High School Physics + more

# 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:

Kf + Uf + ΔEth = Ki + Ui
[final KE +final potential energy + any change in thermal energy = initial KE + initial potential 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.

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.