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In physics and engineering, mechanical advantage (MA) is the factor by which a mechanism multiplies the force or torque applied to it. Generally, the mechanical advantage is calculated as follows:

$MA = \frac{\text{distance over which effort is applied}}{\text{distance over which the load is moved}}$

or more simply:

$MA = \frac{\text{output force}}{\text{input force}}$

The first equation shows that the force exerted IN to the machine multiplied by the distance moved IN will always be equal to the force exerted OUT of the machine multiplied by the distance moved OUT. For example, using a block and tackle with 6 ropes, and a 600 pound load, the operator would be required to pull the rope 6 feet, and exert 100 pounds of force to lift the 600 pound load 1 foot.

The second equation is a simplified formula based just on the forces in and out. Using the example above, 100 pounds of force IN results in 600 pounds of force OUT, an MA of 6. Both of these equations calculate only the ideal mechanical advantage (IMA) and ignore any losses due to friction. The actual mechanical advantage (AMA) includes those frictional losses. The difference between the two is the mechanical efficiency of the system.

## Types

The ideal mechanical advantage (IMA), or theoretical mechanical advantage, is the mechanical advantage of an ideal machine. It is usually calculated using physics principles because there is no ideal machine.

The IMA of a machine can be found with the following formula:

$IMA = \frac {D_E} {D_R}$

where

DE equals the 'effort distance' (the distance from the fulcrum to where the effort is applied)
DR equals the resistance distance (the distance from the fulcrum to where the resistance is applied)

The actual mechanical advantage (AMA) is the mechanical advantage of a real machine. Actual mechanical advantage takes into consideration real world factors such as energy lost in friction.

The AMA of a machine is calculated with the following formula:

$AMA = \frac {R} {E_\text{actual}}$

where

R = resistance force
Eactual = actual effort force

## Simple machines

Beam balanced around a fulcrum

The following simple machines exhibit a mechanical advantage:

• The beam shown is in static equilibrium around the fulcrum. This is due to the moment created by vector force "A" counterclockwise (moment A*a) being in equilibrium with the moment created by vector force "B" clockwise (moment B*b). The relatively low vector force "B" is translated in a relatively high vector force "A". The force is thus increased in the ratio of the forces A : B, which is equal to the ratio of the distances to the fulcrum b : a. This ratio is called the mechanical advantage. This idealised situation does not take into account friction. For more explanation, see also lever.
• Wheel and axle motion (e.g. screwdrivers, doorknobs): A wheel is essentially a lever with one arm the distance between the axle and the outer point of the wheel, and the other the radius of the axle. Typically this is a fairly large difference, leading to a proportionately large mechanical advantage. This allows even simple wheels with wooden axles running in wooden blocks to still turn freely, because their friction is overwhelmed by the rotational force of the wheel multiplied by the mechanical advantage.
• Pulley: Pulleys change the direction of a tension force on a flexible material, e.g. a rope or cable. In addition, a Block and tackle of multiple pulleys creates mechanical advantage, by having the flexible material looped over several pulleys in turn. Adding more loops and pulleys increases the mechanical advantage.
• Screw: A screw is essentially an inclined plane wrapped around a cylinder. The run over the rise of this inclined plane is the mechanical advantage of a screw.[1]

### Pulleys

An example of a rope and pulley system illustrating mechanical advantage.

Consider lifting a weight with rope and pulleys. A rope looped through a pulley attached to a fixed spot, e.g. a barn roof rafter, and attached to the weight is called a single pulley. It has an MA = 1 (assuming frictionless bearings in the pulley), meaning no mechanical advantage (or disadvantage) however advantageous the change in direction may be.

A single movable pulley has an MA of 2 (assuming frictionless bearings in the pulley). Consider a pulley attached to a weight being lifted. A rope passes around it, with one end attached to a fixed point above, e.g. a barn roof rafter, and a pulling force is applied upward to the other end with the two lengths parallel. In this situation the distance the lifter must pull the rope becomes twice the distance the weight travels, allowing the force applied to be halved. Note: if an additional pulley is used to change the direction of the rope, e.g. the person doing the work wants to stand on the ground instead of on a rafter, the mechanical advantage is not increased.

By looping more ropes around more pulleys we can continue to increase the mechanical advantage. For example if we have two pulleys attached to the rafter, two pulleys attached to the weight, one end attached to the rafter, and someone standing on the rafter pulling the rope, we have a mechanical advantage of four. Again note: if we add another pulley so that someone may stand on the ground and pull down, we still have a mechanical advantage of four.

Here are examples where the fixed point is not obvious:

• A velcro strap on a shoe passes through a slot and folds over on itself. The slot is a movable pulley and the MA = 2.
• Two ropes laid down a ramp attached to a raised platform. A barrel is rolled onto the ropes and the ropes are passed over the barrel and handed to two workers at the top of the ramp. The workers pull the ropes together to get the barrel to the top. The barrel is a movable pulley and the MA = 2. If there is enough friction where the rope is pinched between the barrel and the ramp, the pinch point becomes the attachment point. This is considered a fixed attachment point because the rope above the barrel does not move relative to the ramp. Alternatively the ends of the rope can be attached to the platform.
• Block and tackle: MA = 3

### Screws

The theoretical mechanical advantage for a screw can be calculated using the following equation:[2]

$MA = \frac{\pi d_m}{l}$

where

dm = the mean diameter of the screw thread

Note that the actual mechanical advantage of a screw system is greater, as a screwdriver or other screw driving system has a mechanical advantage as well.

## References

### Notes

1. ^ Fisher, pp. 69–70.
2. ^ United States Bureau of Naval Personnel, p. 5-4.