JEE Physics Mechanical Properties Of Fluids Notes Set 06

Density and Relative Density

Density is mass per unit volume, \( \rho=m/V \), with unit kg/m³ (1 g/cm³ = 1000 kg/m³). Relative density (specific gravity) is the density of a substance divided by the density of water at 4 °C, so it has no unit. For a mixture, \( \rho_{mix}=\dfrac{m_1+m_2}{V_1+V_2} \).

Pressure in a Fluid

Pressure is the normal force per unit area, \( P=F/A \), a scalar, measured in pascal. At a point in a fluid at rest the pressure is the same in all directions and the force on any surface is perpendicular to it. At depth h in a liquid of density \( \rho \): \( P=P_0+\rho gh \). Points at the same level in the same connected liquid have equal pressure. The force on a vertical wall uses the average pressure, \( \rho g h_{cm}A \), and acts below the centroid. The force on the base depends only on the height of liquid and the base area, not on the shape of the vessel.

Pascal's Law, Barometer and Manometer

A pressure change in an enclosed liquid is transmitted undiminished to every point. A hydraulic press uses \( F_2=F_1(A_2/A_1) \); work stays the same because the small piston moves further. A mercury barometer balances atmospheric pressure with a column of about 76 cm, \( P_{atm}=\rho_{Hg}gh \), and mercury is used because its high density gives a short column. An open manometer measures gauge pressure from the difference of levels.

Fluids in Accelerated Vessels

With horizontal acceleration a, the free surface of a liquid tilts so that \( \tan\theta=a/g \), and pressure increases in the direction opposite to a: \( \Delta P=\rho a\,\ell \) along the acceleration. With vertical acceleration a the effective gravity is \( g\pm a \). In a vessel rotating at angular speed \( \omega \) the free surface is a paraboloid with height difference \( \omega^2R^2/2g \) between the rim and the axis.

Archimedes' Principle and Floatation

A body in a fluid feels an upward force equal to the weight of the fluid displaced, \( F_B=V_{im}\rho_{fl}g \), acting at the centre of buoyancy. If the body's density is greater than the liquid's it sinks; if equal, it floats fully immersed; if less, it floats with \( V_{im}/V=\rho_b/\rho_{fl} \). Apparent weight = true weight − buoyant force. When ice floating in water melts, the level of water does not change; if a stone is frozen inside, the level falls. For stable floatation the metacentre must lie above the centre of gravity. In an accelerating liquid g is replaced by the effective gravity.

Equation of Continuity

For an ideal (incompressible, non-viscous, steady, irrotational) flow, the volume flow rate through any cross-section of a tube is the same: \( A_1v_1=A_2v_2 \). Speed is higher where the tube is narrower, and streamlines are closer together there.

Bernoulli's Equation

Along a streamline of an ideal fluid, \( P+\tfrac12\rho v^2+\rho gh=\text{constant} \). It expresses conservation of energy per unit volume. Where speed is high, pressure is low, which explains the lift on a wing, the spreading of a vertical hose jet and the working of a Venturi meter.

Applications of Bernoulli's Equation

Speed of efflux (Torricelli): \( v=\sqrt{2gh} \) for a hole at depth h in an open tank. The jet lands at a horizontal range \( x=2\sqrt{h(H-h)} \) when the hole is at depth h in water of height H; the range is maximum (equal to H) for \( h=H/2 \). Time to drain a tank of area A through a hole of area a: \( t=\dfrac Aa\sqrt{\dfrac{2H}{g}} \). The thrust on a vessel from a jet is \( \rho av^2 \). A Venturi meter gives the flow rate from the pressure difference between a wide and a narrow section, and a Pitot tube gives the speed of a gas from \( \tfrac12\rho v^2=\Delta P \). Momentum flux gives the force of a jet: \( F=\rho A(v-V)^2 \) on a plate moving at V.

Worked Example: Venturi Pipe

Water flows through a horizontal tube whose areas are 40 mm² and 20 mm², with 10 cm³/s entering the wide end. The speeds are 0.25 m/s and 0.50 m/s, and the pressure difference is \( \tfrac12\rho(v_2^2-v_1^2)=93.75 \) Pa, higher at the wide end.