One can think of strength and speed as two people sharing a not-large-enough blanket at night; if one person gets more blanket, the other gets colder. There's no other way about it. With VEX motors, one can either have more strength OR more speed, but not both. These are the choices, illustrated below. Student teams will have to decide what the best balance of speed and strength is, for each instance where they are using a motor. Usually it comes down to something like, “What's the fastest speed we can get for the strength we require?” or, “How strong does it need to be to lift the heaviest thing we need to move?”
So what is torque? While the technical definition is “a measure of the rotational influence that a force has on an object,” for VEX motors (which rotate), torque can most easily be thought of as a motor's force or, basically, strength in the diagram above.
So a team has set its priorities are for its given motor use (strength, speed, or [usually] a combination of the two). How to achieve that desired outcome from a VEX motor? With gears! Putting gears in between the motor and the wheel (or turning part) will make the wheel go faster or slower than the motor itself. Looking again at the diagram above, one can also think about this the other way around: the gears will make your wheel/turning part stronger or weaker than the motor by itself.
The phrase “gear ratio” refers to the difference (up or down) in speed between the motor and the end point in a chain of gears (a wheel or other turning part); alternately, it can describe the difference in strength (torque) between the motor and the turning part.
VEX gears come in only 4 sizes: 12-tooth, 36-tooth, 60-tooth, and 84-tooth. Notice that these numbers are all divisible by 12; this attribute is handy for creating gear ratios in easy-to-calculate-and-quantify amounts, as shown below.
To calculate a gear ratio, simply count the teeth on the respective gears (the one connected to the motor, the driving gear; and the one connected to the turning part, the driven gear), place them in a fraction, and simplify the fraction to lowest terms:
If a motor is connected to a 12-tooth gear, and a moving part connected to a 36-tooth gear, as shown at left, then the 12-tooth gear will go around 3 times before the bigger gear goes around 1 time. [Image credit 1)]
So the gear connected to the moving part is moving much more slowly than the motor. What does this accomplish? More strength, or torque. The moving part can lift more, push more, etc. than if it were connected to the motor directly. How much more? 3x more! What's the cost? Speed! The moving part is only going 1/3 the speed that it would turn if it were connected to the motor directly. (See two people sharing too-small-blanket, above.)
In the second example below, the motor is connected to a 60 tooth gear, and the moving part is connected to a 36-tooth gear. When the motor turns its gear around once, the gear connected to the moving part goes around almost 2 times. So the moving part is going almost twice as fast as if it were connected to the motor directly. And what does one have give up in the process? Yes, torque. It can't lift as much, throw as far, push as strongly, etc. But presumably in this case the builders have chosen this gear ratio because they don't need those things, but they do need more speed. [Image credit 2)]
If the biggest gear is 84 teeth, and the smallest one is 12 teeth, then the fastest one can make a wheel spin is 7x the original motor speed (84/12 = 7). What if a robot function needs more than 7x the speed?? Or stronger than 7x? Enter compound gears.
A wide variety of speed and torque can be achieved by stacking the gears in a certain way, called compound gears.
Below is a diagram of one side of a Nothing But Net double-flywheel, which used compound gears to get the flywheel spinning fast enough to launch balls: The drive axle on the left has 2 motors, top and bottom, with a 60-tooth gear; the output axle on the right has a 12-tooth gear and the wheel. The magic happens on the axle(s) in the middle of the sandwich. The key to compound gears is putting 2 gears of different sizes on the same axle, as shown above (a 12-tooth gear on top, being driven by the 60-tooth gear, and an 84-tooth gear below it, driving the 12-tooth gear on the flywheel axle). The calculation is shown below, but it's conceptually useful to think about how each piece of the system is turning, as described in the colored labels in the diagram.
The formula for simple gears, as shown above is calculated as (# Teeth in Driven Gear / # Teeth in Driving Gear). For compound gears, one calculates this same gear ratio for each pair of gears (however many pairs are being used in a given application), and then multiply them together, achieving the same answer as shown in the example above:
Compound gears can be stacked in many different combinations, and by using even more than 3 axles, very high speed or very high torque can be achieved. Notice that last sentence said high speed OR high torque; compound gears can't break the laws of physics. And even after choosing one of these options—speed or torque—there is a limit to how far it can go. What happens if one tries to push the motors too far? Motor Overload (a.k.a. Motor Stall). They will just stop, or they will work for a short period of time, and then fail. A wiki article on motor overload is forthcoming.