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Consider a hydraulic brake system as shown below. The area of the piston in the master cylinder connected to the foot paddel is 1.8 cm2. The area of the piston connected to the brake shoe is 6.4 cm2. The kinetic frictional coefficient between shoe and wheel drum is 0.50. If the wheel's radius is 34 cm, find the frictional torque about the axle when a force of 47 N is exerted on the brake pedal.

Answer solution:

For the hydraulic brake system, first we need to find the force the brake shoe is pushing the wheel drum. This can be done using the hydraulic pressure and force relation. Here P is same in the hydraulic system and the pressure force exerted on the cylinders are ||F=PA||, where A is the area of the cylinder. See the figure below for illustration of the forces and torque on the brake system.

Since pressure in the hydraulic system is the same, we have $$\frac{F_p}{A_1}=\frac{N}{A_2}$$

This will allow you to find the normal force the shoe pushes on the wheel drum.

Then this normal force will create friction ||f=\mu mg||, which then creates a torque ||\tau=fR||.