Fluid at Rest

Fluid at Rest

Fluids at rest are essential to understanding hydrostatics, which deals with the properties and behavior of stationary fluids. This topic covers volume, density, pressure in fluids, and equilibrium of bodies, including principles like Archimedes’ principle and the law of flotation.

1. Volume, Density, and Relative Density

a. Volume

• Definition: Volume is the amount of space occupied by a substance.
• SI Unit: Cubic meters ($\text{m}^3$).
• Formula for Regular Shapes: $$\text{Volume of a cube} = a^3, \quad \text{Volume of a sphere} = \frac{4}{3} \pi r^3$$
• Example: The volume of a water bottle is $2 \, \text{L}$ or $0.002 \, \text{m}^3$.

b. Density

• Definition: Density is the mass per unit volume of a substance.
• Formula: $$\rho = \frac{m}{V}$$ Where $\rho$ = Density ($\text{kgm}^{-3}$), $m$ = Mass ($\text{kg}$), $V$ = Volume ($\text{m}^3$).
• Example: The density of water is $1000 \, \text{kgm}^{-3}$.

c. Relative Density

• Definition: Relative density (specific gravity) is the ratio of the density of a substance to the density of water.
• Formula: $$\text{Relative Density} = \frac{\rho_{\text{substance}}}{\rho_{\text{water}}}$$
• Key Point: Relative density has no unit (it is a ratio).
• Example: The relative density of gold is 19.3.

2. Pressure in Fluids

a. Concept and Definition of Pressure

• Definition: Pressure is the force exerted per unit area.
• Formula: $$P = \frac{F}{A}$$ Where $P$ = Pressure ($\text{Nm}^{-2}$), $F$ = Force ($\text{N}$), $A$ = Area ($\text{m}^2$).
• Key Point: Fluids exert pressure equally in all directions at a given depth.

b. Dependence of Pressure on Depth

• Formula: $$P = \rho g h$$ Where $P$ = Pressure ($\text{Nm}^{-2}$), $\rho$ = Density of fluid ($\text{kgm}^{-3}$), $g$ = Gravitational acceleration ($9.8 \, \text{ms}^{-2}$), $h$ = Depth ($\text{m}$).
• Key Point: Pressure increases with depth.

c. Atmospheric Pressure

• Definition: The pressure exerted by the Earth's atmosphere at sea level.
• Value: $101325 \, \text{Pa}$ or $1 \, \text{atm}$.

d. Pascal’s Principle

• Statement: In a confined fluid, any change in pressure is transmitted uniformly in all directions.
• Applications:
  • Hydraulic Press: Multiplies force using fluid pressure.
  • Car Brakes: Transmits force efficiently through brake fluid.

3. Equilibrium of Bodies in Fluids

a. Archimedes’ Principle

• Statement: A body wholly or partially immersed in a fluid experiences an upward force (buoyancy) equal to the weight of the displaced fluid.
• Formula: $$F_{\text{buoyancy}} = \rho_{\text{fluid}} g V_{\text{displaced}}$$
• Applications:
  • Floating ships.
  • Submarine design.

b. Law of Flotation

• Statement: A floating body displaces its own weight of the fluid in which it floats.
• Applications:
  • Designing boats and rafts.

c. Experimental Determination

1. For Solids: Measure the displaced fluid's weight using a spring balance and determine relative density.
2. For Liquids: Use a U-tube or Hare’s apparatus to compare densities.

4. Tools and Devices in Fluid Studies

a. Simple Barometer

• Purpose: Measures atmospheric pressure.
• Design: A column of mercury in a glass tube inverted in a reservoir.

b. Manometer

• Purpose: Measures pressure difference between two points in a fluid.

c. Siphon

• Purpose: Transfers liquid from one container to another using atmospheric pressure.

d. Syringe and Pump

• Purpose: Utilize fluid pressure for suction or delivery of fluids.

5. Determining Relative Density

a. U-Tube

• Measures relative density by comparing heights of different fluids.

b. Hare’s Apparatus

• Compares densities of two liquids by measuring the rise in their columns.

6. Forces Acting on a Body in Fluid

1. Weight: Downward force due to gravity.
2. Buoyancy: Upward force due to displaced fluid.
3. Net Force: Determines whether the body sinks, floats, or remains neutral.

7. Applications of Fluid Principles

1. Hydraulic Systems: Lifts, brakes, and jacks.
2. Design of Ships and Submarines: Using Archimedes’ principle.
3. Weather Forecasting: Atmospheric pressure readings.

8. Common Misconceptions

1. “Heavier objects always sink”:
   • False; floating depends on density, not just weight.
2. “Liquids exert pressure only downward”:
   • Incorrect; fluids exert pressure in all directions.

Summary

• Volume is the space a fluid occupies.
• Density is mass per unit volume, while relative density compares it to water.
• Pressure in fluids depends on depth, density, and gravity, following Pascal’s principle.
• Archimedes’ principle explains buoyancy, while the law of flotation governs floating bodies.
• Tools like barometers, manometers, and siphons help study fluids.
• Real-world applications range from hydraulic presses to ship design.