91Ó°ÊÓ

Mechanical energy: Mechanical energy is defined as the sum of kinetic energy and potential energy of the system.

Mathematically,

Here $\mathrm{ME}$is the mechanical energy of the system, $\mathrm{KE}$is the kinetic energy of the system, and $\mathrm{PE}$is the potential energy of the system.

When both conservative and nonconservative force acts on a body, the net work done is given as the sum of work done by the conservative force and the work done by the nonconservative force. Mathematically,

$W_{net}=W_c+W_{nc}$ ..…â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦..(1.1)

Here $W_c$is the total work done by all conservative forces and $W_{nc}$is the total work done by all nonconservative forces.

According to the work-energy theorem, the net work done on the system equals the change in kinetic energy of the system. Hence,

$W_{net}=\mathrm{Δ}\text{KE}$ …â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦..(1.2)

From equations (1.1) and (1.2), we get,

$W_c+W_{nc}=\mathrm{Δ}\text{KE}$ …â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦.…..(1.3)

The work done by a conservative force comes from a loss of gravitational potential energy. Hence,

$W_c=−\mathrm{Δ}\text{PE}$ …â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦.(1.4)

From equations (1.3) and (1.4), we get,

localid="1655385932588" $\begin{array}{c}−\mathrm{Δ}\text{PE}+W_{nc}=\mathrm{Δ}\text{KE}\\ W_{nc}=\mathrm{Δ}\text{KE}+\mathrm{Δ}\text{PE}\end{array}$…â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦.(1.5)

From equation (1.5), it is clear that the total mechanical energy changes by exactly the amount of work done by nonconservative force.

Rearranging equation (1.5)

$\begin{array}{c}{W}_{nc}=\left({\text{KE}}_f−{\text{KE}}_i\right)+\left({\text{PE}}_f−{\text{PE}}_i\right)\\ {\text{KE}}_i+{\text{PE}}_i+W_{nc}={\text{KE}}_f+{\text{PE}}_f\end{array}$…â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦.(1.6)

Therefore, from equation (1.6), it is clear that the amount of work done by all nonconservative forces adds to the mechanical energy of the system.

When only a conservative force acts on the system, the work done by all nonconservative forces will be zero, i.e., $W_{nc}=0$. Hence, equation (1.5) reduces to,

$0=\mathrm{Δ}\text{KE}+\mathrm{Δ}\text{PE}$ …â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦.(1.7)

Rearranging equation (1.7), we get,

$\begin{array}{c}0=\left({\text{KE}}_f−{\text{KE}}_i\right)+\left({\text{PE}}_f−{\text{PE}}_i\right)\\ {\text{KE}}_i+{\text{PE}}_i={\text{KE}}_f+{\text{PE}}_f\end{array}$…â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦â¶Ä¦.(1.8)

Therefore, from equation (1.8), it is clear that for a conservative force, the mechanical energy of the system remains constant.