HSC Physics Revision Hub

2.1

Pressure and Hydrostatics

P

Pressure

Force per unit normal area. Scalar; acts equally in all directions.

SI unit: pascal (Pa) = N/m². Also atm, bar, cm of Hg.

P = \frac{F}{A}

Pressure at depth $h$ in a fluid of density $\rho$ with atmospheric pressure $P_0$ at the surface:

P = P_0 + \rho g h

Pascal's law and Archimedes' principle

Pascal's law - pressure applied to an enclosed fluid is transmitted undiminished to every part. Buoyancy - a body in a fluid is acted on by an upward force equal to the weight of the displaced fluid: $F_b = \rho_f V g$ . A body floats/sinks according to its average density.

2.2

Fluid Flow and Bernoulli's Equation

Continuity (incompressible, steady flow)

A_1 v_1 = A_2 v_2 \qquad Q = A v

$A$ is the cross-section area, $v$ the flow speed and $Q$ the volume flow rate (m³/s).

Bernoulli's equation (ideal fluid)

P + \frac12 \rho v^2 + \rho g h = \text{constant}

$P$ pressure (Pa), $v$ speed (m/s), $h$ height (m), $\rho$ density (kg/m³). Applies along a streamline for steady, incompressible, non-viscous flow. Speed increases where the pressure is lower.

Torricelli's law (efflux speed from a tank)

v = \sqrt{2 g h}

$h$ is the height of the free surface above the hole; the emerging speed equals that of free fall through $h$ .

2.3

Viscosity and Stokes' Law

Newton's law of viscosity

F = -\eta A \frac{dv}{dx}

A tangential force opposes relative motion between adjacent layers. $\eta$ is the coefficient of viscosity (SI: Pa·s, or poise for CGS), $A$ the area and $dv/dx$ the velocity gradient.

Stokes' law (viscous drag on a sphere)

F = 6\pi \eta r v

Valid for a small sphere of radius $r$ moving slowly in a fluid of viscosity $\eta$ with speed $v$ , at low Reynolds number (streamline flow).

Terminal velocity

v_t = \frac{2 r^2 (\rho_s - \rho_f) g}{9 \eta}

When the upward viscous drag plus buoyancy balances the weight, the sphere falls with constant speed. $\rho_s$ is the density of the sphere, $\rho_f$ the density of the fluid. If $\rho_s > \rho_f$ the sphere sinks with the speed given (rising if lighter). Applies below the limit set by Re ≈ 0.5.

2.4

Surface Tension and Capillarity

T

Surface tension

The force acting along a unit length of the surface, tangent to it.

SI unit: N/m (also dyn/cm in CGS).

T = \frac{F}{L}

Surface energy

dW = T\, dA

Increasing the free surface area by $dA$ requires work $dW$ , stored as surface energy. For a film with two surfaces (e.g. soap film) the effective area doubles.

Excess pressure across curved surfaces

\Delta P = \frac{2T}{r} \quad (\text{liquid drop, one surface})

\Delta P = \frac{4T}{r} \quad (\text{soap bubble, two surfaces})

Pressure inside a curved surface exceeds that outside where the surface is convex. A bubble has two surfaces, hence the extra factor of 2. $r$ is the radius of curvature, $T$ the surface tension.

Angle of contact and wetting

The angle of contact $\theta$ is the angle between the tangent to the liquid surface and the solid surface, measured inside the liquid. Water on clean glass gives $\theta < 90\degree$ (wetting) and rises; mercury on glass gives $\theta > 90\degree$ (non-wetting) and dips.

Capillary rise

h = \frac{2 T \cos\theta}{r \rho g}

Rise of liquid in a capillary tube of inner radius $r$ . $T$ surface tension, $\theta$ angle of contact, $\rho$ liquid density, $g$ gravity. Valid for complete wetting/interfaces; the meniscus is the surface whose curvature sustains the column.