Calculate the force created at each of the slave cylinders.

Question

A force of 100 N is applied to the brake pedal, which acts on the cylinder called the master through a lever. A force of 500 N is exerted on the master cylinder. The pressure created in the master cylinder is transmitted to four so-called slave cylinders. The master cylinder has a diameter of 0.55 cm, and each slave cylinder has a diameter of 2.50 cm. Calculate the force F₂ created at each of the slave cylinders.

Answer 1

Here we have a cylinder, the "master," squeezing or releasing a liquid or gas inside a closed series of pipes or tubes. This creates a pressure on the fluid, which is felt by whatever is on the other end(s) -- in this case, four other cylinders. Inside the closed system, whatever the pressure is, the pressure is. It is felt everywhere.

We can find the pressure created by the master using Pressure = Force/Area. This equals the pressure the other cylinders feel. Therefore,

F_master/A_master = F_slave/A_slave

or F_slave = F_master*A_slave/A_master

We can either use the total area of all the slave cylinders here and divide the final force by 4 (since it is acting across four equal objects), or just find the force on one slave cylinder directly.

The area of the master cylinder is π*(0.55/2 cm)². The area of a slave cylinder is π*(2.50/2 cm)².

Therefore, F_slave = 500 N * π*(2.50/2 cm)²/[π*(0.55/2 cm)²]

= 500 N * (2.50/0.55)²

= 1.03 * 10⁴ N