energy in simple harmonic motion
Energy in simple harmonic motion is the story of how energy sloshes back and forth between motion and storage as an object oscillates. On a swing, you move fastest at the very bottom, where all the energy is energy of motion; at the top of the arc you pause for an instant, where all the energy is stored up. The two forms keep trading places, over and over.
Precisely, for a mass-spring system the stored (potential) energy is U = 1/2 k x^2 and the energy of motion (kinetic energy) is K = 1/2 m v^2. As the object oscillates, x and v change, so K and U trade back and forth. But if there is no friction, their sum, the total mechanical energy E = K + U = 1/2 k A^2, stays exactly constant. The speed is greatest as it passes through equilibrium (x = 0) and momentarily zero at the turning points (x = plus or minus A), where all the energy is potential.
A direct consequence is that the total energy is proportional to the square of the amplitude, E scales as A^2, so doubling the swing quadruples the energy. One crucial honesty: this constant-energy picture holds only for undamped SHM. In any real oscillation, friction and drag slowly drain the mechanical energy away as heat, and the amplitude shrinks as time goes on.
A spring oscillator with k = 200 N/m and amplitude A = 0.10 m holds total energy E = 1/2 k A^2 = 1.0 J. At the midpoint all of it is kinetic, giving maximum speed v_max = sqrt(2E/m); at the extremes all of it is potential and the speed is zero.
Kinetic and potential energy trade places each cycle; their sum stays fixed if there is no friction.
Total energy is conserved only for undamped SHM; with damping it steadily converts to heat and the amplitude decays.