Energy in Simple Harmonic Motion

A mass on a spring has no friction to drain its energy, so whatever energy it starts with, it keeps. At the extremes of motion (x = A or x = -A), the mass is momentarily still: all the energy is potential, stored in the stretched or compressed spring, equal to one half k A squared. At the center (x = 0), the spring is relaxed, so all the energy is kinetic, one half m v squared. Everywhere in between, the energy is shared. Because the mass passes through x = 0 twice per cycle, it hits pure kinetic energy twice per cycle too, which is why the trade happens twice each period.

The mechanism is just conservation of energy applied position by position. Total energy E = one half k A squared is a constant set entirely by the spring constant and the amplitude, it does not care what x is. At any position x, potential energy is one half k x squared, so kinetic energy must fill in the rest: one half m v squared = one half k A squared minus one half k x squared. Solve that for v and you get the speed at any point without ever touching time or Newton's second law directly.

This energy view sits right next to the position and velocity equations from earlier in the unit, x(t) = A cos(omega t) and v(t) = -A omega sin(omega t). Square and add the right pieces and the energy equation falls right out, which is why both descriptions agree. It also sets up resonance and damping later: damping is literally energy leaking out of this one half k A squared bucket over time.