The time period of an oscillator is the time it takes for the oscillator to complete one full cycle of its motion.
In the scenario you described, where the oscillator moves from an extreme position to half of the amplitude, we can assume that the oscillator follows simple harmonic motion. In simple harmonic motion, the displacement of the oscillator as a function of time can be described by a sinusoidal waveform.
For an oscillator in simple harmonic motion, the time period (T) is related to the frequency (f) by the equation:
T = 1 / f
Now, let's consider the scenario where the oscillator moves from an extreme position to half of the amplitude. Since the amplitude represents the maximum displacement of the oscillator, moving from an extreme position to half of the amplitude means that the oscillator has covered a quarter of its total cycle.
In simple harmonic motion, one complete cycle consists of two extreme positions. Therefore, if the oscillator covers a quarter of its total cycle, it means it has covered half of a single complete cycle. In other words, it has gone from one extreme position to the midpoint (half of the amplitude).
Since the time period represents the duration of one complete cycle, if the oscillator has covered half of a cycle, the time period required to reach this point would be half of the total time period. Hence, the time period (T') from an extreme position to half of the amplitude would be:
T' = T / 2
Where T is the time period of the oscillator in general.
In summary, the time period required for an oscillator to move from an extreme position to half of the amplitude is equal to half of the time period of the oscillator in simple harmonic motion.