Hydrostatic Pressure and Its Applications

15.1 Pressure

  • Definition of Pressure: Pressure is defined as the force acting on a unit area. It can be mathematically expressed as:
    • extPressure=FAext{Pressure} = \frac{F}{A}
      where:
    • FF = Perpendicular force applied
    • AA = Area.
  • Unit of Pressure:
    • The unit of pressure is Newtons per square meter (Nm⁻²), named after the French scientist Blaise Pascal as the Pascal (Pa). Therefore,
    • 1extNm−2=1extPa1 ext{ N m}^{-2} = 1 ext{ Pa}.
  • Characteristics of Pressure:
    • Pressure is a scalar quantity because it has only magnitude and no direction.

Example 1: Pressure Calculation

  • A cubic-shaped box weighs 400 N. If the area of the bottom of the box is 0.2 m²:
    • extPressure=FA=400extN0.2extm2=2000extPaext{Pressure} = \frac{F}{A} = \frac{400 ext{ N}}{0.2 ext{ m}²} = 2000 ext{ Pa}.

Example 2: Soil Pressure Calculation

  • The pressure exerted by a pile of soil distributed over an area of 8 m² is 150 Pa. To find the force exerted on the ground:
    • extForce=extPressureimesextArea=150extNm−2imes8extm2=1200extNext{Force} = ext{Pressure} imes ext{Area} = 150 ext{ N m}^{-2} imes 8 ext{ m}² = 1200 ext{ N}.

15.2 Hydrostatic Pressure

  • Definition: Hydrostatic pressure is the pressure exerted by fluids (liquids) in rest. It operates under principles similar to solid pressure but applies to the weight of liquid columns.

Characteristics of Hydrostatic Pressure:

  1. When a solid object rests on a table, it exerts pressure on the surface because the weight of the object spreads over its contact area.
  2. Similarly, liquids exert pressure not only on the bottom but also on the walls of the container:
    • Example: Polythene bag with holes: Observations show water exits through all holes because water pressure exists at every perforation, indicating pressure acts in all directions.

Experiment of Water Column

  • Setup:
    • A plastic bottle with holes filled with water demonstrates that the horizontal distance traveled by water from each hole at the same level is equal.
  • Conclusion: The pressure at the same level of a liquid is constant.

Effect of Depth on Pressure

  • If you create holes at varying heights in a bottle, you’ll observe:
    • Water streams from lower holes come out faster than those from upper holes because pressure increases with depth.

15.3 Transmission of Pressure Through Liquids

  • Principle: Liquids are incompressible, so pressure applied at one point in a liquid is transmitted throughout the liquid.
  • Hydraulic Press: Explained through a piston scenario where:
    • Area of piston A = 10 cm² and piston B = 200 cm². If a 20 N force is applied to piston A:
    • extPressureatA=20extN10extcm2=2extN/cm2ext{Pressure at A} = \frac{20 ext{ N}}{10 ext{ cm}²} = 2 ext{ N/cm}².
    • This pressure is transmitted to piston B, resulting in a total force on piston B:
      • extTotalForceatB=extPressureimesextArea=2extN/cm2imes200extcm2=400extNext{Total Force at B} = ext{Pressure} imes ext{Area} = 2 ext{ N/cm}² imes 200 ext{ cm}² = 400 ext{ N}.

Applications of Hydraulic Principles

  • Hoists for Vehicles: Use hydraulic pressure to lift heavy loads.
  • Braking Systems: Driver applies force on brake pedal, which is transmitted to brake pads through fluid, generating greater force on brake discs or drums due to larger area in slave cylinder.

15.4 Pressure from Gases

  • Gases exert pressure similarly to solids and liquids. Pressure arises from:
    1. Weight of a column of gas (like atmospheric pressure).
    2. Compressed gases attempting to expand.
  • Atmospheric Pressure: Exists from the weight of air extending above the Earth's surface.

Mercury Barometer

  • Instrument to measure atmospheric pressure using a glass tube filled with mercury.
  • Empty space forms above mercury when tube inverted, with height of mercury column (approximately 76 cm) indicating atmospheric pressure:
    • extAtmosphericPressure=76extcmHgext{Atmospheric Pressure} = 76 ext{ cm Hg}.

Aneroid Barometer

  • A mercury-free device to measure atmospheric pressure using evacuated cavities that react to external pressure changes.

Applications of Atmospheric Pressure

  1. Straw Usage: Sucking air out of a straw lowers pressure inside, allowing atmospheric pressure to push liquid up.
  2. Siphon Method: Water flows from a higher tank to a lower tank due to pressure difference created by atmospheric pressure acting on the different surfaces of the water.
  3. Rubber Suckers: When pressed against a surface, a vacuum forms behind the sucker due to reduced internal pressure, allowing atmospheric pressure to hold it in place.

15.5 Archimedes’ Principle

  • Definition: An object submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced by the object.
  • Application:
    • Measuring density via hydrometers, which float at a level proportional to the liquid’s density, displacing a weight of liquid equivalent to their own weight.

Examples and Exercises:

  1. Calculate pressure at bottom of different container depths.
  2. Describe experiments to validate principles of liquid pressure and buoyancy.