## Mechanical Properties of Fluids: Study Notes ### 1. Introduction to Fluids Fluids are substances that flow when an external force is applied. This includes liquids and gases. A key characteristic is that fluids don't have a fixed shape; they conform to the shape of their container. Think of water in a glass versus a water balloon – the water adapts to the shape of its container in both cases. The total normal force exerted by a fluid at rest on a surface is called **thrust**. The SI unit of thrust is the newton (N). Fluid mechanics is broadly divided into: * **Hydrostatics:** The study of fluids at rest. * **Hydrodynamics:** The study of fluids in motion. ### 2. Pressure in Fluids **Pressure** is the normal force exerted by a fluid per unit area. In a liquid at rest, this is called **hydrostatic pressure**. The pressure exerted by a liquid column of height *h*, density *ρ*, and under acceleration due to gravity *g* is given by: $$p = hρg$$ Consider a water column 1 meter high. The density of water is approximately 1000 kg/m³. The pressure at the bottom of the column is: $p = (1 m)(1000 kg/m³)(9.8 m/s²) = 9800 Pa$ Pressure is a scalar quantity. The mean pressure on the walls of a vessel containing liquid up to height *h* is $\frac{hρg}{2}$. ### 3. Pascal's Law Pascal's Law states that any change in pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and the walls of the containing vessel. Think of squeezing a toothpaste tube – the pressure you apply is transmitted throughout the tube. This principle is fundamental to the operation of hydraulic lifts, hydraulic presses, and hydraulic brakes. ### 4. Atmospheric Pressure **Atmospheric pressure** is the pressure exerted by the atmosphere on Earth. At sea level, it's approximately 100,000 N/m² or 76 cm of mercury column. Many students mistakenly believe atmospheric pressure crushes us. Our internal pressure balances the external atmospheric pressure. Atmospheric pressure is also measured in torr and bar: * 1 torr = 1 mm of mercury column * 1 bar = 10⁵ Pa An aneroid barometer is used to measure atmospheric pressure. ### 5. Buoyancy and Archimedes' Principle **Buoyancy** is the upward force exerted on an object submerged in a fluid. This force acts at the **center of buoyancy**, which is the center of gravity of the displaced fluid. **Archimedes' Principle** states that the buoyant force on an object is equal to the weight of the fluid displaced by the object. A block of wood floating in water displaces a volume of water equal to its own weight. The buoyant force (weight of displaced water) equals the weight of the wood. If *T* is the observed weight of a body of density σ fully immersed in a liquid of density ρ, then the real weight *w* of the body is: $$w = \frac{T}{1 - \frac{ρ}{σ}}$$ ### 6. Laws of Floatation A body will float if the weight of the body is equal to the weight of the liquid displaced. * **W > w:** The body sinks. * **W < w:** The body floats partially submerged. * **W = w:** The body floats completely submerged. The stability of a floating body depends on the relative positions of its center of gravity and metacenter (center of buoyancy): * **Metacenter above center of gravity:** Stable equilibrium. * **Metacenter below center of gravity:** Unstable equilibrium. * **Metacenter coincides with center of gravity:** Neutral equilibrium. ### 7. Density and Relative Density **Density (ρ)** is the mass per unit volume of a substance: $$ρ = \frac{mass}{volume}$$ The density of water is approximately 1 g/cm³ or 10³ kg/m³. Density is a scalar quantity with dimensions [ML⁻³]. **Relative density** (or specific gravity) is the ratio of the density of a substance to the density of water at 4°C. It's a dimensionless quantity. $$Relative\ Density = \frac{Density\ of\ substance}{Density\ of\ water\ at\ 4°C} = \frac{Weight\ in\ air}{Loss\ of\ weight\ in\ water}$$ ### 8. Density of Mixtures For two liquids with masses *m₁* and *m₂* and densities *ρ₁* and *ρ₂*, the density of the mixture (ρ) is: $$ρ = \frac{m_1 + m_2}{\frac{m_1}{ρ_1} + \frac{m_2}{ρ_2}} = \frac{ρ_1 ρ_2 (m_1 + m_2)}{m_1 ρ_2 + m_2 ρ_1}$$ If the masses are equal (*m₁ = m₂ = m*): $$ρ = \frac{2ρ_1 ρ_2}{ρ_1 + ρ_2}$$ If the volumes are equal (*V₁ = V₂ = V*): $$ρ = \frac{ρ_1 + ρ_2}{2}$$ ### 9. Variation of Density with Pressure The density of a liquid varies with pressure: $$p = p_0 [1 + \frac{Δp}{K}]$$ where *p₀* is the initial density, *Δp* is the change in pressure, and *K* is the bulk modulus of elasticity. This concludes the study notes on the mechanical properties of fluids. Remember to review the key concepts, formulas, and examples to solidify your understanding.