A gear ratio is one of the few ideas in mechanical engineering where the physics is simple, the arithmetic is trivial, and the design consequences are still routinely got wrong. The whole subject comes down to one conserved quantity and one exchange rate. This guide works from the definition of torque up to a complete winch calculation, and shows where the energy actually goes.
Torque is force that turns
Torque measures how hard something tends to rotate:
T = F × r
with F in newtons and r in metres, giving newton-metres (N·m). A 20 N force applied at the end of a 300 mm spanner produces T = 20 × 0.30 = 6.0 N·m. The torque calculator handles this and the unit conversions below in one place.
Why "perpendicular" is the load-bearing word
The r in T = F × r is the perpendicular distance from the axis of rotation to the line of the force — not the distance to your hand, and not the length of the lever. This is the single most common misunderstanding in the topic.
Push a wheel at the top of the rim and you apply full torque. Push the exact same force directly at the centre, along the line to the axle, and the torque is zero, however hard you push. The general form makes the geometry explicit:
T = F × r × sin θ
where θ is the angle between the lever and the force. A 50 N push on a 0.40 m lever held at 30° from perpendicular gives T = 50 × 0.40 × 0.5 = 10.0 N·m — you supplied 100% of the force and got half the torque because half of the force went into compressing the spoke rather than turning it. At 90° you get the full 20.0 N·m; at 0° you get nothing.
Units: N·m, kgf·cm and lb·ft
Torque turns up in three systems, and mixing them up is a reliable way to be wrong by a factor of ten:
- 1 N·m = 10.197 kgf·cm — the kilogram-force is the force of 1 kg under gravity, so a mass hangs from 1 kgf·cm when the arm is 1 cm.
- 1 N·m = 0.7376 lb·ft — beware the widely printed 0.7386, which is a transposition error. The correct factor is 1 ÷ 1.35582.
So 240 N·m is 2,447 kgf·cm or 177.3 lb·ft, and 204 N·m is 2,080 kgf·cm or 150.7 lb·ft. Small-number units are the ones that cause accidents: 2,447 is easy to read as 244.7, and the mistake is not obvious afterwards.
Worked example: a 30:1 winch, end to end
Take a motor delivering 8 N·m at 1,500 RPM and drive it through a 30:1 worm reducer. Work through it in four steps.
Step 1 — output speed. A 30:1 reduction means the output turns 30 times slower: 1,500 ÷ 30 = 50 RPM.
Step 2 — ideal output torque. In an ideal gearbox torque multiplies by the same factor: 8 × 30 = 240 N·m.
Step 3 — apply efficiency. A worm drive of this type is typically 85% efficient, and efficiency belongs to the output side: 240 × 0.85 = 204 N·m. Note that this makes the output torque lower than ideal, while output speed is unchanged. This is worth pausing on, because it is the opposite of what people assume.
Step 4 — size the drum. To raise a 500 kg load the drum must supply at least the weight: F = m × g = 500 × 9.81 = 4,905 N. From T = F × r, the radius needed is r = T ÷ F = 204 ÷ 4,905 = 0.0416 m, i.e. a drum radius of about 42 mm (roughly an 83 mm diameter drum).
That is the real answer, and it is worth noticing how modest it is. For comparison, lifting the same 500 kg on a 300 mm lever would need 4,905 × 0.30 = 1,472 N·m — more than seven times the gear output. The gearbox is what makes the load manageable in the first place.
One caution on this calculation: the drum radius is the effective radius at which the rope leaves the drum. As a layer of rope winds on, the effective radius grows and the same drum lifts less at the same torque. Design for the fully wound condition, not the bare drum.
Gear ratio: teeth, and the one that does not matter
The gear ratio is defined from the teeth counts:
i = N_driven ÷ N_driving
A 12-tooth pinion driving a 72-tooth gear gives 72 ÷ 12 = 6:1. Three things are worth being precise about. First, the ratio is the inverse of the speed ratio, so 6:1 means the output turns six times slower. Second, it is also the torque ratio — the output torque is six times the input, in the ideal case. Third, an idler gear in between changes nothing: it transmits the ratio between the first and last gear, and only reverses the direction of rotation. A 12-tooth, 20-tooth idler and a 72-tooth gear still give 6:1.
The gear ratio calculator takes tooth counts and returns speed, torque and the ratio together, which prevents working half the problem and forgetting the other half.
Why a big gear multiplies torque: energy conservation
Nothing creates force. What a gear ratio does is exchange speed for force, and conservation of power is the proof. Power is torque multiplied by angular velocity, P = T × ω. Halve the angular velocity and, to keep power constant, the torque must double.
Check it on the winch. Motor at 1,500 RPM has ω = 2 × π × 1,500 ÷ 60 = 157.08 rad/s, so P = 8 × 157.08 = 1,257 W. Output at 50 RPM has ω = 5.236 rad/s, and 240 × 5.236 = 1,257 W. Identical, as it must be.
Now apply the 85% figure: 204 × 5.236 = 1,068 W, which is exactly 85% of the input. The missing 189 W leaves as heat in the gearbox — and on a worm drive that heat is substantial, which is why worm reducers need cooling fins and why continuous-duty ratings differ so much from intermittent ones.
This is the answer to "where does the force come from?" It comes from the motor running for longer. The gearbox gives you 30 times the torque and charges you 30 times the time, plus the losses.
Gear types and their efficiency
The efficiency differences are not small and they drive real design choices:
- Spur and helical gears — 97–99% per stage. Cheap, quiet, efficient. The default for a reason.
- Planetary (epicyclic) gears — 95–98%. Coaxial input and output, so they fit where a layshaft would not, and ratios from about 3:1 to 10:1 per stage are practical.
- Worm gears — 40–85% depending on lead angle. Enormous ratios in one stage, and the friction is so high that the output cannot back-drive the input: the load cannot overhaul the motor. That self-locking property is worth paying for in a hoist or an electric window regulator.
That last point is easy to miss. A worm drive will hold a load with the power switched off; a planetary or helical gearbox of the same ratio will let the load back-drive the motor. If your application needs that, the inefficiency is not waste — it is the mechanism doing its job.
Torque or speed: what motor selection actually turns on
Power is torque multiplied by speed, and a motor is bought on power. So the design question is not "how much torque do I need" but "at what speed". Two consequences follow that people regularly miss.
You cannot get speed for free. A gearbox can only reduce speed. If you need a faster output than the motor, the motor must already be fast — a 1,500 RPM motor cannot be geared up to 3,000 RPM. This is why high-speed applications use high-speed motors rather than small motors with a step-up. Note also that a speed calculator is only half the tool; the ratio decides the rest.
Overspeeding a gearbox wastes power as heat. If a load needs 50 RPM and 204 N·m, and you have a 1,000 RPM motor delivering 8 N·m, a 20:1 reduction matches. Struggling to get 20,000 RPM out of a 1,500 RPM motor is impossible; running a 1,500 RPM motor down to 50 RPM through a 30:1 and then back up with a chain to 1,500 RPM is possible and entirely pointless, since you converted 1,257 W into 1,068 W and back to 1,068 W and then burned the difference in the second drive.
High-speed motors are smaller and cheaper per watt for the same power, which is why they dominate drives needing modest reduction. Low-speed high-torque motors suit large reductions and sometimes remove the need for a stage entirely — a common reason to revisit a design after the initial calculation.
Three everyday examples
Propellers. Propeller design is where the exchange rate becomes unavoidable. For geometrically similar propellers, thrust scales with roughly the fourth power of diameter and power with roughly the fifth, at constant rpm and efficiency. Double the diameter and you need about 16 times the thrust at 32 times the power. This is why small props are high-speed and large props are low-speed — they are the same physics. The RPM and speed ratios guide works through the diameter/speed trade in detail.
Winches and drum hoists. The calculation above is exactly this. The drum radius is the lever, the rope tension is the force, and the reduction is chosen so the drum torque beats the weight with a margin. Practical designs add a brake, because a self-locking worm holds the load only while it is holding it — it does not control the descent.
Car window regulators. A cable-driven window weighs roughly 5–8 kg. The motor is geared to tens of RPM and a few N·m, driving a small drum that the cable winds on. The window's own weight, plus the glass friction in the channel, is the load that must be lifted — and the motor's job is to exceed it by enough margin to move the glass at a usable speed without fighting it. The regulator drum is a gear ratio in plain sight, and the same T = F × r calculation sizes the drum.
Habits that prevent the standard errors
Always work in consistent units. Mixing N·m with mm gives answers off by a thousand. Convert first, then compute.
Apply efficiency on the output side. Speed is unaffected; torque is reduced. Applying it to the input speed instead is a classic error.
Check the power both ways. If P_in and P_out do not differ by exactly the efficiency, an error has been made somewhere — and finding it takes thirty seconds.
Remember the drum grows. Winding rope increases the effective radius, so lifting capacity falls as the drum fills.
Frequently asked questions
Why does torque depend on the perpendicular distance and not the distance itself?
Because only the part of a force that turns the object does any work. Push straight at the centre of a wheel, along the line to the axle, and the turning effect is zero however hard you push. Push that force at the rim, perpendicular to the spoke, and it gives maximum torque. That is why a spoke wrench works where a fist on the hub does not.
Does a gear ratio increase torque, or does it just trade speed for it?
It trades one for the other, and the exchange is exact in the ideal case: the output turns that many times slower while delivering that many times the torque. A 30:1 reduction giving 30 times the torque gives exactly one thirtieth of the speed. Nothing is created. Efficiency decides how much of the input power arrives at the output.
How much torque is lost in a gearbox?
A well-made spur or helical gearbox runs at 97–99% efficiency per stage. A planetary gearbox is typically 95–98%, and a single-stage worm gear considerably less, often 40–85% depending on the lead angle. That figure is not a defect: a worm trades a large amount of torque for a self-locking property no other gear type provides.
How do I choose between a motor that is fast and weak and one that is slow and strong?
Work out the speed and torque your load needs first, then pick whichever motor lands nearest to it, because overspeeding a gearbox to gain speed wastes power as heat. High-speed motors are cheaper per watt and smaller for the same power, so they suit drives with a low reduction. High-torque low-speed motors suit large reductions and often let you drop a stage entirely.