Mechanical Properties of Fluids

P = P₀ + ρgh — see pressure grow linearly with depth in any fluid.

F₁/A₁ = F₂/A₂ — small input force lifts a heavy load via large piston.

F_b = ρ_fluid V_disp g — sink, float, suspend depending on density.

Compare 4 materials in one fluid — see who floats by density rule.

Re = ρvD/η — sweep velocity to cross from laminar to turbulent flow.

A₁v₁ = A₂v₂ — pipe narrows; particles speed up. Live volume flow rate.

P + ½ρv² + ρgh = const — manometers show pressure drop where speed is high.

Pipe with throat — flow accelerates, pressure plummets. Used in flow meters.

Curved airfoil — air on top is faster, pressure lower → net upward force.

F_drag = 6πηrv — viscous drag on a sphere through fluid.

Sphere in viscous fluid — accelerates until weight = buoyancy + drag at v_t.

h = 2T cosθ/(ρgr) — narrow tube pulls liquid up by surface tension.

Excess pressure inside drops/bubbles — ΔP = 2T/R (drop), 4T/R (bubble).