Final answer:
The conservation of mechanical energy in a system with conservative forces is shown by using Newton's third law and the work-energy theorem, where the net work done changes the kinetic and potential energy without altering the system's total mechanical energy.
Step-by-step explanation:
To demonstrate that the mechanical energy of an isolated system consisting of two bodies interacting with a conservative force is conserved, we begin by considering Newton's third law of motion. This law states that for every action, there is an equal and opposite reaction. When two bodies interact through a conservative force, the work done on one body by this force is equal and opposite to the work done on the other body.
According to the work-energy theorem, the net work done by all forces acting on a system is equal to the change in the system's kinetic energy. For a conservative force, the work done is related to the change in potential energy. In mathematical terms, this means Wnet = ΔKE + ΔPE where ΔKE is the change in kinetic energy and ΔPE is the change in potential energy.
In an isolated system with no external forces, the net work done by internal conservative forces results in energy being exchanged between kinetic and potential forms but does not result in a change in the total mechanical energy, which is the sum of kinetic and potential energies. Thus, mechanical energy is conserved. This is because the work done on one body leads to an increase in potential energy while decreasing kinetic energy, and the opposite occurs for the other body due to Newton's third law. The conservation of mechanical energy principle holds, stating that when only conservative forces act on a system, the total mechanical energy remains constant.