A simple realization of the harmonic oscillator in classical mechanics is a particle which is acted upon by a restoring force proportional to its displacement from its equilibrium position. Considering motion in one dimension, this means
F=−kx
Such a force might originate from a spring which obeys Hooke’s law
The force constant k is a measure of the stiffness of the spring. The variable x is chosen equal to zero at the equilibrium position, The negative sign shows restoring force, always in the opposite sense to the displacement x.
F = m .{d^2 x / dx^2} = −kx
where m is the mass of the body attached to the spring, which is itself assumed massless. This leads to a differential equation of familiar form, although with different variables:
the solution of the differential equation is of the form
x = A sin wt where
w= sqrt(k/m) and A is maximum displacement called amplitude,
if at t=0 x=0
thus the velocity at any time t will be
dx/dt = A.w.cos(wt)
so the kinetic energy at time t
(1/2) m. vel^2 = (1/2) m. {A^2. w^2 Cos^2 (wt)}
The general relation between force and potential energy in a conservative system in one dimension is
F = −(dV / dx) i.e. the force is derived as negative rate of change of potential
Thus the potential energy of a harmonic oscillator is given by
V(x) = (1/2). k .x ^2
which has the shape of a parabola,turning points ±x(max)
where the total energy E equals the potential energy (1/2). k .x ^2 while the kinetic energy is momentarily zero.
In contrast, when the oscillator moves past x=0 the kinetic energy reaches its maximum value while the potential energy equals zero.
therefore the two forms of energy are being shared with total E constant.
Now let at any instant t the two energies are equal
K.E. = P.E.
so (1/2) m. {A^2. w^2 Cos^2 (wt)} = (1/2). k .x(t) ^2 = (1/2). k .( A sin wt )^2
Both are equal to (1/2) total Energy
so its possible if wt= pi/4 where sin and cos function will have same value.
at wt= pi/4 the displacement x will be
x= a sin(wt) = A. sin (pi/4) = A. (1/sqrt(2) = A/1.414
x= 0.714 A
thus the kinetic energy and potential energy will be equal at x =0,714 A value as above