## The Wheel and Axis Principle

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 ratiois 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 ratiois also equal toso