The wheel and axis principle is used to produce a large turning force. It is most frequently used in the gearboxes of cars. A large force is required to start the car moving (low gear), which can be reduced once the car starts to move (high gear).
Assuming the mechanism is 100% efficient, the force done by force F-r is equal to the force needed to move the force F-e .If the axle turns n times, the work done by
is equal to
is equal to
Equating these gives
The ratio
is called the 'mechanical advantage'.
In a set of gears the inner and outer wheels are covered in teeth, which mesh in a 1 – 1 way like the teeth in a zip. This means that driven (inner wheel above) and outer wheel (driving wheel above) rotate so that the teeth intermesh and wheels turn in inverse ratios to their circumference (and radii).
If the outer wheel has N teeth and the inner wheel has n teeth then we may define the velocity ratio as
Each wheel will turn the same distance in the same time so that
and
Dividing the first of these by the second gives
The ratio
is also equal to
so