Energy Trading in an Oscillator

A mass on a spring or a pendulum bob never loses energy because no friction or air resistance drains it away. Instead the energy just changes costume. At the extreme point of the swing, the object is momentarily still but stretched or lifted as far as it goes, so all the energy is potential: elastic potential energy in a spring, PE = (1/2)kx², or gravitational potential energy in a pendulum. At the equilibrium point, the object moves fastest and has zero displacement from center, so all the energy is kinetic, KE = (1/2)mv^2. Anywhere in between, the energy splits between the two forms, but the total stays constant: KE + PE = E_total.

This trade-off is why you can find a maximum speed without touching time or forces directly. Set the total energy at the extreme (all potential) equal to the total energy at equilibrium (all kinetic), and solve for v_max. Energy bar charts make this visible: at the endpoint the potential bar is full and the kinetic bar is empty; at the midpoint they swap; at the quarter point, where the object has covered some but not all of its amplitude, both bars are partially filled but together they still reach the same total height every time.

This connects directly to why period does not depend on amplitude: a bigger pull stores more energy, which produces more speed, which covers the longer distance in the same time. It also sets up a subtlety worth catching: kinetic energy is not a simple sine wave matching position's rhythm, because it depends on velocity squared, so it peaks twice as often as position does.