On the kart: the brake system. The pedal (M01) pulls a rod into the master cylinder, a small piston pushing brake fluid down a line to the caliper, where bigger pistons squeeze the pads onto the rotor. The fluid is a lever with no bar and no pivot.
The principle in one sentence
Pressure in a trapped fluid is the same everywhere in it, so a force on a small piston becomes a bigger force on a bigger piston, in proportion to their areas.
The one equation
pressure = force / area (psi = lb / in^2)Same pressure at both ends, so:
force_out / area_out = force_in / area_in
force_out = force_in x (area_out / area_in)The ratio area_out / area_in is the multiplication, a pure number. That is M01's arm ratio again, with piston areas in place of arm lengths. And the same price: the big piston moves less, by the same ratio, because the fluid volume that leaves the small cylinder is the volume that arrives at the big one.
Math: the area of a circle. A piston is a circle. Its area is pi x radius^2, where pi = 3.1416 and radius is half the diameter. "radius^2" means radius x radius (a square). A 0.625 in bore has radius 0.3125 in; area = 3.1416 x 0.3125 x 0.3125 = 0.307 in^2. On a calculator: radius, times itself, times 3.1416. Here the area is the surface the fluid pushes on: pressure times that area is the force on the piston.
Math: why doubling the diameter quadruples the area. Area goes as radius^2, so doubling the radius gives 2 x 2 = 4 times the area. Tripling gives 9. That is what "squared" does: a change in the input shows up as that change multiplied by itself. It's why piston bores are a strong tuning lever: a small change in diameter is a bigger change in force. On a calculator: multiply the diameter ratio by itself (1.2 x 1.2 = 1.44, so 20% more bore is 44% more area).
Why it can't be free
Push the master piston in by 1 in: it displaces 0.307 in^3 of fluid (area x travel). That fluid has to go somewhere; it pushes the caliper pistons out by 0.307 in^3 / caliper area. Bigger caliper area, less travel. Force x distance is the same at both ends, just as it was for the lever. Hydraulics is not a trick; it's a lever you can route around a corner with a hose.
On our kart, in numbers
Master to caliper. Master cylinder bore 0.625 in (typical; measure ours), caliper with two pistons of 1.0 in (typical). Areas: master 0.307 in^2, caliper 2 x 0.785 = 1.571 in^2. Ratio 1.571 / 0.307 = 5.12. Whatever the rod pushes with, the pads squeeze with 5.12 times that.
Chained to the pedal. M01's pedal at 4:1 with 50 lb at the foot gives 200 lb on the rod. Pressure in the line = 200 / 0.307 = 652 psi. Force on the pads = 652 x 1.571 = 1024 lb. Fifty pounds at the foot became a thousand at the rotor: 4 x 5.12 = 20.5 times. The rod moved 1/4 in for 1 in of pedal; the caliper pistons moved 0.25 x 0.307 / 1.571 = 0.049 in. That's all the pad has to travel, which is why the pads sit so close to the rotor and why a warped rotor drags.
Air in the line. Brake fluid barely compresses; air does, a lot. Say 0.1 in^3 of air is trapped in the line (a bubble the size of a pea). At 652 psi it shrinks to 0.1 x 14.7 / (652 + 14.7) = 0.0022 in^3. The other 0.098 in^3 of that space has to be filled by fluid before the pressure can build, which is 0.098 / 0.307 = 0.32 in of extra master piston travel. The rod only moved 0.25 in to begin with. The pedal goes to the floor before the pads bite. That is what "spongy brakes" means, physically, and why bleeding the system is a safety item and not maintenance.
Two things to notice
- The multiplication is fixed by the parts. Unlike the pedal, where you can move the rod between holes, the hydraulic ratio is set by the bores. Changing "brake feel" means changing a master cylinder or caliper, or the pedal ratio. Pad compound (M03) changes the bite, not the force.
- Pressure doesn't care about hose length. A longer line just holds more fluid; the pressure at the caliper equals the pressure at the master (ignoring the tiny weight of the fluid). That is the whole reason hydraulics beat rods and cables for brakes: the force arrives at full strength wherever the hose goes.
Problems
1. Three master cylinders. Keep the caliper at 1.571 in^2 and the rod force at 200 lb. For master bores of 0.50, 0.625 and 0.75 in: what is the area of each, and how many pounds squeeze the pads in each case? Which bore gives the hardest bite for the same foot? For that bore, how many times farther does the master piston move than the caliper pistons (caliper area / master area), and so what happens to pedal travel?
2. Doubling the piston. A caliper piston of 0.5 in bore has an area of 0.196 in^2. What is the area of a 1.0 in piston? By what factor did the force at that piston change for the same line pressure?
3. The bubble. Redo the air calculation for a bubble half the size, 0.05 in^3, at the same 652 psi. How many inches of extra master piston travel does it cost? Compare that with the 0.25 in the rod normally moves: is the pedal still usable?
Go look: find the master cylinder (the small cylinder the brake rod enters) and read the bore if it's cast into it; find the caliper and count its pistons. Then find the bleed nipple on the caliper. That's where the pea-sized bubble leaves.