Archimedes' and Pascal's Principles: Buoyancy and Fluid Pressure

Archimedes and the Principle of Buoyancy

• Archimedes was a Greek mathematician, physicist, engineer, inventor, and astronomer from Syracuse who is regarded as one of the greatest scientists of the ancient world.
• He discovered the principle of buoyancy while determining the purity of a gold crown for a king.
• The principle states that an object partially or completely immersed in a fluid (liquid or gas) experiences an upward buoyant force equal to the weight of the fluid displaced by that object.
• Floating and Sinking: An object floats if the buoyant force is greater than its weight ($F_{b} > W$) and sinks if the buoyant force is less than its weight ($F_{b} < W$).

Archimedes' Principle Formula

• $F_{b} = \rho \times g \times V$
• $\rho$ = density of the fluid ($kg/m^3$)
• $g$ = acceleration due to gravity ($9.81\,m/s^2$)
• $V$ = volume of displaced fluid ($m^3$)
• Unit of buoyant force: $\text{Newton (N)}$

Applications of Archimedes' Principle

• Home: Life jackets and swimming aids help people float by increasing buoyant force; bathtubs and sinks show water level rises relative to displaced volume.
• Community: Rescue boats are engineered to float safely while carrying heavy loads; swimming pools utilize buoyancy to assist human flotation.
• Transportation: Ships and ferries float when the buoyant force equals their weight; submarines rise or sink by adjusting the amount of water in ballast tanks to change buoyancy.
• Atmosphere: Helium balloons rise because the buoyant force exerted by the air is greater than the total weight of the balloon.

Blaise Pascal and Fluid Pressure

• Blaise Pascal (1623-1662) was a French mathematician, physicist, inventor, philosopher, and writer.
• He discovered Pascal's Principle, which explains the behavior of pressure in enclosed, confined fluids.
• The SI unit of pressure, the pascal ($Pa$), is named in his honor.

Pascal's Principle and Hydraulic Systems

• Pascal's Principle states that when pressure is applied to a confined fluid, it is transmitted equally and undiminished in all directions to every part of the fluid.
• This allows a small input force to be multiplied into a much larger output force within hydraulic systems, making it possible to lift or move heavy objects with ease.

Pascal's Principle Formulas

• Pressure Formula: $P = \frac{F}{A}$
• Hydraulic System Ratio: $\frac{F_{1}}{A_{1}} = \frac{F_{2}}{A_{2}}$
• Variables:
  • $P$ = Pressure ($\text{Pascals, Pa}$)
  • $F$ = Force ($\text{Newtons, N}$)
  • $A$ = Area ($\text{square meters, } m^2$)
  • $F_{1}, A_{1}$ = Input force and input piston area
  • $F_{2}, A_{2}$ = Output force and output piston area

Applications of Pascal's Principle

• Home: Hydraulic bottle jacks used to lift vehicles for tire changes; hydraulic office or barber chairs that adjust height via fluid pressure.
• Community and Industry: Garbage truck compactors that compress waste; hydraulic presses and manufacturing machines used in factories to cut or shape heavy materials.
• Transportation: Hydraulic braking systems in cars, buses, motorcycles, and airplanes; hydraulic lifts in auto shops for vehicle inspections.

Pressure Calculation Example

• Problem: Find the pressure when a force of $300\,N$ is applied over an area of $0.06\,m^2$.
• Solution:
  • $P = \frac{F}{A}$
  • $P = \frac{300}{0.06}$
• Answer: $5,000\,Pa$