Gear Ratios & Mechanical Advantage — Proportions at Work
Gear Ratios
Every bicycle, car transmission, clock, and wind turbine uses gears — and every gear system is governed by a simple ratio.
$$\text{Gear Ratio} = \frac{N_{\text{driven}}}{N_{\text{driver}}} = \frac{\omega_{\text{driver}}}{\omega_{\text{driven}}}$$
where $N$ is the number of teeth and $\omega$ is the angular speed. A gear with more teeth turns slower but with more torque.
This is a proportion — the same concept from arithmetic. If gear A has 20 teeth and gear B has 60 teeth, the ratio is $60/20 = 3$, meaning gear A must rotate 3 times for gear B to rotate once.
Torque Multiplication
$$\tau_{\text{out}} = \tau_{\text{in}} \times \text{Gear Ratio}$$
A car's first gear has a high ratio (~3.5:1) for maximum torque. Top gear has a low ratio (~0.7:1) for speed.
A bicycle's front sprocket has 48 teeth and the rear has 16 teeth. The cyclist pedals at 80 RPM. What is the wheel RPM and effective speed (wheel diameter = 0.7 m)?
Gear ratio: $\frac{48}{16} = 3$
Wheel RPM: $80 \times 3 = 240$ RPM
Wheel circumference: $C = \pi d = \pi \times 0.7 \approx 2.20$ m
Speed: $240 \times 2.20 = 528$ m/min $= 31.7$ km/h
Ratios, multiplication, and $\pi$ — that's all you need to calculate bicycle speed.
Gear ratios are proportions — the same math as scaling recipes or converting units. Every mechanical system from watches to wind turbines uses this fundamental concept.