Crash Course Physics: The Physics of Fluids

Introduction to Fluids

• Fluids are defined as substances that flow, which includes liquids and gases.

• Examples of fluids: water, air, corn syrup.

• Importance of fluid understanding for physicists and engineers: allows for the design of pressure sensors, hydraulic pumps, airplanes.

Properties of Fluids at Rest

Density

• Density is represented by the Greek letter rho (ρ).

• Formula for density:

  • $\text{Density (ρ)} = \frac{\text{Mass (m)}}{\text{Volume (V)}}$

• Units of density: kilograms per cubic meter (kg/m³).

• An object or fluid made of heavier atoms or with particles packed closely has a higher density.

Pressure

• Pressure is described as applied force divided by area.

• Formula for pressure:

  • $\text{Pressure (P)} = \frac{\text{Force (F)}}{\text{Area (A)}}$

• Units of pressure: Newtons per meter squared (N/m²), also known as Pascals (Pa).

• Fluids exert pressure in all directions.

• Average air pressure at sea level: 101,325 Pa.

• Pressure increases with depth in a fluid due to the weight of the fluid above.

Calculating Pressure at Depth

• Pressure at a given depth in a fluid can be calculated by:

  • $P = ρgh$

  • where:

  • ρ = fluid density (kg/m³)

  • g = acceleration due to gravity (≈9.81 m/s²)

  • h = depth (m)

• Example: In a pool 3 meters deep with the top at a quarter of a meter depth:

  • Density of water (ρ) = 1000 kg/m³

  • Change in depth (h) = 2.75 m

  • Resulting pressure difference at the bottom of the pool:

  • $\Delta P = 1000 kg/m³ \times 9.81 m/s² \times 2.75 m = 27,000 Pa$

Confined Fluids and Pascal's Principle

• Confined Fluid: A fluid that has no room to move when pressure is applied (e.g., water in a sealed container).

• Pascal’s Principle:

  • If pressure is applied to a confined fluid, the pressure increases equally throughout the fluid.

  • Example: Applying 10,000 Pa of pressure to a piston in a cup increases the pressure throughout the fluid by the same amount.

Applications of Pascal’s Principle

• Use in hydraulic systems, e.g., hydraulic lifts, where a difference in area can assist in moving heavy objects.

• The relationship for force based on area is:

  • $\frac{F<em>{out}}{F</em>{in}} = \frac{A<em>{out}}{A</em>{in}}$

  • where:

  • $F_{out}$ = output force

  • $F_{in}$ = input force

  • $A_{out}$ = output area

  • $A_{in}$ = input area

• If a piston with an area of 1 m² exerts a pressure of 10,000 Pa, and the output piston has an area of 2 m², the output force is:

  • $F_{out} = 10,000 Pa \times 2 m² = 20,000 N$

Measuring Pressure

Manometers

• A manometer is a U-shaped tube containing a fluid used to measure pressure.

• To measure tire pressure, one side of the manometer is placed in the tire.

• Fluid height difference in the U-tube indicates pressure difference:

  • If tire pressure is higher than atmospheric pressure, fluid height will be lower on that side.

  • Gauge pressure = difference between atmospheric pressure and tire pressure.

  • Absolute pressure = atmospheric pressure + gauge pressure.

  • Formula:

  • $P<em>{tire} = P</em>{atmosphere} + (ρ imes g imes h_{difference})$

  • where:

    • $P_{tire}$ = absolute pressure inside the tire

    • $ρ$ = density of the manometer fluid

    • $g$ = acceleration due to gravity

    • $h_{difference}$ = height difference in fluid levels.

Barometers

• A barometer measures atmospheric pressure using a vertical tube closed at one end, usually filled with mercury.

• At standard atmospheric pressure, mercury height in the tube is 76 cm.

• Changes in atmospheric pressure cause corresponding changes in the height of mercury in the tube.

Archimedes’ Principle and Buoyancy

Archimedes' Principle

• Archimedes discovered that the volume of water displaced by an object in a fluid equals the volume of the object itself.

• This discovery relates to the concept of buoyancy, which involves:

  • Buoyant Force: The upward force exerted on an object submerged in a fluid, equal to the weight of the fluid displaced.

Applications of Archimedes' Principle

• Example: Placing a billiard ball and a racquetball in water:

  • Billiard Ball: Denser than water.

  • Sinks because the gravitational force exceeds the buoyant force.

  • Racquetball: Less dense than water.

  • Floats because the buoyant force exceeds the weight of the ball.

  • When completely submerged, the racquetball displaces a volume of water weighing the same as the ball.

Conclusion

• Reviewed properties of fluids at rest: density and pressure.

• Discussed Pascal’s Principle and its importance in hydraulics.

• Reviewed methods of measuring pressure with manometers and barometers.

• Explained Archimedes' Principle and buoyancy, detailing how they dictate whether objects float or sink.