Gears are toothed wheels that are meshed together to transmit a twisting force (torque) and motion. They are usually attached to a shaft and can be considered a rotating lever. Utilizing leverage principles, gears can enhance or inhibit effort (force) or change effort (force) direction. Gears are either turned by a shaft or they turn the shaft. A large gear can apply more twisting force on a shaft to which it is attached than a smaller gear.
Gears that have straight teeth (perpendicular to their facing) and that mesh together in the same plane with axles parallel are known as spur gears. Spur gears provide an important way of transmitting a positive motion between two shafts. They give a smooth and uniform drive.
In the diagram, one gear, labeled the driver (also known as the driving gear) is turned by a motor. As it turns, it turns the other gear, known as the driven gear. A basic rule concerning gears states that each gear in a series of gears reverses the direction of rotation of the previous gear.
Another basic rule of gears is that when you have a pair of meshing gears, and the smaller gear with less pitch diameter (number of teeth) is the driver, torque output will be enhanced, with the trade-off being a decrease in speed of rotation. Conversely, when a larger gear with greater pitch diameter is the driver, torque output will be inhibited, but the trade-off is the speed of rotation will be enhanced.
Example: The driving (driver) gear has 9 teeth while the driven gear has 36 teeth. Find the gear ratio and mechanical advantage (torque) of this gear system.
Note: When computing gear ratio, always compare the larger gear rotating once to the smaller gear regardless of whether it is a driver or driven gear.
Gear ratio formula:
Input movement (Driver): Output movement (Driven)
1: 4
OR
The driver gear rotates 4 times faster than the driven gear (decrease in speed).
Mechanical advantage formula:
Example: The driving (driver) gear has 90 teeth while the driven gear has 15 teeth. Find the gear ratio and mechanical advantage (torque) of this gear system. Gear ratio formula:
Input movement (Driver): Output movement (Driven)
1: 6
OR
The driver gear rotates 6 times slower than the driven gear (increase in speed).
Mechanical advantage formula:
Revolutions Per Minute
As a general rule, if the number of revolutions per minute (rpm) of the driver gear is given and the driver gear is smaller than the driven gear, divide the gear ratio number of the driver gear into the rpm of the smaller gear (driver).
Example: In a two-gear system, the driver gear with 25 teeth revolves at 60 rpm; what is the rpm for the driven gear with 75 teeth?
Gear ratio formula:
Input movement (Driver): Output movement (Driven)
3: 1
OR
The driver gear rotates 3 times faster than the driven gear (decrease in speed).
Divide the gear ratio number of the driver into the rpm for the driver gear.
(60 rpm/3) = 20 rpm (agrees with driver gear rotating 3 times faster than the driven gear)
This adheres to the general rule of gears, which states that when a smaller gear drives a larger gear the speed should decrease.
Another general rule is that if the driver rpm is given and the driver gear is larger than the driven gear, multiply the gear ratio number of the driven gear by the rpm of the larger gear (driver).
Example: In a two-gear system, the driver gear with 60 teeth revolves at 120 rpm; what is the rpm for the driven gear with 30 teeth?
Gear ratio formula:
Input movement (Driver): Output movement (Driven)
2: 1
OR
The driver gear rotates 2 times slower than the driven gear (increase in speed).
Multiply the gear ratio number of the driven gear by the driver gear's rpm.
2 × 120 rpm = 240 rpm (agrees with driver gear 2 times slower than the driven gear)
This adheres to the general rule of gears, which states that when a larger gear drives a smaller gear the speed should increase.
Gear Trains
When solving questions using more than two gears (gear trains) concentrate on two gears at a time. Give each gear a letter designation. Focus on the driver gear and the direction in which it is rotating (clockwise or counterclockwise). Gears on either end of the driver gear will rotate in the opposite direction. These gears in turn will cause the gears abutting them to rotate in opposite directions. A gear train component used to allow adjacent gears to rotate in a desired direction is known as an idler gear.
In the illustration shown, the driver (gear B) rotates in a clockwise direction. Gears A and C, abutting gear B, rotate in a counterclockwise direction. Gears D and E, which are abutting gear C, rotate in a clockwise direction. 
Pedal and Sprocket Gears
Special gears with elongated teeth are connected with a chain. These gears (pedal and sprocket) are found quite commonly in bicycles. They operate the same way as gears except that the direction of the rotating gears is not reversed. When the large pedal gear toward the front of the bicycle revolves, the chain pulls round the sprocket gear wheel at the rear in the same direction. Gear ratios and mechanical advantage are calculated via the same formulas used for gear systems.
Example: If the pedal (driver) gear with 30 teeth in the illustration below revolves once, how many times will the sprocket (driven) gear with 15 teeth revolve? Find the gear ratio as well as the mechanical advantage of this gear system.
Gear ratio formula:
Input movement (Driver): Output movement (Driven)
1: 2
OR
The pedal (driver) gear is 2 times slower than the sprocket (driven) gear (increase in speed).
Mechanical advantage formula:
Belt Drive
In machinery, a pair of pulley wheels attached to parallel shafts and connected by a flexible band of flat leather, rubber, or similar material is known as a belt drive. Like gear trains, a belt drive can be used to increase or reduce the speed and mechanical advantage (torque) of the pulley wheels they are attached to and modify rotational motion from one shaft to the other. Unlike gears, however, belt drives rotate in the same direction. A belt's top surface can also be used to convey materials across it, such as on a conveyor belt. Belts are installed under tension in order to create friction, which allows the belt to grip and turn the pulley wheels. Substantial tension also keeps the belt from slipping off the pulley wheels when rotating. Belt drives are better for greater distances with smaller forces since they are not directly joined.
There are two types of belt drives, positive and nonpositive. Positive belt drives (gear belts) consist of belts with teeth that mesh with pulley wheels that also have teeth. This type of belt drive does not allow the belt to slip. They are used in applications requiring a higher horsepower or torque capacity. A nonpositive belt drive system utilizes smooth belts and pulley wheels. The design of a nonpositive belt is in the shape of a "V.' These belts require less tension than do flat belts because they have more surface area in contact with the pulley wheels. Nonpositive belt drives are useful for connecting shafts that are in close proximity to one another. The V-belt found inside of an automobile engine is an example of the nonpositive type. Crossed (twisted) belts cause shafts to rotate in opposite directions.
The mechanical principles and applications that apply to gears, in general, also apply to belt drives. To calculate the mechanical advantage of a belt drive system, instead of counting the teeth as you would when working with gears, divide the diameter of the driven pulley wheel by the diameter of the driver pulley wheel. Whenever the driven pulley wheel diameter is larger than the driver pulley wheel diameter, you will obtain a mechanical advantage.
Example: A belt drive system has a 9-inch diameter driven pulley wheel and a 3-inch diameter driver pulley wheel. What is the mechanical advantage of the system? 
To calculate the speed of the belt drive pulley system above, which has a driver pulley speed of 1200 rpm, use the following formula:
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