Under Pressure: Brake Master Cylinder Math
Today, in the spirit of keeping up with my fluids themed summer blog, I’m going to dive into a very important topic in the automotive industry: how to stop. I’m sure that a majority of people out there are like me and don’t give stopping and going a second thought when they’re driving in their car; you step on the gas pedal, you go. You step on the brake pedal, you stop. Its something that just happens. BUT, the underlying math behind stopping a 2000 lbs+ vehicle is interesting and lead me to ask the question, just how much pressure is generated by the brake fluid within the master cylinder?
I feel like I should take a moment before I delve in to reiterate a key factor in the problem:
Brake fluid is incomprehensible, meaning that the fluid maintains a constant density.
A.k.a. the fluid can’t be squished together; it acts like a solid but has the properties of a liquid. Noted. Now, let’s get into it. Typically, the average person can apply approximately 60–70 lbs of force to the brake pedal. The pedal acts like a mechanical leaver to the brake master cylinder transferring the force to said cylinder (There is vacuum device called a brake booster that amplifies the force applied to the pedal to assist in braking, but for this post, we will exclude this…). The pedal ratio, or the over all pedal length (distance from the pedal pivot to the center of the pedal) divided by the distance from the pivot point to the push rod, provides a mechanical advantage to increase the force applied by the driver’s foot. According to a few online sources I browsed, an optimal pedal ratio is 6.2:1. That is, if 70 lbs of force is applied by the driver, multiply that by 6.2 to yield the force acting on the push rod in the master cylinder. Roughly 434 lbs.
A force of 434 lbs is now applied to the push rod within the master cylinder. When this happens, the force is transferred from the rod, through the brake fluid, to the primary and secondary pistons. When force is applied to the two pistons, they simultaneously close off the inlet and compensating ports, which are filled with brake fluid. When they close, the the brake calipers force the brake pads against both sides of the rotor, slowing the car down. So, the question is, how much pressure is generated in the master cylinder?
The problem isn’t as bad as you may think. We calculated the force applied to the push rod, so were already half-way there. The only thing left to do is calculate the area of the cylinder. Then we’ll have our variables to solve for pressure.
Let’s assume a 1" master cylinder with a 7/8" bore (0.875). To find the area, you take the radius of the bore as 7/16" (0.4375), square it, and multiply by pi. You should get 0.60"² for you area.
We rearrange the equation Force = pressure x Area (F = pA) to find pressure in pounds per square inch:
p = 0.60"² x 484 lbs = 290.4 psi
There you have it! The pressure in the master cylinder is close 300 pounds per square inch, equivalent to the same pressure generated in the boiler of an old steam locomotive. What’s the point of all this you might ask? In drag racing and stock car racing, understanding pressure, bore area, and force applied to the brakes helps crews better adjust the brakes to get the car to handle better; a small but crucial tweak to gain an advantage in a fairly uniform sport.
Thanks for ready, feel free to drop a comment! Tell me what you liked, disliked, what I messed up, or how you would’ve approached the problem below.
