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# Conservative and Non-Conservative Force

## Introduction

**Conservative forces and non-Conservative forces** are the classification in which the forces are divided in Physics. Forces for which the Work done is free of the path through which it is carried are supposed to be Conservative forces. The forces for which the work done is reliant on the path are said to be non-Conservative forces. For Conservative forces, the aggregate sum of **mechanical energy** always remains **constant** but for non-Conservative forces, the mechanical energy retained by the body decreases by some amount and is converted to other forms like heat, sound, etc.

## What is a Conservative Force?

**Conservative force** is the force in which work done is determined by the path taken by the object. Here, Work is said to be Conservative if the **Work done** over a **closed track is Zero**. The total energy (sum of kinetic and potential) of any system is Conserved. Gravitational force, Buoyancy force, and Electrostatic forces are some examples of Conservative forces.

## What is Potential Energy?

The energy in a system or a body is due to its position or configuration is known as potential energy. It is stored or completely recoverable energy. A force is said to be Conservative if the Work done by or against it, doesn’t depend on the path. For **Potential energy**, the Work done depends only on starting and ending points and doesn’t depend on the path. So, it is a Conservative force. Potential energy is not associated with non-Conservative forces.

## Which Forces Are Conservative?

Forces for which the work done is depend on initial and final position, and energy is conserved are considered conservative. There are many conservative forces in nature such as electrostatic force, gravitational force, Buoyancy force, elastic spring restoring force, etc.

## What is a Non-Conservative Force?

A **Non-Conservative force** is a force in which work done relies on the track of the object. So, in this type of force, the starting and end position of the object is a matter of the work done. Here the transfer of mechanical energy takes place in some other form in which performing work is not possible. So, this force is also called **dissipative force or degradation force**. Viscosity, resistance to the flow of electric current, air resistance, and friction are some examples of non-conservative force.

Air resistance degrades or dissipates energy in the form of waves or turbulence.

Friction degrades energy in the form of heat and sound.

## Work Done by A Conservative Force

If the Work done is the same irrespective of the track taken by the object is known as a conservative force.

Consider the figure below −

**Fig1. Diagram showing the paths taken by the object**

Here the work done for both path-1 and path-2 are the same.

Work done

$$\mathrm{W=F×r}$$

Where F is force, r is the distance moved in the direction of force

Suppose A and B are the interconnecting points for the track, and the work done by the conservative force is represented as

$$\mathrm{W_{AB,path\:1}=\int F_{AB,path\:1} dr=W_{AB,path\:2}=\int F_{AB,path\:2} dr}$$

## Difference Between Conservative Force and Non-Conservative Force

Conservative force and non-conservative force can be differentiated as follows −

Conservative Force | Non-conservative Force |
---|---|

The force in which work done is independent of the track of the object. | The force in which work done depends on the track of the object. |

As shown in figure-1, the work done for both path-1 and path-2 is the same in a conservative force. | As shown in figure-1, the work done by the object is different for both path-1 and path-2 in a non-conservative force. |

In this type of force, one form of energy is transferred to another completely by applying or removing external force. | In this type of force, the transfer of mechanical energy takes place into some other form in which performing work is not possible. |

The total energy is conserved. | There will be energy loss due to dissipation. |

Work done can be regained. | Work done cannot be regained. |

In a closed track, work done by this force is zero. | In a track path, work done by this force is non-zero. |

Examples: Gravitational force, Buoyancy force, electrostatic forces | Examples: Viscosity, resistance to the flow of electric current, air resistance, friction |

**Table-1: Difference between Conservative forces and non-conservative forces**

## Conservative Force Formula

The work done by the conservative force is represented as

$$\mathrm{W=\int F.dx=-dU}$$

where F is force, x changes in distance, U is the potential energy

Then we can write −

$$\mathrm{F.dx=-dU}$$

On rearranging, we get the conservative force formula as

$$\mathrm{F=-\frac{dU}{dx}}$$

## Conclusion

In Physics, force is divided into two classifications namely conservative force and non-conservative force. The force in which work done is the same irrespective of the track taken by the object is called **conservative force**. Here, one form of energy is transferred to another completely by applying or removing external force. A **non-conservative force** is a force in which work done relies on the track of the object. Here the transfer of mechanical energy takes place in some other form in which performing work is not possible.

So, this force is also called dissipative force or degradation force. When a body moves in a direction opposite to that of a conservative force, the energy stored inside that body is called potential energy. There are many conservative forces in nature namely electrostatic force, gravitational force, Buoyancy force, elastic spring restoring force, etc. Conservative force and non-conservative force are differentiated based on their nature.

## FAQs

**Q1. What happens to stored energy if the force does productive work on the object?**

Ans. If the force does productive work on the object, the stored energy decreases, and vice versa. The stored energy is also called potential energy. The difference in stored energies in-between the positions of the two objects is the sum of work needed to displace the object from one site to another site.

**Q2. Water drag on a moving boat is an example of which type of force?**

Ans. Water drag on a moving boat is an illustration of non-conservative force. In a non-conservative force, the work done relies on the track of the object.

**Q3. What is the work done in a round trip?**

Ans. The work done in a closed track is zero.

**Q4. Write the expression for work done in a non-conservative force.**

Ans. Work done in non-conservative is given by −

$$\mathrm{ W=\Delta KE+\Delta PE }$$

Where KE is kinetic energy, PE is potential energy

**Q5. What is the potential energy reserve in an elastic spring?**

Ans. The potential energy reserve in an elastic spring is given by −

$$\mathrm{PE=\frac{1}{2}kx^2}$$

where k is spring constant, x is the change in length.