One property of gears is they change the rotational speed of the axles that hold the gears. In this picture, there is a 40 tooth spur gear on the output axle meshed with an 8 tooth spur gear on the input axle.

The gear ratio of this gear pair is 40 to 8 or, by simplifying, 5 to 1. That means it takes 5 revolutions of the input gear to make 1 revolution of the output gear. This results in a slowdown of the output gear by a factor of 5. If the input and output axles are reversed, then the gear ratio would be 1 to 5. That means the output axle would rotate 5 times faster than the input axle.

Why is this so? The answer is because of the different diameters of the gears, and the relationship between rotational speed and linear speed. Both gears are moving at the same linear speed at the point where the teeth of the gears mesh but in opposite directions.

The relationship between linear speed and rotational speed is, v = w x r, where v is linear speed, w is rotational speed, and r is the radius of the gear. The radius is half the gearís diameter.

Since v40 = -v8, we can substitute the above equation for v and get w40 x r40 = -(w8 x r8). Rewriting this equation gives w40 = -w8 x (r8 / r40). The ratio of r8 to r40 is 1 to 5, or w40 = -w8 x (1/5). The rotational speed of the 40 tooth gear is 1/5 of the 8 tooth gear, or 5 times slower. The negative sign means the rotational speed is in the opposite direction.