In a simple harmonic oscillator, the displacement of the oscillator as a function of time is given by the equation:
x(t) = A * cos(ωt + φ)
Where: x(t) is the displacement at time t, A is the amplitude of the oscillator, ω is the angular frequency of the oscillator (ω = 2πf, where f is the frequency), t is the time, and φ is the phase constant.
For a simple harmonic oscillator starting from an extreme position, the phase constant φ is either 0 or π, depending on whether the initial displacement is in the positive or negative direction.
If the time is 3/2 T, where T is the period of the oscillator, we can substitute t = 3/2 T into the equation to find the displacement at that time:
x(3/2 T) = A * cos(ω * (3/2 T) + φ)
Since the cosine function has a period of 2π, and the period of the oscillator is T, we can rewrite ω * (3/2 T) as 3π:
x(3/2 T) = A * cos(3π + φ)
If the phase constant φ is 0, then:
x(3/2 T) = A * cos(3π) = -A
If the phase constant φ is π, then:
x(3/2 T) = A * cos(3π + π) = A
So, depending on the phase constant, the displacement covered by the simple harmonic oscillator starting from an extreme position in a time of 3/2 T with amplitude A will be either -A or A.