JEE Physics Mechanical Properties Of Fluids Notes Set 01

Elasticity

Stress is the restoring force per unit area, and strain is the fractional deformation. Within the elastic limit stress is proportional to strain (Hooke's law). Young's modulus \( Y=\dfrac{F/A}{\Delta l/l} \), bulk modulus \( B=\dfrac{-\Delta P}{\Delta V/V} \) (its reciprocal is compressibility) and rigidity modulus \( \eta=\dfrac{\text{shear stress}}{\text{shear strain}} \). Poisson's ratio \( \sigma \) is the ratio of lateral strain to longitudinal strain and lies between -1 and 0.5; the moduli are related by \( Y=3B(1-2\sigma)=2\eta(1+\sigma) \).

The energy stored in a stretched wire is \( \tfrac12 F\,\Delta l=\dfrac{YA(\Delta l)^2}{2l} \), and per unit volume \( \tfrac12 \times\text{stress}\times\text{strain}=\dfrac{S^2}{2Y} \). For a wire hanging under its own weight, \( \Delta l=\dfrac{\rho g l^2}{2Y} \). If a wire of a fixed volume is drawn longer, the extension for a given force varies as \( l^2 \). The force constant of a wire is \( YA/l \), and between atoms \( k=Y r_0 \).

Hydrostatics

Pressure at depth h is \( P=P_0+\rho gh \), the same at all points at the same level, and it is transmitted equally in all directions (Pascal's law), which is used in hydraulic lifts and presses. The thrust on the base of a vessel is \( \rho g h A \), independent of its shape. The mean pressure on a vertical wall is \( \rho gh/2 \).

Archimedes' principle: the buoyant force on a body equals the weight of the liquid displaced. A body floats if its density is less than that of the liquid, with the fraction \( \rho_s/\rho_l \) of its volume immersed. Apparent weight is \( W\left(1-\dfrac{\rho_l}{\rho_s}\right) \). When floating ice melts, the water level does not change. A ball at the interface of two liquids has \( \rho_1\lt\rho\lt\rho_2 \).