Q7.4-11CQ Question: Define mechanical ener... [FREE SOLUTION]

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Chapter 7: Q7.4-11CQ (page 258)

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

Short Answer

Expert verified

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

Step by step solution

Step 1: Definition of Concept

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

Mathematically,

ME=KE + PE

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

Explain the relationship between Mechanical Energy to Nonconservative Force

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,

$W_{n e t} = W_{c} + W_{n c}$ ..鈥︹赌︹赌︹赌︹赌�..(1.1)

Here $W_{c}$ is the total work done by all conservative forces and $W_{n c}$ 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,

$W_{n e t} = 螖 KE$ 鈥︹赌︹赌︹赌︹赌︹赌︹赌�..(1.2)

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

$W_{c} + W_{n c} = 螖 KE$ 鈥︹赌︹赌︹赌︹赌︹赌�.鈥�..(1.3)

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

$W_{c} = 鈭� 螖 PE$ 鈥︹赌︹赌︹赌︹赌︹赌︹赌︹赌︹赌�.(1.4)

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

localid="1655385932588" $\begin{matrix} 鈭� 螖 PE + W_{n c} = 螖 KE \\ W_{n c} = 螖 KE + 螖 PE \end{matrix}$ 鈥︹赌︹赌︹赌︹赌︹赌︹赌︹赌︹赌�.(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)

$\begin{matrix} W_{n c} = ({KE}_{f} 鈭� {KE}_{i}) + ({PE}_{f} 鈭� {PE}_{i}) \\ {KE}_{i} + {PE}_{i} + W_{n c} = {KE}_{f} + {PE}_{f} \end{matrix}$ 鈥︹赌︹赌︹赌︹赌︹赌︹赌︹赌︹赌�.(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.

Explain when an only a conservative force acts on the system

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

$0 = 螖 KE + 螖 PE$ 鈥︹赌︹赌︹赌︹赌︹赌︹赌︹赌︹赌�.(1.7)

Rearranging equation (1.7), we get,

$\begin{matrix} 0 = ({KE}_{f} 鈭� {KE}_{i}) + ({PE}_{f} 鈭� {PE}_{i}) \\ {KE}_{i} + {PE}_{i} = {KE}_{f} + {PE}_{f} \end{matrix}$ 鈥︹赌︹赌︹赌︹赌︹赌︹赌︹赌︹赌�.(1.8)