Nothing new to derive here — a system in SHM has both kinetic and potential energy at any given moment, and its total mechanical energy is simply their sum:
For a spring-object system, potential energy is elastic potential energy, U = ½kx² — a formula you already know from Unit 3. What's new in this lesson is what happens to that sum as the object moves through a full cycle.
Ideal SHM has no friction or dissipation, so conservation of energy applies directly: E_total stays exactly constant throughout the entire cycle, from one turning point to the other and back again.
Because E_total = U + K is fixed, K and U can't both grow or shrink independently — whenever one goes up, the other must go down by exactly the same amount.
And that minimum kinetic energy isn't just small — it's exactly zero. At the instant the object is at maximum displacement, it is, for that instant, completely at rest.
The stacked bar always fills completely — that's conservation of energy made visual. What changes is the split between K (bottom) and U (top), and — when you move the amplitude slider — the total energy itself.
Drag amplitude and watch Total Energy climb — then drag it back down and watch the stacked bar still always fill completely, no matter what A is.
A 0.40 kg block oscillates on a spring with spring constant k = 25 N/m and amplitude A = 0.15 m. Find the block's speed when it is at x = 0.10 m from equilibrium.
Amplitude is the one place SHM's amplitude-independence rule stops applying. Changing the amplitude changes how far the object travels from equilibrium, which changes the maximum potential energy — and since all the energy is potential at the turning point, that also changes the total energy of the system:
A spring-mass oscillator's amplitude is doubled, with mass m and spring constant k unchanged. What happens to (a) the period, (b) the maximum potential energy, (c) the total energy, and (d) the maximum speed?