Torque, speed, and gearing

Twisting effort is torque. A screwdriver, bicycle crank, toy-car wheel, and vacuum blower shaft all trade force at a radius for rotation. For a force perpendicular to a lever, τ = rF. More generally, τ = |r × F| = rF sin θ. Torque uses newton-metres; angular speed uses radians per second. Mechanical power connects them: P = τω.

For two meshing ideal gears, tooth pitch makes tangential speeds equal. Define reduction ratio G = ω_in/ω_out. Then ω_out = ω_in/G and τ_out = Gτ_in. Real gearboxes lose power, so τ_out ≈ ηGτ_in, where efficiency η is below one. The motor also needs torque to accelerate inertia: τ = Iα. A 10:1 reducer may offer roughly ten times stall torque at one-tenth no-load speed, but it does not create energy.

Start ratio selection from the output. Convert the required push or lift to torque, add allowance for friction and acceleration, then compare the reflected demand with the motor's usable operating region—not merely stall torque. For a toy car, wheel radius converts tractive force into axle torque, while desired vehicle speed converts to wheel rpm. Those two constraints bound the gear ratio. This is the build decision: choose a ratio that satisfies both with margin, or change wheel size or motor rather than forcing an impossible compromise.

Common failure modes are mixing rpm with rad/s, using diameter where radius belongs, and designing at stall. At stall, speed and mechanical output power are zero and heating is severe. Backlash also means direction changes can move the motor briefly without moving the load. The optional DC-motor simulation is a bounded linear torque-speed and gearbox model; it omits temperature, nonlinear friction, tooth compliance, and controller current limits.