# Conservative Forces: What is it & Why it Matters (w/ Examples)

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When someone lifts a coffee cup off the table, they are working against gravity. Once the cup is no longer on the table, but midair, it has stored potential energy. If let go, it would fall.

The force of gravity is an example of a conservative force. Other forces around us similarly store energy in a system by holding objects in a particular configuration that will result in motion, transforming the stored energy to kinetic energy, once released.

The total kinetic and potential energy in a system is referred to as the system's total mechanical energy. The total energy is the sum of the mechanical energy and thermal and other forms of energy.

#### TL;DR (Too Long; Didn't Read)

Conservative forces result in stored potential energy_._

## Definition of a Conservative Force

Take a look at that coffee cup again. Whether the cup is tossed upwards higher than the person's waist before being caught in their hands at waist height, or whether it is simply lifted straight up to waist height, the amount of gravitational potential energy the cup contains at the end stays the same. It will have the same fall if it drops from waist height in either case.

Put another way, the amount of potential energy an object gains from experiencing a conservative force is path-independent: it depends only on the object's final displacement (the difference between final position and initial position), not the total distance traveled. While in the first case the cup traveled farther in total from point a to point b – moving up, stopping instantaneously and then falling down to the hands that caught it – it still started and ended in the same positions as the cup that was simply lifted up. That makes the total change in potential energy from the force of gravity for each cup the same.

In contrast, a non-conservative force does not result in stored energy. Instead, the energy used by a non-conservative force to do work dissipates into the surrounding environment as heat (thermal energy) and cannot be reused.

Another common definition of a conservative force is one in which the net work done in a closed path is zero. A closed path is one in which the object starts and ends in the same position (its displacement is zero). Because work in physics is the force F acting on an object times its displacement d (W = F × d), it follows that if the displacement of an object is zero, so is the total work. The force that moved it, then, must have been conservative.

## Examples of Conservative Forces

Like gravity, other conservative forces cause a change in an object's position that results in stored energy. For example, the spring's force in a coiled toy like a Slinky or an arrow being pulled back on a bow are conservative forces, as are the electrostatic forces holding two charged particles at some distance from one another, or the magnetic forces holding the North and South poles or two bar magnets apart.

As soon as the objects are "let go" in any of these cases, they will move. The Slinky will pop up, the arrow will fly to the target or the particles will attract or repel.

## Why Conservative Forces Matter

Conservative forces allow humans to store energy that can later be put to useful work. They are a source of reusable energy – an object can be raised off the ground for the gravitational force to act on it over and over, and a bow-and-arrow can be set and released multiple times.

Of course, in the real wold, conservative forces do not exist in isolation. In each of the situations described here, some energy is always dissipated as heat from the force of friction, a non-conservative force, and no longer able to return directly to stored energy in that system.

Nor are conservative forces themselves limitless. At some point, a stretched spring will reach its elastic limit and break. A coffee cup dropped from too high will smash apart beyond repair. But with a solid understanding of these object's limits, a physicist or engineer can make dams, grandfather clocks, car batteries and myriad other devices that can do useful things.