A gear question usually asks for one of three things: how fast the last gear turns, which way it turns, or how much turning force it delivers. All three follow from counting teeth.
The Tooth-Count Ratio
When two gears mesh, their teeth interlock, so the same number of teeth pass the contact point on each gear in any moment. A gear with fewer teeth must therefore turn faster to keep up. The speed of the driven gear is the driver speed times driver teeth over driven teeth.
Worked example: an 18-tooth driver turns at 150 rpm and meshes with a 54-tooth gear. The driven gear turns at 150 x 18 / 54 = 50 rpm. It turns three times slower because it has three times as many teeth.
A quick sense check beats any formula: the small gear always spins faster, and the large gear always spins slower. If your answer breaks that rule, you have turned the ratio upside down.
Direction: Each Mesh Reverses
Two gears in direct contact turn in opposite directions. In a row of meshed gears, direction alternates: if the first turns clockwise, the second turns anticlockwise, the third clockwise again.
A gear placed between the driver and the driven gear is an idler. It flips the direction and cancels out of the ratio. Gear A (20 teeth) drives idler B (35 teeth), which drives gear C (40 teeth): the speed of C relative to A is 20 / 40, whatever B is, but C turns the same way as A because two reversals cancel.
Speed Down, Turning Force Up
Ignoring losses, a gear train cannot create energy, so power in equals power out, and power is torque times speed. When a train slows the output by some factor, it raises the torque by the same factor.
In the 18-tooth to 54-tooth example, the output runs three times slower, so a 6 N m torque on the driver becomes 18 N m on the driven gear. The machine trades speed for turning force, the same bargain as the lever.
Compound Trains: Same Shaft, Same Speed
In a compound train, two gears sit on one shaft. They turn together, so the speed is carried from the first stage to the second without change.
Worked example: gear A (15 teeth) turns at 300 rpm clockwise and meshes with gear B (45 teeth). Gear C (20 teeth) is fixed on the same shaft as B and meshes with gear D (40 teeth).
- Stage one: B turns at 300 x 15 / 45 = 100 rpm, anticlockwise.
- The shaft: C turns with B, so C is also 100 rpm anticlockwise.
- Stage two: D turns at 100 x 20 / 40 = 50 rpm, clockwise.
The overall ratio is the product of the stage ratios, 3 x 2 = 6, which matches 300 / 50. Writing each stage on its own line is the surest way to avoid dropping a factor.
Worm Gears: A Big Reduction in One Step
A worm is a screw-shaped gear that meshes with a toothed wheel. One full turn of a single-start worm moves the wheel on by exactly one tooth. So a worm driving a 40-tooth wheel is a 40-to-1 reduction: the worm turns 40 times for each single turn of the wheel.
That makes worm gears a compact way to slow a motor down and raise its turning force a great deal. The tooth-count rule still works; the worm simply counts as a gear with one tooth. Many worm drives are also self-locking, meaning the wheel cannot turn the worm backwards, which is useful where a load must stay where it is put.
Try These Three
- A 30-tooth gear turns at 400 rpm and drives a 10-tooth gear. How fast does the small gear turn? Answer: 400 x 30 / 10 = 1,200 rpm.
- A gear train slows a shaft down by a ratio of 5 to 1. If the input torque is 4 N m and losses are ignored, what is the output torque? Answer: five times as large, 20 N m, because speed is traded for turning force.
- A single-start worm drives a 40-tooth wheel at the rate of 1,200 rpm for the worm. How fast does the wheel turn? Answer: 1,200 / 40 = 30 rpm.
Chains and Belts Keep the Direction
A chain or belt joining two wheels makes them turn the same way, unlike meshed gears. The tooth-count ratio still applies. A bicycle with a 40-tooth chainring and a 20-tooth rear sprocket turns the rear wheel twice for each turn of the pedals, because the smaller sprocket spins faster.
Practise this on the JobCannon mechanical reasoning test, whose gear questions use exactly these steps, or see more on the mechanical reasoning practice page. For the force-for-distance idea behind it, read how levers work.