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Q7.4-11CQ

Expert-verifiedFound in: Page 258

Book edition
1st Edition

Author(s)
Paul Peter Urone

Pages
1272 pages

ISBN
9781938168000

**Question: Define mechanical energy. What is the relationship of mechanical energy to nonconservative forces? What happens to mechanical energy if only conservative forces act?**

When only a conservative force acts on the system, the total mechanical energy of the system remains conserved.

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

Mathematically,

$\text{ME=KE}+\text{PE}$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,

${W}_{net}={W}_{c}+{W}_{nc}$ ..……………..(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,

${W}_{net}=\Delta \text{KE}$ …………………..(1.2)

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

${W}_{c}+{W}_{nc}=\Delta \text{KE}$ ……………….…..(1.3)

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

${W}_{c}=-\Delta \text{PE}$ ……………………….(1.4)

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

localid="1655385932588" $\phantom{\rule{0ex}{0ex}}\begin{array}{c}-\Delta \text{PE}+{W}_{nc}=\Delta \text{KE}\\ {W}_{nc}=\Delta \text{KE}+\Delta \text{PE}\end{array}$……………………….(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)

$\begin{array}{c}{W}_{nc}=\left({\text{KE}}_{f}-{\text{KE}}_{i}\right)+\left({\text{PE}}_{f}-{\text{PE}}_{i}\right)\\ {\text{KE}}_{i}+{\text{PE}}_{i}+{W}_{nc}={\text{KE}}_{f}+{\text{PE}}_{f}\end{array}$……………………….(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,

$0=\Delta \text{KE}+\Delta \text{PE}$ ……………………….(1.7)

Rearranging equation (1.7), we get,

$\begin{array}{c}0=\left({\text{KE}}_{f}-{\text{KE}}_{i}\right)+\left({\text{PE}}_{f}-{\text{PE}}_{i}\right)\\ {\text{KE}}_{i}+{\text{PE}}_{i}={\text{KE}}_{f}+{\text{PE}}_{f}\end{array}$……………………….(1.8)

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

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