In a disc-brake system, the driver’s foot moves the brake pedal, which in turn displaces the master-cylinder piston of diameter 15 mm. The brake lines lead to four wheel-cylinder pistons, each of diameter 40 mm, pressing brake pads against the discs.
a. (i) Derive an expression that relates the displacement xm of the master-cylinder piston to the common displacement xp of each wheel-cylinder piston, assuming the brake fluid to be incompressible and the connecting lines to be rigid.
b. (ii) Using the diameters given, calculate the master-cylinder piston displacement required to achieve a pad movement of 0.50 mm at each wheel.
c. (iii) The pedal linkage is such that the master-cylinder piston moves through only one-third of the driver’s foot movement. Determine the foot travel needed to obtain the pad movement stated in part (ii).
d. (iv) Briefly explain how a small compressibility of the brake fluid would alter the foot travel calculated above, stating whether it would increase or decrease and why.