Understanding Fluids and Pressure

Fluids

• Definition: A substance that can flow.
• Examples: Liquids and gases
• Technical Difference from Solids: A fluid cannot bear a shear stress
• Concept of Pressure: Pressure is the perpendicular force component applied per unit area.
• Formula for Pressure: 𝑃 = 𝑑𝐹/ 𝑑𝐴 or 𝑃 = 𝐹┴ / 𝐴
• Units for Pressure: N/m² or Pascal (Pa)
• Pressure is a Scalar Quantity: It has no direction.
• Understanding Pressure and Impact: A smaller area with a larger force results in higher pressure, increasing impact.
• Example: A sharp knife has a smaller impact area than a blunt object, causing more pressure even with similar force.
• Pressure Varies with Depth: Pressure in a fluid increases with depth.
• Relationship between Pressure Change and Depth: (P₁ - P₂) = ρgh (where ρ is the fluid density, g is acceleration due to gravity, and h is the depth difference).
• Atmospheric Pressure: Approximately 1 atm or 10⁵ Pa.

Pressure in Fluids

• Pressure is measured in Pascals (Pa) or Newtons per square meter (N/m²).
• Pressure increases with depth in a fluid.
• The formula for pressure at a depth h in a fluid is: P₂ = P₁ + ρgh, where:
  • P₂ is the pressure at depth h
  • P₁ is the pressure at the surface
  • ρ is the density of the fluid
  • g is the acceleration due to gravity
  • h is the depth
• Pressure is the same in all directions at a given point in a static fluid.
• Pressure is the same at all points at the same level within a static fluid, regardless of the vessel's shape.
• The pressure difference between two points in a static fluid is called the gauge pressure.

Example Problem

• Question: At a depth of 1000 meters in the ocean, what is the absolute pressure and the gauge pressure? Assume the density of seawater is 1.03 g/cm³ and the atmospheric pressure is 10⁵ Pa.

• Solution:

  • Absolute pressure (P₂) = P₁ + ρgh = 10⁵ Pa + (1.03 × 10³ kg/m³ × 9.8 m/s² × 1000 m) = 1.04 × 10⁷ Pa
  • Gauge pressure (P_gauge) = P₂ - P₁ = (1.04 × 10⁷ Pa) - (10⁵ Pa) = 1.03 × 10⁷ Pa

• To calculate the force on a window of a submarine at this depth:

  • Force = Pressure × Area
  • Net pressure (P_net) = P₂ - P₁ = 1.03 × 10⁷ Pa
  • Area = 20 cm × 20 cm = 4 × 10⁻⁴ m²
  • Therefore, Force = (1.03 × 10⁷ Pa) × (4 × 10⁻⁴ m²) = 4120 N

Key Points To Remember

• The pressure exerted by a fluid due to its weight is called hydrostatic pressure.
• The pressure in a static fluid at a given depth is independent of the shape of the container.
• The pressure at a point in a fluid is dependent on the density of the fluid, the acceleration due to gravity, and the depth of the point.

Specific Gravity and Relative Density

• Specific Gravity is the ratio of the density of a substance to the density of a reference substance (usually water at 4°C).
• Specific gravity is also known as relative density.

Pressure in Liquids

• Pressure is the force exerted per unit area.
• Pressure increases with depth in a liquid.
• Pressure at the same level in a connected liquid container is the same.

Calculating Specific Gravity

• The experiment uses a U-tube.
• One arm has water (10 cm) and the other arm has spirit (12.5 cm).
• Pressures at the same level are equal.
• The pressure at the bottom of the water column is P_atm + ρ_water * g * h_water.
• The pressure at the bottom of the spirit column is P_atm + ρ_spirit * g * h_spirit
• Equating pressures gives the specific gravity (ratio of densities).
• Specific gravity = 10/12.5 in the example.

Bulk Modulus and Volume Change

• The bulk modulus measures a substance's resistance to compression, applicable to solids, liquids, and gases.
• The equation for bulk modulus is B = - ΔP / (ΔV/V)
• To calculate the depth to compress a rubber ball by 0.2%, use the formula
  • ΔP = B * (ΔV / V)
  • Then, use the equation ΔP = ρ * g * h:
    • B * (ΔV/V) = ρ * g * h
    • Solve for the depth, h.

Barometer and Atmospheric Pressure

• A barometer measures atmospheric pressure.
• It's based on the height of a liquid column being proportional to the pressure.
• A common barometer uses mercury.
• Standard atmospheric pressure is 760 mm Hg or 1 atm.
• Atmospheric pressure is determined by the air column's weight.

Exploring the Concept of a Barometer

• A simple barometer uses an inverted test tube filled with mercury, immersed in a mercury reservoir.
• The mercury level drops until the column pressure equals atmospheric pressure.
• The mercury column height measures atmospheric pressure.

Comparing Mercury and Water in a Barometer

• Water in a barometer rises higher than mercury due to its lower density.
• The water column height in a barometer is approximately 10.3 m.
• Liquid column height is inversely proportional to liquid density.

Hydraulics and Fluids Applications

• Barometers use low pressure differences with mercury for practicality. A water barometer would require a much larger tube.
• Manometers measure gas pressures in closed containers. Liquid rises on one side, where the head difference is proportional to the gas pressure.
• Gas pressure equals atmospheric plus head pressure.
• Pascal's law: Pressure changes in an enclosed fluid are transmitted equally throughout.
• In a closed vessel, a non-compressible fluid experiences the same pressure change at every point, when an external force is applied. This change in pressure, not the absolute pressure, is identical.
• Applications include hydraulic lifts, hydraulic brakes, hydraulic presses, pressure gauges, and sprayers.
• A manometer directly applies Pascal’s law, where changing gas pressure in the container creates a pressure change, and this is transmitted equally to the liquid in the manometer.
• Pascal’s law amplifies force in hydraulic lifts. A small force on a small piston creates a larger force on a larger piston.
• The force ratio in a hydraulic lift equals the area ratio.
• The smaller the area of the first piston, the smaller the required force.
• In a hydraulic lift, the system has interconnected cylinders with pistons and liquid, with the smaller piston called the master cylinder and the larger piston connected to it. When the smaller piston is pushed down, the liquid is forced to the larger piston. Fluid pressure, therefore, is transferred from the smaller piston.
• Hydraulic brakes transfer pressure from the pedal to brake pads, clamping them onto rotors for vehicle deceleration.

Pascal’s Law and Toothpaste:

• Squeezing a toothpaste tube applies pressure, transmitting it throughout the toothpaste to expel it.

Question Solved:

• The toothpaste question verifies the statement and reasoning regarding Pascal's Law. Both are correct, and the reasoning correctly explains the statement.

Practice Question:

• A hydraulic lift lifts a 1000 kg car. The smaller piston has an area of 1 cm² and the larger piston has an area of 5 cm². What force must be applied to the smaller piston?