Wheel and Axle Systems and Gears

A wheel and axle is a large wheel fixed to a smaller axle so that they turn together, with an ideal mechanical advantage equal to the ratio of the radii, R ÷ r; gears are toothed wheels that mesh and transmit rotation, changing speed and torque according to the ratio of their numbers of teeth. In brief: a force applied to the rim of the large wheel gives a larger force at the axle. For two meshing gears, the speed ratio is inversely proportional to the number of teeth: Ndriver × ωdriver = Ndriven × ωdriven. A small gear driving a large one increases torque and decreases speed.

Door handles, steering wheels, windlasses, bicycles and clocks use wheels, axles and gears. This guide explains the wheel and axle and its mechanical advantage, describes gears, gear trains and chain drives, compares gear arrangements in a table, works through calculations and corrects common mistakes.

How does a wheel and axle work?

The wheel and axle is a kind of continuous lever, with the center as the fulcrum. When the effort is applied at the rim of the wheel (radius R) and the load acts at the axle (radius r), the moments are balanced when Fe × R = Fl × r. The ideal mechanical advantage is IMA = R ÷ r. For example, a door knob (wheel) turns a small spindle (axle). Turning the wheel through a full turn moves the effort through 2πR, while the axle rope moves through 2πr, so the effort moves farther than the load. If the effort is applied to the axle and the load at the wheel (a car wheel driven by its axle), the machine reduces force but increases speed and distance, like a third-class lever. Examples: steering wheel, screwdriver, winch, water wheel, pasta machine handle.

How do gears work?

Gears are wheels with teeth that mesh, so that one drives the other. Meshing gears turn in opposite directions. The teeth have the same size (pitch) so that the number of teeth is proportional to the circumference. The speed ratio follows from the fact that the teeth that pass the contact point in a given time must be the same for both gears:

N1 × ω1 = N2 × ω2, so ω2 = ω1 × (N1 ÷ N2)

and the torque scales inversely, so for an ideal pair τ2 = τ1 × (N2 ÷ N1). A small driving gear turning a large driven gear gives a lower speed and higher torque (a speed-reducing, force-multiplying gear), and a large driving gear turning a small driven gear gives a higher speed and lower torque. Power (torque times angular speed) is conserved in the ideal case, apart from friction losses.

What are gear trains and bicycle gears?

In a gear train, several gears are connected in sequence, and the overall ratio is the product of the individual ratios. An idler gear placed between two gears reverses the direction, but does not change the ratio. A chain and sprockets (bicycle) act as gears whose directions are the same. In a bicycle, the front chainring (for example 48 teeth) drives the rear sprocket (for example 16 teeth), so the wheel turns three times for each turn of the pedals. A low gear (large rear sprocket, small chainring) gives more torque for climbing, and a high gear gives more speed on the flat. Gearboxes in cars, wind turbines, clocks and mechanical watches use gear trains.

Side view of a windlass with a crank radius of 0.40 m and a drum radius of 0.050 m R = 0.40 m r = 0.050 m Effort 50 N 400 N Load IMA = R ÷ r = 0.40 ÷ 0.050 = 8
Side view of the windlass in the worked example. The effort acts at the large radius R = 0.40 m, and the rope winds on the small drum of radius r = 0.050 m, so the ideal mechanical advantage is R ÷ r = 8. An ideal effort of 50 N lifts the 400 N bucket. The drum is drawn larger than its true proportion for clarity.

Gear arrangements compared

Effects of gear size
Arrangement Speed of driven gear Torque of driven gear Use
Small driver, large drivenLowerHigherClimbing hills, winches
Large driver, small drivenHigherLowerHigh speed cycling, drills
Equal gearsSameSameReversing direction

Worked example: a windlass and a gear pair

  1. Windlass. A hand crank turns on a circle of radius 0.40 m (wheel radius R). The rope winds on a drum (axle) of radius 0.050 m. IMA = 0.40 ÷ 0.050 = 8. To lift a 400 N bucket, the ideal effort is 400 ÷ 8 = 50 N. With an efficiency of 80 percent, the effort = 400 ÷ (8 × 0.80) = 62.5 N.
  2. Distance. For one turn of the crank, the effort moves 2π × 0.40 = 2.51 m, and the bucket rises 2π × 0.050 = 0.314 m, a ratio of 8.
  3. Gear pair. A driving gear with 10 teeth turns a driven gear with 40 teeth at 240 rpm. The driven gear speed = 240 × (10 ÷ 40) = 60 rpm. The speed ratio is 4:1, and the ideal torque is multiplied by 4.
  4. Bicycle. A chainring of 48 teeth and a rear sprocket of 16 teeth: ratio 48 ÷ 16 = 3. At a cadence of 80 pedal revolutions per minute, the rear wheel turns at 240 rpm. For a wheel of circumference 2.1 m, the speed is 240 × 2.1 = 504 m/min = 30 km/h (504 × 60 ÷ 1000 = 30.2 km/h).
0.40 m 0.050 m 50 N 400 N 10 40 240 rpm 60 rpm

How were they developed?

The potter’s wheel and cart wheel appeared in Mesopotamia about 3500 BCE. Gears were used by the Greeks, and the Antikythera mechanism (about 100 BCE) used at least 30 bronze gears to model the motion of the Sun, Moon and planets. Water mills and windmills used gear trains in the Middle Ages, and the chain-driven bicycle was developed in the 1880s.

Which mistakes are common?

Mistake 1: Mixing up which gear is the driver

The ratio depends on the direction of drive. Define driver and driven carefully.

Mistake 2: Expecting both a speed gain and a torque gain

A gear pair trades speed for torque. It cannot increase both.

Mistake 3: Forgetting that meshing gears turn in opposite directions

An idler gear can be added to restore the direction.

Mistake 4: Using diameters of the wheel instead of the radii on one side and the axle on the other

Use consistent measures: either both radii or both diameters. The ratio is the same.