Gravity doesn't make a pendulum stop. Air resistance and friction in the pivot are the
things that rob its energy. If you could eliminate those and leave it all up to gravity,
the pendulum would never stop.
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Yes. Given a constant for gravity, the period of the pendulum is a function of it's length to the center of mass. In a higher gravity, the period would be shorter for the same length of pendulum.
The period of a pendulum is give approximately by the formula t = 2*pi*sqrt(l/g) where l is the length of the pendulum and g is the acceleration (not accerlation) due to gravity. Thus g is part of the formula for the period.
Height does not affect the period of a pendulum.
It messes up the math. For large amplitude swings, the simple relation that the period of a pendulum is directly proportional to the square root of the length of the pendulum (only, assuming constant gravity) no longer holds. Specifically, the period increases with increasing amplitude.
For a simple pendulum, consisting of a heavy mass suspended by a string with virtually no mass, and a small angle of oscillation, only the length of the pendulum and the force of gravity affect its period. t = 2*pi*sqrt(l/g) where t = time, l = length and g = acceleration due to gravity.