# Law of Conservation of Energy and how to derive its equation | Isolated systems – examples

Last updated on October 2nd, 2023 at 04:14 pm

The * Law of conservation of energy states that the total energy of an isolated system remains constant.* Now in this post, we will discuss this in detail, write its statement, and then derive an equation expressing this law of conservation of energy. So let’s begin.

**Law of Conservation of Energy – Explanation**

As per the work-energy equation the total energy of a system changes by the amount of work done on it. That means to conserve the energy of a system no work can be done on it. To discuss the Law of conservation of energy we need to maintain this condition.

Now let’s consider an isolated system that is separated from its surrounding environment in such a way that no energy is transferred into or out of the system. This means that *no work is done on the system*.

The energy within this isolated system may be transformed from one form into another, but it is a remarkable fact of nature that, during these transformations, the total energy of an isolated system i.e. the *sum *of all the individual kinds of energy remains *constant.*

**We say that the total energy of an isolated system is conserved.**

## State the Law of Conservation of Energy

The **Law of conservation of energy **states that the total energy of an isolated system remains constant.

## How to derive the Law of Conservation of Energy equation?

To derive the equation expressing the Law of conservation of energy we need to start from the work-energy equation that goes like this: **ΔE = ΔK + ΔUg + ΔUs + ΔEth + ΔEchem + …= W…………….(1)**

[ K denotes Kinetic energy, U stands for potential energy, Eth is the thermal energy and Echem is the chemical energy]

For an isolated system, we must set *W *= 0 in Equation (1), leading to the following statement of the law of conservation of energy: The law** of conservation of energy states that the total energy of an isolated system remains constant.**

As said earlier, to get the Equation expressing the Law of conservation of energy we need to set W=0 and that gives us the following equations: **ΔE = ΔK + ΔUg + ΔUs + ΔEth + ΔEchem + …= 0**

=> **ΔE = 0 **[This is the Law of Conservation of Energy Equation that states that for an isolated system, total energy remains constant. In other words, change in the total energy equals zero.]

This equation can also be written in a bit different way. As said earlier, in an isolated system no work is done on the system and no energy is transferred into or out of the system. **So the final energy, including any change in thermal energy, equals the initial energy.** And when we write the equation for this it looks like this:

**K _{f} + U_{f} + ΔE_{th} = K_{i }+ U_{i}**

[final KE +final potential energy + any change in thermal energy = initial KE + initial potential energy]

*Applications of the Law of conservation of energy*

*Applications of the Law of conservation of energy*

We can apply the law of conservation of energy in different isolated systems and formulate a problem-solving approach to solve different numerical problems.

This law relates a system’s final energy to its initial energy.

We can solve for initial and final heights, speeds, and displacements from these energies.

The following table lists a few such isolated systems where this energy conservation law is applied.

Isolated System Name | Isolated System description | Law of conservation of energy Applied (Yes/No) |
---|---|---|

An object in free fall | We choose the falling object and the earth as the system so that the forces between them are internal forces.There are no external forces to do work, so the system is isolated. | Yes |

An object compressing a spring | We choose the object and the spring to be the system. The forces between them are internal forces, so no work is done. | Yes |

An object sliding down a frictionless ramp | The external force (Normal force) the ramp exerts on the object is perpendicular to the motion, and so does no work. The object and the earth together form an isolated system. | Yes |

An object sliding along a surface with friction | The block and the surface interact via kinetic friction forces, but these forces are internal to the system. There are no external forces to do work, so the system is isolated | Yes |

**List of isolated systems where the Law of conservation of energy can be applied to take a problem-solving approach.**

Note: **Numerical problems**: Here is the link to one useful post in this portal where you will find a good collection of **solved numerical problems on energy conversion or transformation**.

## Summary | Suggested Reading

In this post, we have stated and explained the law and derived the equation for the law of conservation of energy using the work-energy equation. We have also seen the concept of the isolated system. Now here is a small list for your reading.

S**uggested Reading:** **Work-Kinetic Energy Theorem****Work done by a force**