Page 44 - The Unofficial Guide to Lego Mindstorms Robots

33

Do the Math

The mathematics of gears can be described in a high school physics class. The two
important equations have to do with torque and angular velocity.

H ere's the equation for torque, which is a measure of the power in a turning shaft:

τ = Fr

In this case, τ is torque, F is force, and r is the distance from the center of the rotation
to the point where the force is applied. For a gear, this is the distance from the center
(where the shaft runs through) to the teeth. This is the same as the radius of the gear.
Suppose, then, that you have an 8t gear driving a 24t gear.

The equation for the torque of the 8t gear's shaft is this:

τ 8 = Fr

The radius of the 24t gear is exactly three times the radius of the 8t gear. The force is the
sa me where the teeth of the two gears meet. Therefore, the torque on the shaft on the 24t
gear is exactly three times the torque on the 8t gear's shaft:

τ 24 = 3Fr = 3τ
8

Angular velocity is the measure of how fast a shaft rotates. The angular velocity of a shaft
can be expressed in terms of the velocity of a point on the gear as follows:

v
ω =
r

Here, ω is the angular velocity, v is the velocity of the point on the gear, and r is the distance
between the point and the center of the gear. For the example I just described (an 8t gear driving
a 24t gear), the angular velocity of the 24t gear is exactly one third of the angular velocity of
the 8t gear. You can figure this out because the velocities of the gear teeth must be the same:

v
ω =
8
r

v ω
ω = = 8
24
3r 3

In general, then, it's easy to figure out the ratios of torque and angular velocity for two mating gears,
just by figuring out the ratios of gear teeth. If you use an 8t gear to drive a 40t gear, you'll end up with
fives times the torque and one fifth the angular velocity.