2.4 - Pascal’s Principle

Uniform Transmission of Pressure

Pascal’s principle: Pressure applied to an enclosed fluid is transmitted uniformly in all directions throughout the fluid.
In Pascal’s piston, pushing the piston applies pressure to the enclosed water. Water spurts from every hole because the pressure is transmitted in all directions.

A hydraulic system uses a liquid to transmit pressure. A small input force on a small piston can produce a larger output force on a larger piston.
With input force $F_{1}$ , input area $A_{1}$ , output force $F_{2}$ and output area $A_{2}$ :

P_{1} = \frac{F _{1}}{A _{1}}, P_{2} = \frac{F _{2}}{A _{2}}

P_{1} = P_{2} \Rightarrow \frac{F _{1}}{A _{1}} = \frac{F _{2}}{A _{2}}

F_{2} = \frac{A _{2}}{A _{1}} F_{1}, Multiplying factor = \frac{A _{2}}{A _{1}}

If $A_{2} > A_{1}$ , then $F_{2} > F_{1}$ . Force increases because the same pressure acts over a larger area.

Coconut milk extractor: Pressure transmission in an enclosed fluid produces a large force to press grated coconut.
Hydraulic brake: Force on the brake pedal acts through the master cylinder and brake fluid. Transmitted pressure acts at the slave cylinders to apply the brake pads or brake shoes.
Hydraulic jack: A handle operates the small piston; pressure transmitted through hydraulic fluid acts on the large piston to lift the load. The arrangement includes a fluid reservoir, valves A and B and a release valve.

Force multiplication: With $F_{1} = 12 N$ , $A_{1} = 10 c m^{2}$ and $A_{2} = 50 c m^{2}$ :

\frac{A _{2}}{A _{1}} = \frac{50}{10} = 5, F_{2} = 5 (12) = 60 N

Bicycle brake: An input force of $60 N$ on $0.80 c m^{2}$ must produce $840 N$ :

A_{2} = \frac{F _{2} A _{1}}{F _{1}} = \frac{840 ( 0.80 )}{60} = 11.2 c m^{2}