JEE Physics Mechanical Properties Of Fluids Notes Set 04

Pressure and Density

Pressure is the normal force acting on a unit area of a surface: \( P=F/A \) (unit pascal, 1 Pa = 1 N/m²). It is a scalar. Density is mass per unit volume, \( \rho=m/V \), and relative density is the ratio of the density of a substance to that of water at 4 °C.
Pressure at depth h in a liquid of density ρ at rest is \( P=P_0+\rho g h \). The part \( \rho g h \) is the gauge pressure. All points at the same level in a connected liquid at rest have the same pressure, and the shape of the vessel does not matter (hydrostatic paradox).

Pascal's Law and Hydraulic Machines

A change of pressure applied to an enclosed liquid is passed on undiminished to every part of the liquid and to the walls. In a hydraulic lift or brake, equal pressure on pistons of areas \( A_1 \) and \( A_2 \) gives \( F_2=F_1\,\frac{A_2}{A_1} \). The force is multiplied while the work done stays the same, because the small piston moves through a larger distance.

Atmospheric Pressure and Barometers

The atmosphere exerts about \( 1.013\times10^5 \) Pa (76 cm of mercury). A mercury barometer balances this pressure with a column of height h, so \( P_{atm}=\rho_{Hg}gh \). An open-tube manometer measures the gauge pressure from the difference in liquid levels.

Buoyancy and Archimedes' Principle

A body partly or fully immersed in a fluid experiences an upward force equal to the weight of the fluid it displaces: \( F_B=\rho_{fluid}V_{immersed}\,g \). It acts at the centre of buoyancy, the centroid of the displaced fluid.
Apparent weight in a liquid = true weight − buoyant force. Relative density of a body = weight in air / loss of weight in water.

Floating, Sinking and Stability

1. If the body's density is greater than the liquid's, it sinks.
2. If equal, it floats fully immersed in any position.
3. If smaller, it floats with immersed fraction \( V_{in}/V=\rho_{body}/\rho_{liquid} \).
For a floating body of uniform cross-section the exposed height fraction is \( 1-\rho_{body}/\rho_{liquid} \). A body is in stable equilibrium when the metacentre lies above its centre of gravity.
In an accelerating container the effective gravity \( g_{eff} \) replaces g in both weight and buoyancy. In free fall the buoyant force is zero.

Liquid in Accelerated and Rotating Vessels

For a vessel with horizontal acceleration a, the free surface tilts through angle θ with \( \tan\theta=a/g \). For a vessel with vertical acceleration a, the pressure at depth h is \( \rho h(g\pm a) \).
For a vessel rotating about its vertical axis with angular speed ω, the free surface is a paraboloid with height at distance r above the lowest point \( y=\frac{\omega^2r^2}{2g} \).

Equation of Continuity and Bernoulli's Theorem

For steady flow of an incompressible fluid, the volume flow rate is constant: \( A_1v_1=A_2v_2 \). Narrow sections carry faster flow.
For an ideal fluid (non-viscous, incompressible, steady, irrotational) along a streamline: \( P+\tfrac12\rho v^2+\rho g h=\text{constant} \). The three terms are the pressure energy, kinetic energy and potential energy per unit volume. Dividing by ρg gives the pressure head, velocity head and gravitational head.

Applications of Bernoulli's Theorem

1. Torricelli's theorem: speed of efflux from a hole at depth h is \( v=\sqrt{2gh} \). The horizontal range from a hole at depth h in a tank of height H is \( 2\sqrt{h(H-h)} \), which is greatest (equal to H) when h = H/2.
2. Venturi meter: the flow rate is found from the pressure difference between a wide and a narrow section, \( P_1-P_2=\tfrac12\rho(v_2^2-v_1^2) \).
3. Dynamic lift: faster air over the upper surface of a wing means lower pressure there and a net upward force.
4. Time to drain a tank through a bottom hole of area a: \( t=\frac Aa\sqrt{\frac{2H}{g}} \).
5. Reaction force of a jet leaving at speed v through area a is \( \rho a v^2 \); this explains the recoil of tanks and rotating sprinklers.