Question

Derive an expression for the law of conservation of energy.

Answer 1

Hint: Conservation of energy means that the energy in a system should remain constant. There can be many types of energy like kinetic energy, potential energy, vibrational energy, etc.. the sum of all these energies should be a constant for a system. The magnitude of different types of energy can change with time, but their sum should always be constant.

Complete step by step answer:
In order to prove the law of conservation of energy in a system, let us consider a conservative force $\left( F \right)$ acting on a particle. The force displaces the particle by a small distance $dr$, so the work done in displacing the particle through a small distance $dr$ is,
$W=\int\limits_{{{r}_{1}}}^{{{r}_{2}}}{\overrightarrow{F}\centerdot \overrightarrow{dr}}$ … equation(1)
Where ${{\text{r}}_{1}}\text{ and }{{\text{r}}_{2}}$ are the initial and final positions of the particle.
We know that in a conservative system the work done can be found out by finding the change in kinetic energy of the particle, so we can write,
$W=K.{{E}_{f}}-K.{{E}_{i}}$ … equation (2)
Where,
$K.{{E}_{f}}$ is the final kinetic energy of the particle after the force is applied.
$K.{{E}_{i}}$ is the initial kinetic energy of the particle before the force is applied.
From equation (1) and (2), we can write,
$\int\limits_{{{r}_{1}}}^{{{r}_{2}}}{\overrightarrow{F}\centerdot \overrightarrow{dr}}=K.{{E}_{f}}-K.{{E}_{i}}$ .. equation (3)
Also, the work done by a conservative force is the negative of the change in potential energy of the system. So, we can write,
$W=-\left( P.{{E}_{f}}-P.{{E}_{i}} \right)$ … equation (4)
$P.{{E}_{f}}$ is the final potential energy of the particle after the force is applied.
$P.{{E}_{i}}$ is the initial potential energy of the particle before the force is applied.
From equation (4) and equation (1), we can write,
$\int\limits_{{{r}_{1}}}^{{{r}_{2}}}{\overrightarrow{F}\centerdot \overrightarrow{dr}}=-\left( P.{{E}_{f}}-P.{{E}_{i}} \right)$ .. equation (5)
From equation (3) and (5), we can write,
$K.{{E}_{f}}-K.{{E}_{i}}=-\left( P.{{E}_{f}}-P.{{E}_{i}} \right)$
$\therefore \left( P.{{E}_{f}}+K.{{E}_{f}} \right)=\left( P.{{E}_{i}}+K.{{E}_{i}} \right)$
So, the sum of initial kinetic energy and the potential energy is equal to the sum of the final kinetic and potential energy of the system. So, energy is conserved in this case.
So, for a system acted upon by a conservative force, its sum of potential and kinetic energy remains a constant.

Note: The work done by a conservative force only depends on the initial and final position of the body and not on the path taken.
Gravitational force and Electrostatic force are some popular examples of the conservative force.
If the work done by a force depends on the path taken by the body, then we call these forces as non-conservative forces. The frictional force is an example of a non-conservative force.