Question

What happens to the potential energy of the body in the presence of conservative forces, if the kinetic energy is increased by an amount?

Answer 1

Presence of conservative forces:

1. The work of a conservative force is unaffected by the course taken.
2. In other words, a conservative force does the same amount of work on any path linking two places.
3. The potential energy of a thing reduces as it falls.
4. According to the law of conservation of energy, as kinetic energy increases, the decrease in potential energy is exactly equal to the increase in kinetic energy.
5. Therefore, if the kinetic energy increases it is equal t to the decrease in potential energy.
6. When conservative forces act on a body or a system, the sum of the kinetic and potential energy (mechanical energy) of the body or system is conserved.
7. If the kinetic energy of the body is increased by an amount $∆\mathrm{K}$ in the presence of conservative forces, the potential energy of the body will drop by the same amount, i.e. $∆\mathrm{U}$.

So,

$$\begin{gathered} ∆\mathrm{K}=∆\mathrm{U} \\ ∆\mathrm{K}={\mathrm{K}}_2-{\mathrm{K}}_1 \\ \mathrm{and} \\ ∆\mathrm{U}={\mathrm{U}}_2-{\mathrm{U}}_1 \\ {\mathrm{K}}_2-{\mathrm{K}}_1=-\left({\mathrm{U}}_2-{\mathrm{U}}_1\right) \\ {\mathrm{K}}_2+{\mathrm{U}}_2={\mathrm{K}}_1+{\mathrm{U}}_1 \end{gathered}$$

Thus, in the presence of conservative forces, a body's total kinetic and potential energy will be conserved.