simple harmonic motion energy
Watch the mass go back and forth and look closely at the two end points. At that instant the mass stops dead, and that is exactly when the spring is stretched the most. Now look at the center. The mass never stops there. It passes at its fastest, and the spring is fully slack and holds nothing. When the motion stops at the ends the energy went nowhere. Only its form changed. At the ends it is all stored in the stretched spring, and at the center the running mass carries all of it. The same energy is handed from one form to the other without end, and if there is no friction the sum never changes. A child on a swing stops for one beat at the highest point and passes fastest at the lowest. This scene looks closely at that one exchange. The high place is storage and the low place is running. The child on the swing already knows the trade. A stop is storage and a passing is running, and the sum of the two stays as one line. The swing already teaches the trade before any formula.
The bar under the spring is the whole energy budget of this system, and its whole length never changes. The blue part is the energy stored in the spring, potential energy, and the green part is the energy of motion, kinetic energy. Watch the boundary slide as the mass moves. At the ends the bar is all blue. The spring is full and the mass is stopped. At the center it is all green, the mass is running and the spring is empty. By exactly as much as one side fills, the other empties, and the sum stays. Whichever way it goes, every time the mass passes the center the energy is all kinetic. Blue and green trade places inside one bar. The budget line does not grow or shrink. Each round trip passes the center twice, and each time the bar fills with green. The budget does not leak while the colors trade.
Now pull off only the spring side. The energy stored in a spring stretched or shortened by x is PE = ½kx². Drag the slider and the value climbs along a parabola, a bowl, not a straight line. Because of the square, pull twice as far and the stored energy is four times, not twice, and pull three times as far and it is nine times. This bowl is the hill the mass has to climb as it leaves the center, and the stiffer the spring (the larger k) the steeper the hill. At the two ends x = ±A the mass sits on the top of the bowl, and every bit of the system's energy is gathered here as ½kA². The steeper the bowl, the more expensive the same distance. A stiff spring charges more for the same pull. The farther you pull, the more the hill charges as a square, and the returning force grows with it. The bowl is the price list of distance.
Now the other half, the energy of motion, KE = ½mv². The mass stops for an instant at the ends, and there the speed is 0 so it carries no kinetic energy. When it crosses the center it is fastest, and the speed peaks at vmax = Aωₙ. Draw kinetic energy against position and you get an upside-down bowl, highest in the middle and 0 at the ends, the mirror of the spring bowl. The arrow on the mass shows the speed, longest at the center and shrinking to 0 toward the ends. On the way down to the center it spends spring energy to buy speed, and on the way up the other side it spends that speed to fill the spring again. The longer the arrow the more motion, and the shorter the arrow the less motion. The center is the richest in speed. The stop at the ends and the sprint at the center are the two faces of one period. Speed is bought on the way down and spent on the way up.
Put both halves on the same picture and move the position with the slider. The blue bowl is the spring energy PE = ½kx², and the green upside-down bowl is the energy of motion KE = ½k(A²−x²). Wherever you place x, the two heights add to the flat gold line on top, E = ½kA². When one side rises the other falls by exactly that much, and the small bar on the right stacks the two into the same sum. That flat line is conservation of energy drawn as a picture. In a vibration with no friction, energy is neither created nor lost. It only travels between the spring and the mass. ½kx² + ½mv² = ½kA² is the tool that gives the speed at any position without ever bringing out the time. A flat gold line is the picture of a sum that does not change. The two bowls are complements of one budget. You can read the speed from position alone, without ever bringing out the time. The gold line is the promise that nothing was lost.
The size of the whole budget is set by the amplitude. The total energy is E = ½kA², so it grows as the square of how far you pull. Tap the mass lightly or pull it hard and the beat is the same. Energy is a different story. Move A and while the wave on the left grows taller the energy bar on the right shoots up much faster, and the widening gap is the square at work. Double the amplitude and you have put in four times the energy. Pluck harder and the string is louder. Push harder and the swing goes higher. Resonance keeps pushing energy in, the amplitude grows, and the stored energy climbs as that square until something breaks. Four times the budget fits inside twice the amplitude. Loudness and height are this square made visible. The beat stays and only the budget swells as a square, and that is what amplitude means. Loudness, height, and danger are this square.