r₁r₂FFT₁T₂T₁ = F · r₁T₂ = F · r₂same F, so T₂ / T₁ = r₂ / r₁ = z₂ / z₁
Force the driver's tooth puts on the driven gearEqual and opposite force back on the driverAt the contact point the two teeth push on each other with one force F, equal and opposite. Each gear turns that force into torque with its own lever arm, its pitch radius. The big gear has the longer lever, so the same F makes more torque on it: T₂/T₁ = r₂/r₁, which is the tooth ratio again. Torque goes up by exactly the factor that speed goes down, so power (torque × speed) comes through unchanged, minus 1–2% of friction.