A sprocket is a toothed wheel that engages a chain. A gear is something you can get by having two sprockets sharing one chain.
Another definition of a gear is a cogwheel that engages another cogwheel.
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A sprocket is like a pulley, the difference being that a sprocket has gear-like teeth, to transport a chain rather than a belt. A bicycle has two sprockets, one attached to the pedals, and one attached to the rear wheel.
The "driving" or drive gear is the source of power or rotation. The driven gear is turned or moved by the drive gear. Example: The pulley on a motor is the drive pulley and a pulley on a pump is the driven pulley. Example: The pinion gear in a differential is the drive gear and the ring is the driven gear.
The "driving" or drive gear is the source of power or rotation. The driven gear is turned or moved by the drive gear. Example: The pulley on a motor is the drive pulley and a pulley on a pump is the driven pulley. Example: The pinion gear in a differential is the drive gear and the ring is the driven gear.
It depends on whether you are talking about the front sprocket (the one at the pedals, called a chainring) or the sprocket at the rear derailleur. The lowest gear at the front is the smallest sprocket/chainring. The lowest gear at the rear is the largest sprocket. So if you combine the smallest sprocket at the front with the largest sprocket in the rear you have the lowest gear available on your bike.
The sprocket on the nose of the crankshaft that drives the cam sprocket.
On a bike there's one(or more) front sprocket(s) by the pedals connected by a chain to one(or more) rear sprocket(s) by the rear wheel. As the wheel isn't driven directly by the pedals, and there's usually a difference in size between the front & rear sprocket this constitutes a gear.
The function of a cassette sprocket in a bicycle's gear system is to provide different gear ratios, allowing the rider to adjust the resistance and speed when pedaling.
Think about your ten speed bicycle. The bike is in the lowest gear when the front sprocket is on the smallest diameter and the rear is on the largest. On the contrary, when the bike is at high speed, the rear is on the small diameter sprocket and the front is on the large diameter sprocket. Therefore, increasing the drive pulley size will increase the driven pulley speed (assuming the drive pulley stays at the same RPM). Similarly, increasing the driven pulley size will result in a slower speed (again, assuming the drive pulley stays at the same RPM).
The name 'sprocket' applies generally to any wheel upon which are radial projections that engage a chain passing over it. It is distinguished from a gear in that sprockets are never meshed together directly, and differs from a pulley in that sprockets have teeth and pulleys are smooth. A sprocket can be used for many things, depending on what you need it to do. Sprockets is most commonly used for tracked vehicles.
Chain is on the smallest sprocket on the crank and on the largest sprocket on the rear wheel