# Relationships in Levers

by Ron Kurtus (revised 26 September 2016)

There are relationships in levers between the lengths of the arms in a lever and the distances moved, speeds of objects, and the forces applied. These can be derived.

Questions you may have include:

• What is are the distance relationships?
• What is are the speed relationships?
• What is are the force relationships?

This lesson will answer those questions.

## Distance relationships

When the lever is moved about its fulcrum, the relationship between the effort distance and load distance is dependent on the ratio of the arms of the lever, according to the equation:

DO/DI = dO/dI

where

• DO is the output distance the load force is moved
• DI is the input distance the effort forces moves
• dO is the length of the load or output arm
• dI is the length of the effort or input arm

(See Increasing Distance Moved with a Lever)

Class 1 or Class 3 levers are used to increase the output distance.

## Speed relationships

You can find the relationship of the speeds with the arms of the lever by comparing with the distance relationships. Since speed equals distance divided by time, the output and input speeds are:

DO = SOt

DI = SIt

Divide by DI and SIt:

DO/DI = SOt/SIt = SO/SI

Since:

DO/DI = dO/dI

Therefore:

SO/SI = dO/dI

(See Increasing Speed with a Lever)

Class 1 or Class 3 levers are used to increase the output speed.

## Force relationships

Class 1 or Class 2 levers are used to increase the output force. The derivation of the force relationships starts with the equation for the lever distance relationships (from above):

DO/DI = dO/dI

Now, the definition of work is force times the displacement:

W = FD

where

• W is work
• F is the force
• D is the displacement

According to the Law of Conservation of Energy, the output energy or work equals the input energy or work:

WO = WI

WO = FODO

WI = FIDI

where

• WO is the work done by the load or output work
• WI is the work done by the effort or input work
• FO is the output force or the load (can also be the weight lifted)
• DO is the output distance the load is moved
• FI is the effort or input force
• DI is the input distance the effort moves

Thus:

FODO = FIDI

Divide by FO and DO:

FO/FI = DI/DO

Since DO/DI = dO/dI, you get:

FO/FI = dI/dO

Note that the force relationship is the reciprocal of the distance relationship.

(See Increasing Force with a Lever)

## Summary

The relationships between the lengths of the arms in a lever, and the distances moved, speeds of objects, and the forces applied are:

DO/DI = dO/dI

SO/SI = dO/dI

FO/FI = dI/dO (Notice that the force relationship is the reciprocal of the others)

Think logically

## Resources and references

Ron Kurtus' Credentials

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## Questions and comments

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www.school-for-champions.com/machines/
levers_relationships.htm

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