18 5.3 Applications
- Hydraulic brakes in cars: the pedal pushes a piston in the master cylinder; the pressure is transmitted through the brake fluid to the pistons of the wheel cylinders, which push brake shoes against the drum (rear drum brakes) or pads against the disc (front disc brakes). A small foot force produces large friction forces at all four wheels, and the pressure is the same to all wheels.
- Hydraulic lift (car lift in a garage).
- Hydraulic excavator and other hydraulic machines (jacks, presses, aircraft control systems).
Example 1. In a hydraulic press the small piston has diameter 4 cm and the large piston has diameter 40 cm. A force of 50 N is applied to the small piston. (g = 10 m/s²)
- Area ratio A/a = (40/4)² = 100, so η = 100.
- F = 50 × 100 = 5000 N; the largest mass it can lift = 5000/10 = 500 kg.
- To raise the large piston by 2 cm: y₁ = η y₂ = 100 × 2 = 200 cm.
Example 2 (car lift). A car of mass 1200 kg (weight 12 000 N) rests on a piston of radius 20 cm; the small piston has a radius of 2 cm. f = F(a/A) = 12 000 × (2/20)² = 12 000/100 = 120 N.
Example 3 (pressure). What pressure is transmitted if 120 N acts on a small piston of area 4 × 10⁻⁴ m²? P = 120/4 × 10⁻⁴ = 3 × 10⁵ N/m².
Example 4 (efficiency idea). A press has η = 40. A force of 250 N acts on the small piston, which is pushed 10 cm. F = 10 000 N; the large piston rises y₂ = 10/40 = 0.25 cm. Work input = 250 × 0.10 = 25 J; work output = 10 000 × 0.0025 = 25 J (equal).
Example 5 (given η and F). A hydraulic press lifts a 2000 N load on the large piston of area 0.05 m² with a small-piston area of 2.5 × 10⁻³ m². η = 0.05/2.5 × 10⁻³ = 20, so f = 2000/20 = 100 N.