On the kart: the throttle return spring, the brake pedal return spring, the two valve springs inside the sealed engine, the nylon in a nylock nut, the air in each tire (lesson 1's addendum), and the frame itself (lesson 13). Anything that bends and comes back is a spring, and they all obey the same one-line law.
The principle in one sentence
A spring pushes back with a force proportional to how far you have moved it: twice the squash, twice the push.
The one equation
F = k x xF is the force in pounds, x is how far the spring has been compressed or stretched from its rest length in inches, and k is the spring's stiffness in pounds per inch: how many pounds it takes to move it one inch. That's Hooke's law. Rearranged, k = F / x: measure a force and a movement and you have the stiffness.
Math: proportionality and slope. "F is proportional to x" means F = (some constant) x x. Double x and F doubles; triple it and F triples. On a graph of F against x that's a straight line through zero, and the constant k is its slope: rise over run, pounds per inch. If a spring gives 4 lb at 1 in and 8 lb at 2 in, slope = (8 - 4) / (2 - 1) = 4 lb/in. Any time a page says "goes as" or "scales with", it's this idea: one thing is a fixed multiple of another. Here the multiple is k, the spring's stiffness: the pounds it pushes back for every inch you move it.
Why it works
Every solid is a spring. Pull on a steel wire and the atoms move apart a tiny bit and pull back; the pull is proportional to the stretch until you go too far and something slips (that's yield, and it's why a bent pedal stays bent). A coil spring is just a long wire wound so a small twist of the wire adds up to a big movement of the ends. The material sets how hard each bit of wire resists; the winding (wire thickness, coil diameter, number of coils) sets k.
On our kart, in numbers
The valve springs. Each valve is held shut by a coil spring. The cam pushes the valve open against it; the spring has to slam the valve shut again in time for the next cycle. At 6100 rpm the cam turns at 3050 rpm (half engine speed, mechanism 7), so one valve opening takes about 5 thousandths of a second. Moving a valve 0.2 in in that time takes roughly 40 g of acceleration (typical). If the spring is too soft the valve can't keep up, "floats", and the engine loses power right at the limiter. That's why the rule set fixes the valve spring to one stock part (B&S 26826), with a wire diameter of 0.103-0.107 in, 4.00- 4.75 coils and a max free length of 0.940 in, all tech items. A stiffer spring is a few horsepower at the top; the rule takes it off the table for everyone.
The throttle return spring. Say it's 4 lb/in (typical; measure ours) and the pedal pulls it 1.5 in: 6 lb of return force. That's the force snapping the slide shut if the cable sticks. The "throttle snaps shut" line on the pre-departure checklist is a spring check.
The tire as an air spring. Lesson 1's addendum found the balloon tire deflects about 0.043 in under 90 lb at 12 psi. k = 90 / 0.043 = about 2100 lb/in. Add air and k goes up: a tire at 14 psi is a stiffer spring than at 12. That's the physical meaning of "pressure changes the ride".
The frame. Lesson 13 used a frame stiffness of 5 lb per mm of twist (typical), which is 127 lb/in. Softer than the tires by a lot - which is why the chassis, not the tires, is the spring that decides when the inside rear lifts.
Two things to notice
- Springs add differently in series and in parallel. Two springs side by side (parallel: tire air + tire carcass) add their k's. Two springs end to end (series: tire on top of frame) combine to something SOFTER than either. The kart's corner stiffness is tire and frame in series, so the frame dominates.
- Preload is not stiffness. A spring installed already compressed starts at some force before it moves at all (the valve spring's seat pressure). Turning a preload adjuster changes where the force starts, not how fast it rises. Both matter; they're different knobs.
Problems
1. Measure a spring. Hang the throttle return spring from a hook, add a known weight (a full 14 oz oil bottle is about 1 lb), measure how far it stretches in inches. What is k in lb/in? With that k, how many pounds does it pull when the pedal stretches it 1.5 in?
2. Two in series. The tire's air spring is 2100 lb/in and the frame's is 127 lb/in. For springs in series, 1/k_total = 1/k_1 + 1/k_2 ("1/k" is one divided by k, the 1/x key on a calculator; add the two, then take 1/x of the sum). What is k_total in lb/in? Which of the two is it closest to, and why does that mean the frame is the spring that matters for lifting the inside rear?
3. Float. The valve spring rule sets max free length at 0.940 in. A spring that has sagged to 0.900 in from heat has lost some of its preload. With k = 40 lb/in (typical), how many pounds of seat force did that 0.040 in of sag cost? Would the engine notice it at 3000 rpm or at 6100 rpm, and why?
Go look: press the brake pedal by hand with the kart on the stand and feel the return spring; then push down on a rear tire with your whole weight and watch how little it moves; then lift one front corner an inch and watch the opposite rear lift. Three springs, three stiffnesses, all Hooke's law.