Energy in oscillatory motion — Ep and Ek

What just happened?

A mass on a spring stores energy in two ways: when it is displaced, the energy sits in the stretched spring (potential); when it races through the middle, it sits in the motion (kinetic). The sum is always the same.

Oscillation energy balance:

— potential energy grows with the square of displacement, and kinetic is the rest up to the constant total. k — spring constant (N/m), x — displacement (m), A — amplitude.

Invariant — what stays constant?

At the extreme position ($x = \pm A$) the speed is zero, so $E = E_p$ — all the energy is potential. At equilibrium ($x = 0$) the spring is slack, so $E = E_k$ — all of it is kinetic. Between them the energy just "sloshes" from one form to the other.

Why you need this for the exam:

Conservation of mechanical energy solves most oscillation problems — without computing forces and accelerations. Trick: at the extreme point compute from $E = \tfrac{1}{2}kA^2$, and find the maximum speed from $\tfrac{1}{2}mv_{max}^2 = \tfrac{1}{2}kA^2$.

Remember: in oscillations the energy circulates between Ep and Ek, but their sum E = ½kA² is constant. Extreme = pure potential, middle = pure kinetic.