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Fig. 3-20.
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Wheel.
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If you draw a line from the center of the wheel to the ground, it looks a lot like a rst- or second-class lever. The catch is that the point of e ort on a powered wheel is at the same spot as the fulcrum, or pivot point. This gives an e ort arm of zero, and yet the wheel moves. If we try to spin the wheel from its edge, the point of resistance is at the center. A zero resistance arm is even harder to solve for in our mechanical advantage equations. So how do we deal with this impossibility We add a new force. The force of a wheel turning around a single point is called torque, and is usually represented by the Greek letter  (tau). The raw calculation of torque was given at the head of this section,  kg m2/s2, which is the same relationship that de nes energy in Joules. In other terms, it is  m (kg m2/s2), also known as  m F. The m is for the radius of the wheel in meters, and F is the force applied at the edge of the wheel. The same equations work if the wheel isn t round but is in fact a lever attached to a shaft. The wheel is a continuous, rotating lever. If you push on the edge of a wheel with a force F, the torque you create depends on the radius of the wheel:  F m 3-8 If the wheel is being powered with a given input torque , the push at the edge of the wheel is:  3-9 F m where m is still the radius in meters.
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A gear is a wheel with teeth in it so it won t slip as it rubs against another gear. Two or more gears connected together make a gear train. Gears may also push against a toothed bar, called a rack, making a rack and pinion. The gear is called a pinion in this application (Fig. 3-21). Two gears in a train convert torque from one shaft to force where the gears meet, and back to torque in the other shaft. This makes the gear train a rotary lever, with its mechanical advantage calculated from the radius of the gears. Any number of gears can be paired together like this, in many clever arrangements. Some of these are explored in 9. When you select gears and calculate the mechanical advantage of a gear train, you don t use the gear radius, you use the tooth count. The number of teeth on a gear is directly related to its radius by way of the circumference. The circumference of a circle, including gears, is: c 2  r 3-10 The Greek symbol  (pi) is a magic number that represents the ratio of a circle s circumference to its diameter. Pi has a value of about 3.14, though the numbers after the decimal point never come to an end. The diameter of a circle is simply the distance all the way across, or twice the radius.
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Gears and sprocket.
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The mechanical advantage generated by two gears is the ratio of the number of teeth on the output gear A to the teeth on the input gear B: MA A : B MA A B 3-11
Since the number of teeth is directly related to the gear s circumference, MA Cout Cin 3-12
If you stretch the circumference of the gear out at, you have a rack. While it is at, it is easy to see that the number of teeth you can t onto the circumference depends on the width of the teeth and the distance between the teeth. The size of the teeth is called the pitch of the gear. The larger the teeth, the stronger they will be but the less you can t onto the gear. The pitch on two meshed gears must be the same. The shape of a gear s teeth is specially designed to give the gear a smoothly adjusting contact between the meshed teeth at all times. Many gears can be squeezed into a small space, often with two gears sharing a single shaft. These gears can change huge amounts of distance, in terms of the rotation of the input gear, into huge amounts of torque on the output gear. A sprocket is a gear designed to mesh with the links of a chain instead of with another gear. The chain travels through space and meshes with a different sprocket. Sprockets and chains are one method of sending force a long way away. They are commonly used on bicycles. Sometimes a pair of pulleys are used like sprockets, with a tightly- tting rubber belt stretched between them instead of a chain. These are easy to make and are used on power tools and other equipment to transmit force. Pulleys and belts are quieter than chains, though they can slip. Sometimes you want things to be able to slip, so if the machine gets stuck force is lost in the slipping instead of breaking your machine. Inside your car you can see a pulley and belt arrangement. The pulley and belt both have large, square teeth. These prevent the belt from slipping, so you get many of the best features of chains and toothless belts. These toothed belts are often called timing belts, while the toothless ones are v-belts, since they tend to have sloping edges giving them a V shape.
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