Final answer:
The time taken for a simple harmonic oscillator to travel from the extreme position to half the amplitude is T/4, where T is the time period of the oscillator.
Step-by-step explanation:
The time taken by a simple harmonic oscillator to travel from the extreme position to half the amplitude can be found by understanding that the motion from the extreme position to the equilibrium, and then to the opposite extreme position is symmetric. The oscillator takes the same amount of time to travel from the extreme to the equilibrium as from the equilibrium to the extreme, which is half the period (T/2).
To find the time it takes to reach half of the amplitude, we must consider that it moves quicker as it approaches the equilibrium point. However, since the question only asks for the time taken to travel from extreme to half the amplitude, this describes a quarter of the complete oscillation. Since one complete oscillation takes a period T, a quarter oscillation would take T/4.
It is also important to remember that the frequency f and period T of a simple harmonic oscillator are independent of amplitude. This characteristic means the time interval calculated is not affected by the amplitude of the oscillation.
Thus, the time taken by the simple harmonic oscillator to travel from the extreme position to half the amplitude is T/4, where T is the time of oscillation.