Take 2 circles. One of size 1, and One of size 2. Cut one of the biggest circle and roll it flat. Now take the smaller circle and roll it along the flat circle.

How many times that circle rolls is going to be your gear ratio. If you turn(roll) the smaller gear one revolution, youll find that it has only gone halfway around the smaller gear.

as for gears and RPMs is the same concept but in terms of speed. One circle with a radius of 1 meter and one circle with a radius of 2 meters. No matter how large the circle is, if you spin it once, its going to get spun once. But in order for the larger one to make that spin in same amount of time, it has to spin proportionally faster. So the smaller gear spins at 3.14 meters per second and the larger one at 6.28 meters per second.

A gear with 20 teeth is connected to a smaller gear with 10 teeth. The bigger gear turns one full circle, making the smaller gear turn by 20 teeth, or two full circles.

Gear with 20 teeth is connected to a bigger gear with 100 teeth. Now a single rotation only spins the bigger gear by 1/5 of a full circle.

Cara and bicycles both work by transferring a spinning motion (driveshaft or pedals) to wheels. With different gear ratios, we can change how fast the same original spin turns the wheels. In the first example, you could get your wheels spinning much faster (and this *go* faster), but it’s also harder accelerate that second wheel. Because the teeth of the small gear are physically closer to the center, you’re exerting much less turning power (torque) per tooth. This is a high gear.

The second example is a low gear, where you’d have to pedal like a madman to get the second gear spinning fast, but that also means normal pedaling can exert a lot of torque on the gear, and give it lots of low-speed power. You can feel the difference using it to pedal uphill, or in difficult terrain.

Bikes and cars have a range of gears that overlap, and you start in low and shift up to high as your wheel speed increases.

Then again, most cars now have Continuous Variable Transmissions, which don’t have discrete gears at all, but that’s something I barely understand anyway

As to the *why* they do that, gears are bunches of levers. Each tooth it a lever that starts at the center of the gear and ends at the tip of the tooth.

http://www.dynamicscience.com.au/tester/solutions1/hydraulicus/gears1mechanicalad.htm#:~:text=When%20two%20gears%20are%20meshed,the%20shaft%20of%20the%20follower.

An internal combustion engine must be turning to operate. It is not very efficient at very low and very high speeds.

So to get around this, firstly you need a clutch, to disengage the engine from the gearbox.

Next, the gearbox has a bunch of selectable ratios that fit the best power from the engine to various speeds of the vehicle.

You start at zero speed vehicle with (say) 800 RPM engine and the clutch held in. Rev the engine to (say) 2000 RPM, let the clutch out and the car starts to move forward, until the clutch is fully engaged and the car is rolling at a speed matching the current RPM of the engine vie the currently selected gear ratio. Continue to accelerate, and eventually the engine gets close to its maximum RPM, and the vehicle’s is traveling (say) 25 Kph.

Press the clutch, drop the RPM, engage the next gear. That get works to (say) 45 Kph. And so on up the range.

As you go up gears you reduce the torque to the wheels but you increase the speed.

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