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2.9Simple machines
A machine in the broad sense of the word is any device designed to translate some form of energy into useful work. A “simple” machine is one where both the input energy and the output energy are mechanical in nature (i.e. both are forces acting along displacements). Examples of simple machines include levers, pulleys, ramps, wedges, gears, and chain/sprockets. More complex machines include such examples as electric motors, heat engines, pumps, compressors, and refrigerators.
The e ciency of any machine (symbolized by the Greek letter “eta” η) is defined as the ratio of output energy to input energy:
Einput Eoutput
Eoutput
Machine Efficiency = η =
Einput
(Useful work)
Ewaste
(Usually heat)
Ideally, these two will be equal, with all of the input energy translated losslessly into output energy. However, no machine is perfectly e cient although some simple machines come very close to achieving 100% e ciency. It is physically impossible to achieve an energy e ciency greater than 100%, as that would violate the Law of Energy Conservation.
2.9.1Levers
Perhaps the most basic type of machine is the lever : a rigid beam pivoting on an axis. This axis may be something as simple as a round cylinder, a pointed wedge, or even a sophisticated bearing. In any case, the general term for the pivot point on a lever is fulcrum:
Fulcrum Lever |
Fulcrum Lever |
Fulcrum |
Lever |
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If we look at the lever’s motion at each end, we see that the distance the “output” end moves is a function of how far the “input” end moves as well as the ratio of lengths from each end to the fulcrum. Showing examples using three di erent classes of lever, each one with an 83 length ratio:
First-class lever
(Output and input on opposite sides of the fulcrum)
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Second-class lever |
Third-class lever |
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and the fulcrum) |
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and the fulcrum) |
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The ratio of output force to input force ( Fout ) is called the mechanical advantage16 of the machine.
Fin
This ratio is always the reciprocal of the output versus input motion: if the output of the lever moves less than the input moves, the output force must be greater than the input force, and vice-versa. This makes perfect sense if you view a lever as a perfectly e cient machine where the output energy (work) must equal the input energy (work): since output energy is output force multiplied by output motion, and input energy is input force multiplied by input motion, in order for force to be multiplied, motion must be diminished.
Levers abound in everyday life. A shovel, for example, functions as either a first-class lever or a second-class lever, depending on its use. In either case, it is being used as a force multiplier, the trade-o being that the person must move the handle a farther distance than the rock moves, thus exchanging motion for force:
Shovel as a first-class lever
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Shovel |
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Rock |
Dirt |
Fulcrum |
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Shovel as a second-class lever
Shovel
Rock
Dirt
Fulcrum
16Technically, mechanical advantage should be defined by the ratio of input motion to output motion, rather than being defined in terms of force. The reason for this is if friction happens to exist in the machine, it will cause a degradation of force but not of motion. Since “mechanical advantage” is supposed to represent the ideal ratio of the machine, it is always safest to define it in terms of motion where friction will not a ect the calculation. For a frictionless machine, however, defining mechanical advantage in terms of force is perfectly legitimate, and in fact makes more intuitive sense, since a larger mechanical advantage always corresponds with force multiplication from input to output.
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2.9.2Pulleys
Another simple and useful machine is a pulley and rope. A “pulley” is nothing more than a wheel with a groove cut around its circumference to guide a rope or cable, a bearing and axle supporting the wheel and allowing it to freely turn. A single pulley hung from an overhead support has the ability to convert downward motion of a rope into upward motion to hoist a load:
Pulley
Fout xout xin
Fin
Weight
A single-pulley system such as this exhibits no mechanical advantage, because Fout = Fin. If we get creative with multiple pulleys, however, we can achieve a mechanical advantage su cient to hoist very heavy loads with modest input force:
xout
xin Fin
Fout
Weight
Here, the weight is being supported by the tension within two ropes, not just one rope. Since the person’s force on the rope is what generates the rope’s tension, Fin is equal to rope tension, while Fout is equal to twice the rope’s tension. Thus, this simple machine has a mechanical advantage equal to 2. It also means the person’s motion while pulling the rope will be exactly twice the motion of the hoisted weight. Remember that we cannot cheat the Law of Energy Conservation: work
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in cannot be less than work out. If the output force is twice as much as the input force due to mechanical advantage, the output motion can only be half as much as the input motion.
The mechanical advantage of a pulley system may be extended beyond two by adding even more pulleys. This pulley system has a mechanical advantage of 4, since the weight is being supported by the tension of four ropes, while the person pulling only feels the tension of a single rope:
xout xin
Fin
Fout
Weight
Here is where one must be careful in analyzing pulley systems with regard to mechanical advantage. The mechanical advantage in each of these examples was based on the number of ropes supporting the weight. So far, this also happened to equal the number of pulleys in the system. Lest anyone be tempted to determine mechanical advantage by simply counting pulleys, here is an example that breaks the pattern:
Fin
Here there is only one pulley in the system, yet the weight is being supported by the tension in two ropes and the person pulling on the rope only feels the tension of one rope, which means the system has a mechanical advantage of 2.
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This simple technology is commonly used on cranes to provide huge amounts of lifting force with modest amounts of cable tension. In this photograph you can see the multiple pulleys and lifting cable of a large industrial crane: