Mathematics · Middle school

The gearbox

Change the number of teeth: teeth and turns are inversely proportional, and their product stays the same.

In the learning section below the figure, predict, try, complete the tasks and check yourself with short questions.

Fig. 1The gearbox

Change the teeth on the right gear, watch both speeds and the product

n₂ × z₂ = 720 teeth/min

Teeth z₂12 teeth

Crank speed n₁30 rpm

Your own values

z₁ (teeth)

z₂ (teeth)

n₁ (rpm)

Teeth are whole numbers from 6 to 200, crank speed 1–1,000 rpm. A value applies when you press Enter or leave the box.

Product stays the same

k = 720 teeth/min

n₁ × z₁: 30 × 24 = 720
n₂ × z₂: 60 × 12 = 720

The ratios are inverted: z₂ : z₁ = 1/2 → n₂ : n₁ = 2

n₂ – z₂ graph

Shaded area = n₂ × z₂

Learn

Progress is kept on this page only; no points are given.

1Predict
2Try
3Explain

If you double the teeth on the right gear, from 12 to 24, what happens to its turns per minute?

Because the gears mesh without slipping, the same number of teeth passes the mesh point each minute on both gears: n₁ × z₁ = n₂ × z₂. With the crank speed and z₁ fixed, this number (k) does not change; double the teeth on the driven gear and its speed halves. That is why the graph is not a straight line but a curve that never reaches zero (n₂ = k ÷ z₂). The gears are drawn with a 20° involute profile and equal module.

Direct and inverse proportion: understanding recipes, map scales and speedRead the article