by Ron Kurtus (22 June 2016)

There are three types of mechanical advantages in machines: force, distance, and speed. For a given machine, there are relationships between these mechanical advantages.

Note: Most science books only consider force mechanical advantage, but we will discuss all three, since they are equally important.

Questions you may have include:

• What are the equations for the mechanical advantages?
• What is the distance-speed relationship?
• What is the distance-force relationship?

This lesson will answer those questions. Useful tool: Units Conversion

The mechanical advantage of a machine is defined as its output divided by its input.

where

• DO is the distance the load moves or the output distance
• DI is the distance the effort moves or the input distance

Note: In this notation, MA is NOT M times A. It simply stands for Mechanical Advantage.

The equation for the speed mechanical advantage is:

MAS = SO/SI

where

• MAS is the speed mechanical advantage
• SO is the output speed of the load
• SI is the input speed of the effort

The force mechanical advantage equation is:

MAF = FO/FI

where

• MAF is the force mechanical advantage
• FO is the output force or load
• FI is the input force or effort required to move the object

## Distance-speed relationship

Since distance equals speed times time, there is a distinct relationship between distance mechanical advantage and speed mechanical advantage. For a given amount of time, t:

DO = SOt

DI = SIt

Divide output distance by input distance:

DO/DI = SOt/SIt = SO/SI

Thus:

Distance MA = Speed MA

## Distance-force relationship

According to the Law of Conservation of Energy, the output energy or work equals the input energy or work. Work is defined as a force times the distance moved. Thus:

WO = WI

WO = FODO

WI = FIDI

where

• WO is the output work
• WI is the input work

Thus:

FODO = FIDI

Divide to get:

FO/FI = DI/DO

Therefore:

Thus, Force MA = 1/Distance MA = 1/Speed MA.

### Example

In jacking up a car, you must move the jack-handle a greater distance than the car is lifted.

## Summary

The relationships between the three mechanical advantages is:

Distance MA = Speed MA

and

Force MA = 1/Distance MA = 1/Speed MA

## Resources and references

Ron Kurtus' Credentials

### Websites

The Mechanical Advantage of Machines - Carolina Curriculum (PDF)

Machines Resources

### Books

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