Final answer:
The period of a pendulum is inversely proportional to the square root of the acceleration due to gravity, and the time rate of change of mechanical energy for a damped oscillator is always negative due to energy dissipation.
Step-by-step explanation:
The relationship between the acceleration due to gravity (g) and the period (T) of a simple pendulum is given by the formula T = 2π√(L/g), where L is the length of the pendulum. This shows that the period is directly proportional to the square root of the length of the pendulum and inversely proportional to the square root of the acceleration due to gravity.
As for the time rate of change of mechanical energy of a damped oscillator, it is always negative because damping causes the mechanical energy of the system to decrease over time. A damped oscillator loses energy to its surroundings, resulting in a dissipation of energy which is shown in the negative sign of its time rate of change of mechanical energy.