Energy Trades Endlessly Between Kinetic and Potential
Watch the mass swing back and forth and look closely at the two turning points. For an instant the mass stops dead, and yet that is exactly the moment the spring is stretched the most. Now look at the center: the mass never pauses there, it races through at full speed, and yet the spring is completely relaxed, holding nothing. So when the motion stops at the ends, where did the energy go? Nowhere. It only changed form. At the ends it is all stored in the stretched spring; at the center it is all carried by the rushing mass. Back and forth, over and over, the same energy is handed from one form to the other and back again, and with no friction the total never changes. A child on a swing feels this without any equations: highest point, still for a heartbeat; lowest point, rushing fastest. This lesson is that single trade, looked at closely.
The bar below the spring is the system's entire energy budget, and its total length never changes, the law of the whole lesson in one picture. The blue part is energy stored in the spring (potential), the green part is energy of motion (kinetic). Watch the divider slide as the mass moves. At the two ends the bar is all blue: the spring is fully loaded and the mass is frozen. At the center it is all green: the mass is flying and the spring is empty. One side fills by exactly as much as the other drains, so the total stays put. Notice the swap happens twice on every round trip, the energy becoming all kinetic each time the mass passes the center, no matter which way it is heading.
Now take the spring's half on its own. The energy stored in a spring stretched (or compressed) by an amount x is PE = ½kx². Pull the slider and watch the value climb a parabola, a bowl, not a straight line. That curve is the whole point: because of the square, pulling twice as far stores four times the energy, not twice. Three times as far stores nine times. The bowl is the hill the mass has to climb as it moves away from the center, and a stiffer spring (larger k) makes a steeper bowl. At the turning points, x = ±A, the mass sits at the very top of the bowl, and every bit of the system's energy is parked here as ½kA².
Now the other half, the energy of motion: KE = ½mv². The mass is frozen for an instant at each end, where its speed is zero and it carries no kinetic energy at all, and it is fastest as it whips through the center, where the speed peaks at vmax = Aωₙ. Plot the kinetic energy against position and you get an upside-down bowl: highest in the middle, zero at the edges, the exact mirror image of the spring's bowl from the last step. The arrow on the mass shows its speed, longest at the center and shrinking to nothing at the ends. As the mass slides down the spring's hill toward the center, it spends stored spring energy and buys speed; climbing the far side, it spends that speed back and reloads the spring.
Put the two halves in the same picture and slide the position. The blue bowl is the spring's energy PE = ½kx², the green upside-down bowl is the motion's energy KE = ½k(A²−x²), and wherever you place x, the two heights add up to the same flat gold line at the top, E = ½kA². As one rises the other drops by precisely the same amount; the little bar on the right shows them stacking to a constant total. That flat line is conservation of energy turned into a picture: in an ideal, frictionless oscillator, energy is never made and never lost, only swapped between spring and mass. And this one statement, ½kx² + ½mv² = ½kA², is a tool you will use constantly: set it up at any position and you can solve for the speed there without ever mentioning time.
So what sets the size of the whole budget? The amplitude. The total energy is E = ½kA², which means it grows with the square of how far you pull. Remember from the last lesson that amplitude does not change the frequency: the beat is the same whether you nudge the mass or yank it. The energy is a completely different story. Slide A and watch the wave on the left grow taller in step with A, while the energy bar on the right shoots up far faster; the tick marks spread apart, and that widening gap is the square at work. Double the amplitude and you have put in four times the energy. This is why a harder pluck makes a louder string, a bigger push sends a swing higher, and why resonance is so dangerous: it keeps quietly feeding energy in, the amplitude creeps up, and the stored energy climbs as its square until something gives.