A cork floats on the surface of a liquid of density $\rho_2$. The density of cork is $\rho_1$. If the cork is pushed down and released, it undergoes SHM. The ratio $\rho_2/\rho_1$ at which the cork just sinks (fully submerged but in equilibrium at any depth) is:

Pressure in a fluid: $P = P_0 + \rho g h$ (Pascal\'s law). Pressure is same at same horizontal level. Manometer: measures gauge pressure ($P - P_{atm}$). Hydraulic press: $F_1/A_1 = F_2/A_2$ (same pressure transmitted). Buoyancy: $F_b = \rho_{fluid} V_{displaced} g$ (Archimedes). Object floats if $\rho_{obj} \leq \rho_{fluid}$.

Equation of continuity: $A_1 v_1 = A_2 v_2$ (mass conservation for incompressible fluid). Bernoulli\'s equation: $P + \frac{1}{2}\rho v^2 + \rho g h = $ const (energy conservation). Applications: airplane lift (Bernoulli + angle of attack), Venturi meter ($v_2 > v_1$ where $A_2 < A_1$, so $P_2 < P_1$), spray gun, Pitot tube (measures aircraft speed).

Viscosity: $F = \eta A (dv/dy)$; $\eta$ = coefficient of viscosity (Pa s). Stokes law: drag on sphere = $6\pi\eta r v$; terminal velocity: $v_t = 2r^2(\rho-\rho_f)g/9\eta$. Surface tension: $T$ = force per unit length; pressure inside bubble: $\Delta P = 4T/r$ (soap bubble has 2 surfaces), $\Delta P = 2T/r$ (liquid drop, 1 surface). Capillary rise: $h = 2T\cos\theta/\rho g r$.

Stress = Force/Area. Strain = deformation/original dimension. Young\'s modulus: $Y = (F/A)/(l/L)$; stiffness for longitudinal deformation. Bulk modulus: $K = -V(dP/dV)$; resistance to volume change. Modulus of rigidity (shear modulus): $G = (F/A)/\theta$. Poisson\'s ratio: $\sigma = -(\Delta d/d)/(\Delta l/l)$. Energy stored in stretched wire: $U = \frac{1}{2}YAL\epsilon^2 = \frac{1}{2}\times stress \times strain \times volume$.

Bernoulli's equation, $P + \frac{1}{2}\rho v^2 + \rho gh = $ constant, is energy conservation per unit volume along a streamline. It assumes the flow is steady, incompressible, non-viscous and irrotational — conditions worth stating, because a question that mentions viscosity or turbulence is telling you Bernoulli does not apply. Where the speed is high the pressure is low, which explains aerofoil lift, the Venturi meter and why a shower curtain is pulled inward by the flowing water.

A sphere falling through a viscous fluid experiences drag $F = 6\pi\eta rv$. It accelerates until drag plus buoyancy balances weight, after which it falls at constant terminal velocity $v_t = \frac{2r^2(\rho - \sigma)g}{9\eta}$. Note the $r^2$ dependence: doubling the radius quadruples the terminal speed, which is why fine dust stays suspended in air for hours while a pebble drops immediately.

Surface tension arises because molecules at a liquid surface have unbalanced attractive forces pulling them inward, making the surface behave like a stretched membrane. The excess pressure inside a liquid drop is $2T/r$, but inside a soap bubble it is $4T/r$, because a bubble has two surfaces. Since excess pressure varies as 1/r, a small bubble has higher internal pressure than a large one — which is why, when two bubbles are connected, the smaller one empties into the larger.