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## Relationship between Work and Mechanical Energy

by Ron Kurtus (revised 7 October 2013)

Work is the measurement of the force on an object that *overcomes a resistive force* times the distance the object is moved. In certain situations there is a relationship between that work and mechanical energy (as opposed to heat or radiant energy).

When you accelerate the object, you are doing work against inertia, such that the work equals the change in kinetic energy of the object.

When you are doing work against continuous resistive forces, such as gravity or spring tension, work done equals the change in potential energy of the object.

Questions you may have include:

- What is the equation for work?
- How is work the change in kinetic energy?
- When is work the change in potential energy?

This lesson will answer those questions.

Useful tool: Metric-English Conversion

## Work is force times distance

The definition of work is that it equals a force against some resistance times the distance traveled while that force is being applied. The equation is:

W = Fd

where:

**W**is the work in joules (J or kg-m²/s²)**F**is the force applied to an object in newtons (N or kg-m/s²)**d**is the distance the object moves in the direction of the force in meters (m)**Fd**is**F**times**d**

Thus, if you would apply a force of **F** = 3 newtons (3 N) to move an object a distance of **d** = 5 meters (5 m), the work done would be **W = Fd** = 15 joules (15 J).

## Work as change in kinetic energy

When you accelerate an object, you are doing work against inertia plus any resistive forces such as gravity or friction over the distance that the object is accelerated. This means that the object's velocity—and thus its kinetic energy—changes over the distance moved.

The kinetic energy of an object is:

KE = mv^{2}/2

where

**KE**is the kinetic energy in kg-m^{2}/s^{2}**m**is the mass of the object in kg**v**is the velocity in m/s

The work done in changing the velocity against some resistance is then equal to the change in kinetic energy of the object:

ΔKE = mv_{f}^{2}/2 − mv_{i}^{2}/2

W = ΔKE

where

**Δ**means change (Greek letter capital delta)**v**is the final velocity of the object_{f}**v**is the initial velocity_{i}

Thus, in certain situations, work is the change in kinetic energy.

(

See Proof that Work Can be the Change in Kinetic Energy for more information.)

## Work as the change on potential energy

Some resistive forces, such as gravity and spring tension, act continuously on an object, such that the object has a potential of moving from its present position. This is called its potential energy (**PE**).

Moving the object from one **PE** to a higher one requires work, which can be measured by the change in potential energy (**ΔPE**).

### Example with gravity

For example, the **PE** of an object due to the force of gravity is:

PE = mgh_{i}

where

**m**is the mass of the object**g**is the acceleration due to gravity**h**is the initial height above the ground_{i}

If you lifted the object to another height (**h _{f}**), the new potential energy would be:

PE = mgh_{f}

The amount of work done would be

W = Fd

where

**F = mg****d = h**_{f}− h_{i}

Thus:

W = mg(h_{f}− h_{i})

W = mg_{f}− mgh_{i}

W = PE_{f}− PE_{i}= ΔPE

W = ΔPE

Thus in certain situations, work is the change in potential energy.

## Work as a combination of KE and PE

If you project an object upward, you are doing work both against gravity and inertia. In this situation, the total work done is:

W = ΔPE + ΔKE

## Summary

When you apply enough force on an object to *overcome a resistive force*, such that you move that object, you are doing* *work on that object. There is a relationship between that work and mechanical energy.

When you accelerate the object, you are doing work against inertia, such that the work equals the change in kinetic energy of the object.

When you are doing work against continuous resistive forces, such as gravity or spring tension, you change the potential energy of the object.

Physical Science rules

## Resources and references

### Websites

### Books

**Top-rated books on Physics of Work**

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## Relationship between Work and Mechanical Energy