A spring with spring constant K is oscillating on a frictionless horizontal surface with a mass m attached to its end. The distance x from its mean position may be specified as:
x = A cos (wt+Ф), A is the amplitude and Ф is the phase constant, t is time.
Force on mass m = - k x = ma But for SHM - w² x = a
So w = √(k/m) or m = k/w²
v = dx/dt = - Aw sin (wt+Ф)
Potential Energy = 1/2 k x² = 1/2 k A² cos² (wt+Ф)
Kinetic Energy = 1/2 m v² = 1/2 m A² w² sin² (wt+Ф) = 1/2 k A² sin² (wt+Ф)
So if PE = KE, then sin² (wt+Ф) = cos² (wt+Ф)
But sin² (wt+Ф) + cos² (wt+Ф) = 1
So, sin² (wt+Ф) = 1/2, sin (wt+Ф) = 1/√2
wt+Ф = π/4
NOW, x = A /√2, So, when the body is at 1/√2 th of its amplitude then the PE = KE.