# Conservation Of Energy Formula

Energy is the ability to do work. Practically, energy is required to perform every activity or work that you do in your day to day life. Energy is also required for vehicles to move, machines to run, bulbs to glow, etc. Energy is very crucial for the existence of human beings. And it exists in various forms. Different forms of energy are electrical energy, tidal energy, light energy, chemical energy, gravitational energy and nuclear energy, heat energy. One form of energy can easily be transferred to another. In this section, we will learn about the law of conservation of energy, the conservation of energy formula and its derivation along with some examples.

## Law of Conservation of Energy

According to the law of conservation of energy, the total energy of an isolated system remains conserved over time. In this definition, the isolated system refers to as a thermodynamic system so designed that no matter or energy can pass through it. Â Â Â

### Conservation of Energy Formula

Energy spent in one act = Energy gained in the related act

For a given system, we can write,

$$\begin{array}{l}E_{in}\;-\;E_{out}=\Delta E_{sys}\end{array}$$

As we know, the net amount of energy transfer into or out of any system occurs in the form of heat (Q), mass (m) and work (W). Hence, we can rewrite the aforementioned equation as:

$$\begin{array}{l}E_{in}\;-\;E_{out}=Q\;-\;W\end{array}$$

Upon dividing all the terms into both sides of the equation by the mass of the system, the equation represents the law of conservation of energy on a unit mass basis, as shown below:

$$\begin{array}{l}Q\;-\;W=\Delta u\end{array}$$

Thus, we can write the conservation of energy rate equation as:

$$\begin{array}{l}Q\;-\;W=\frac{dU}{dt}\end{array}$$

### Real-Life Example

We can take the example of a windmill. The mechanical energy of wind rotates the turbines of a windmill, which in turn rotate the shaft of an electric generator, thus generating electrical energy. Here, we observe that the mechanical energy of wind gets converted into electrical energy.

### Solved Examples

Example 1:Â The initial energy and the final energy of a system can be given by 2.95 Ã— 10-3Â and 5.86 Ã— 10-3Â respectively. Find the energy conservation of the system.

Solution:

We can use the following formula to compute the energy conservation of the system:

$$\begin{array}{l}\Delta E_{sys}=E_{in}-E_{out}\end{array}$$

$$\begin{array}{l}\Delta E=(5.83\times 10^{-3})-\left ( 2.95\times 10^{3} \right )\end{array}$$

$$\begin{array}{l}Thus, \Delta E=2.91\times 10^{-3}\end{array}$$

Example 2:Â A particle of charge equal to that of electron and the charge is 1.67 Ã— 10-27Â and mass of the particle is 1.30 Ã— 10-27Â kg. Compute the law of conservation of energy.

Solution:

As per the law of conservation of energy formula, we have:

$$\begin{array}{l}Q-W=\Delta E_{sys}\end{array}$$

$$\begin{array}{l}Or\Delta E_{sys}=Q-W=\left ( 1.67\times 10^{27} \right )-\left ( 1.30\times 10^{27} \right )=0.37\times 10^{27}\end{array}$$

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