Physics Notes Class 11 CHAPTER 10 MECHANICAL PROPERTIES OF FLUIDS part 1
Chapter: Mechanical Properties of Fluids
Definition of Fluids
Fluids: Substances that can flow when an external force is applied; includes liquids and gases.
Characteristics:
Do not have a finite shape; take the shape of their container.
Key Concepts in Fluid Mechanics
Hydrostatics: Study of fluids at rest.
Hydrodynamics: Study of fluids in motion.
Pressure in Fluids
Pressure of Liquid: Normal force exerted by a liquid per unit area of the surface in contact.
Formula for hydrostatic pressure: ( p = h \rho g )
(p): Pressure
(h): Height of liquid column
(\rho): Density of liquid
(g): Acceleration due to gravity
Mean Pressure on Vessel Walls: ( \frac{h \rho g}{2} )
Pascal’s Law
States that an increase in pressure at a point in an enclosed liquid in equilibrium is transmitted equally in all directions and to the walls of the container.
Applications include:
Hydraulic lift
Hydraulic press
Hydraulic brakes
Atmospheric Pressure
Defined as the pressure exerted by the atmosphere on earth (approximately 100,000 N/m² or 10 tons on 1 m²).
Equivalent to 76 cm of mercury column at sea level:
(\text{Atmospheric pressure} = hdg = 76 \times 13.6 \times 980, \text{dyne/cm}^2)
Measured in torr and bar:
(1, \text{torr} = 1, ext{mm of mercury column})
(1, \text{bar} = 10^5, ext{Pa})
Aneroid Barometer: Used to measure atmospheric pressure.
Buoyancy
Buoyant Force: Upward force acting on a body partially or fully immersed in a fluid.
Acts at the center of gravity of the displaced liquid; known as the center of buoyancy.
Archimedes’ Principle
A body immersed in a fluid loses weight equal to the weight of the liquid displaced by it.
If (T) is the observed weight of the body, and (σ) is its density, then:
Real weight of body (w = \frac{T}{1 - \frac{p}{σ}})
(p): Density of the liquid
Laws of Floatation
A body will float in a liquid if its weight equals the weight of the displaced liquid.
Scenarios:
If (W > w): Body sinks.
If (W < w): Body floats partially submerged.
If (W = w): Body floats with full volume immersed.
Equilibrium Conditions:
Stable Equilibrium: Meta-center above center of gravity.
Unstable Equilibrium: Meta-center below center of gravity.
Neutral Equilibrium: Meta-center coincides with center of gravity.
Density and Relative Density
Density: Ratio of mass to volume.
Density of liquid = ( \frac{Mass}{Volume} )
Density of water = 1 g/cm³ or 1000 kg/m³.
Is a scalar quantity with dimensional formula ([ML^{-3}]).
Relative Density (Specific Gravity): Ratio of a substance's density to the density of water at 4°C:
( \text{Relative density} = \frac{\text{Density of substance}}{\text{Density of water}} = \frac{Weight, of, substance, in, air}{Loss, of, weight, in, water} )
No unit, no dimensions.
Density of a Body: For a solid body, density equals density of substance. For a hollow body, density is less than that.
Density of Mixtures:
When mixing two liquids of mass m₁ and m₂ with densities ρ₁ and ρ₂:
(p = \frac{m_1 + m_2}{\frac{m_1}{p_1} + \frac{m_2}{p_2}})
When liquids of same mass mixed together:
(p = \frac{2p_1p_2}{p_1 + p_2})
When liquids of same volume mixed:
(p = \frac{p_1 + p_2}{2})
Density Variation with Pressure
Density changes with pressure:
(p = p_0 [1 + \frac{\Delta p}{K}])
(p_0): Initial density
(K): Bulk modulus of elasticity
(\Delta p): Change in pressure.