The wheel and axle can be regarded as a form of lever: the radius of the wheel corresponds to the force arm, to which force is applied; the radius of the axle corresponds to the resistance arm; and the axis of the axle corresponds to the fulcrum. A wheel and axle has a theoretical mechanical advantage (ratio of the force delivered by the machine to the force put into the machine, disregarding friction) equal to the radius of the wheel divided by the radius of the axle.
 

Mechanical advantage

The simple machine called a wheel and axle refers to the assembly formed by two disks, or cylinders, of different diameters mounted so they rotate together around the same axis. Forces applied to the edges of the two disks, or cylinders, provide mechanical advantage. When used as the wheel of a cart the smaller cylinder is the axle of the wheel, but when used in a windlass, winch, and other similar applications (see medieval mining lift to right) the smaller cylinder may be separate from the axle mounted in the bearings. It cannot be used separately.

Assuming the wheel and axle does not dissipate or store energy, the power generated by forces applied to the wheel must equal the power out at the axle. As the wheel and axle system rotates around its bearings, points on the circumference, or edge, of the wheel move faster than points on the circumference, or edge, of the axle. Therefore a force applied to the edge of the wheel must be less than the force applied to the edge of the axle, because power is the product of force and velocity.

Let a and b be the distances from the center of the bearing to the edges of the wheel A and the axle B. If the input force F_A is applied to the edge of the wheel A and the force F_B at the edge of the axle B is the output, then the ratio of the velocities of points A and B is given by a/b, so the ratio of the output force to the input force, or mechanical advantage, is given by

The mechanical advantage of a simple machine like the wheel and axle is computed as the ratio of the resistance to the effort. The larger the ratio the greater the multiplication of force (torque) created or distance achieved. By varying the radii of the axle and/or wheel, any amount of mechanical advantage may be gained. In this manner, the size of the wheel may be increased to an inconvenient extent. In this case a system or combination of wheels (often toothed, that is, gears) are used. As a wheel and axle is a type of lever, a system of wheels and axles is like a compound lever.

Ideal mechanical advantage

The ideal mechanical advantage of a wheel and axle is calculated with the following formula:

Actual mechanical advantage

The actual mechanical advantage of a wheel and axle is calculated with the following formula:

where

R = resistance force, i.e. the weight of the bucket in this example.

E_actual = actual effort force, the force required to turn the wheel.

A 2.5-ton screw jack. The jack is operated by inserting the bar (visible lower left) in the holes at the top and turning.

A jackscrew operates this automotive scissor jack.

Antique locomotive screw jack

Antique wooden jackscrew for repair of cart and wagon wheels(Ethnographic Museum of Elhovo,Bulgaria)

Advantages

An advantage of jackscrews over some other types of jack is that they are self-locking, which means when the rotational force on the screw is removed, it will remain motionless where it was left and will not rotate backwards, regardless of how much load it is supporting. This makes them inherently safer than hydraulic jacks, for example, which will move backwards under load if the force on the hydraulic actuator is accidentally released.

Mechanical advantage

The mechanical advantage of a screw jack, the ratio of the force the jack exerts on the load to the input force on the lever, ignoring friction, is

where

 is the force the jack exerts on the load

 is the rotational force exerted on the handle of the jack

 is the length of the jack handle, from the screw axis to where the force is applied

 is the lead of the screw.

However, most screw jacks have large amounts of friction which increase the input force necessary, so the actual mechanical advantage is often only 30% to 50% of this figure.

Applications

A jackscrew's threads must support heavy loads. In the most heavy-duty applications, such as screw jacks, a square thread or buttress thread is used, because it has the lowest friction. In other application such as actuators, an Acme thread is used, although it has higher friction.

The large area of sliding contact between the screw threads means jackscrews have high friction and low efficiency as power transmission linkages, around 30%–50%. So they are not often used for continuous transmission of high power, but more often in intermittent positioning applications.

The ball screw is a more advanced type of leadscrew that uses a recirculating-ball nut to minimize friction and prolong the life of the screw threads. The thread profile of such screws is approximately semicircular (commonly a "gothic arch" profile) to properly mate with thebearing balls. The disadvantage to this type of screw is that it is not self-locking.

Jackscrews form vital components in equipment. For instance, the failure of a jackscrew on a McDonnell Douglas MD80 airliner due to a lack of grease resulted in the crash of Alaska Airlines Flight 261 off the coast of California in 2000.

The jackscrew figured prominently in the classic novel Robinson Crusoe. It was also featured in a recent History Channel program as thesaving tool of the Pilgrims' voyage – the main crossbeam, a key structural component of their small ship, cracked during a severe storm. A farmer's jackscrew secured the damage until landfall.