely useful capability! Here are some of the devices that use these simple principles:

- Car jack (lever or threaded gear)
- Fingernail clippers (lever)
- Automobile transmission (gears)
- Come-along (block and tackle, gear)
- Can opener (gear, lever)
- Crowbar (lever)
- Hammer claw (lever)
- Bottle opener (lever)
- Car brakes (hydraulics)
- Hydraulic shop lift
- Elevator (block and tackle)
- Etc...

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