Final answer:
The time rate of change of mechanical energy for a damped oscillator is always negative, indicating a loss of energy. Oscillators lose energy through damping as thermal energy, and an isolated system is one where no external forces are acting. The correct option is A.
Step-by-step explanation:
Damping and Oscillation
The time rate of change of mechanical energy for a damped oscillator is always negative. This is because the presence of damping results in a continuous loss of mechanical energy in the form of thermal energy due to friction or resistance. As the damped oscillator moves, it loses kinetic energy and potential energy, with no external work being done on the system, leading to a decrease in the total mechanical energy over time.
In terms of equation representation, if we denote the mechanical energy of the system as E, then the time derivative dE/dt will be less than zero (dE/dt < 0).
Considering an isolated system, it is defined as a system in which the net external force is zero. This means no energy is transferred in or out of the system by work or heat transfer, and it is often used in discussions of conservative forces and energy conservation in physics.