1) The total energy in an oscillator is the maximum PE (k xmax^2/2) [which occurs when x =+or-xmax] or the max KE which occurs as the oscillating mass passes through the equilibrium point (Mveq^2/2). These numbers are the same.
2) What makes a harmonic oscillator harmonic is the fact that the restoring force ( the force pulling the mass back to the equilibrium position) is proportional to the distance from the equilibrium, i.e. the displacement; so F=-k x. Then if this is true the acceleration toward the center is k x/m.
For the x direction of circular motion, the acceleration = omega^2 x and if we set this acceleration= kx/m we can use the x component of the circular motion equations as long as we set omega ^2 = the same constant i.e. omega ^2 = k/m or omega ^2 = (Frestoring/x)/m. Monday, I am hoping to show you that the same equation holds for a pendulum, as long as it swings through a small angle.
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