No, for a motion to be periodic, it does not necessarily need to have the same or unchanging amplitude over time. Periodicity refers to the repetition of a motion or pattern over a specific time interval, known as the period. The key aspect of periodic motion is that the pattern repeats itself at regular intervals.
Amplitude refers to the maximum displacement or magnitude of the motion. In periodic motion, the amplitude can remain constant, but it can also vary. For example, consider a pendulum swinging back and forth. The pendulum's motion is periodic, as it repeats the same pattern of swinging back and forth over time. However, due to factors such as air resistance or the energy input into the pendulum, the amplitude of its swings may decrease gradually over time. Despite the changing amplitude, the motion remains periodic as long as the pattern repeats at regular intervals.
So, while periodic motion can have a constant or unchanging amplitude, it can also have a varying amplitude over time without affecting its periodicity.