Fluids formulas

Water 1000 kg/m³, seawater 1025, ice 917, air 1.29, mercury 13600. g/cm³ × 1000 = kg/m³. 1 L = 0.001 m³. 1 cm³ = 1E−6 m³. Volume: box L*W*H, ball 4/3*π*r^3 (e.g. 4/3*π*0.1^3), cylinder π*r²*h. Know the weight W? Type W/9.8 for m. Specific gravity = ρ / 1000.

Hydraulic lift: the pressure is the same on both pistons. Type F₁/A₁ in p, e.g. 300/(π*0.01²), the other piston's area in A, solve F. Round area: π*r² (diameter d: π*d²/4). Window, lid or roof: type the pressure difference in p. Hatch at depth h (air inside): type ρgh in p, e.g. 1025*9.8*30. Dam or tank wall: type ρgh/2 in p (average), wall area in A. 1 atm = 1.013E5 Pa = 760 mmHg = 14.7 psi. Weight: F = m*9.8.

p₀ = pressure on top of the fluid (open to air: 1.013E5, filled in). Gauge pressure (tire, blood, "above air"): p₀ = 0. Two points in one fluid: p₀ = pressure at the higher point, p = at the lower, h = how much deeper. Seawater 1025, mercury 13600. mmHg × 133.3 = Pa. Barometer (vacuum on top): p₀ = 0, p = air pressure. U-tube, two liquids: type ρgh of one side in p (e.g. 1000*9.8*0.05), p₀ = 0, the other ρ, solve h.

ρ = density of the FLUID (water 1000, seawater 1025, air 1.29), not the object. V = volume under the fluid (the whole object if it is fully under). Floating: Fb = its weight. Type m*9.8 in Fb, solve V (the part under). Weighed in air (W) and in water (Wₐ): type W−Wₐ in Fb (e.g. 25−21), solve V. Fb does not change with depth.

Wₐ = string tension or scale reading while it is under the fluid. ρ = fluid density, V = object volume. Know its density? Type m/ρobj in V, e.g. 2/2700. Negative answer = it floats up; the string pulls down that hard. Floating on its own: Wₐ = 0, V = part under. Weighed in air (W) and in water (Wₐ): type W/9.8 for m, solve V, then ρ = m/V. Raft/balloon max load: Wₐ = 0, solve m, minus its own mass (+gas). Air 1.29.

Only for a FLOATING object. f = fraction under the fluid (0.8 = 80%). Part above = 1 − f. Floats with 2/3 under: type 2/3 for f. ρ = fluid density (water 1000, seawater 1025). Ice: ρobj = 917. Block: depth under = f × its height.

Pipe gets narrower or wider (A₁v₁ = A₂v₂): Q is the same everywhere. Type A₁v₁ in Q, e.g. π*0.02²*1.5, the new A, solve v. Round pipe: A = π*r² (diameter d: π*d²/4). Several holes (shower head): A = n*π*r². Volume per time given: type V/t in Q, e.g. 0.1/72. 1 L = 0.001 m³. 1 min = 60 s.

Time to fill a tank, or volume that flows in a time. Know the area and speed? Type A*v in Q, e.g. π*0.01²*3. 1 L = 0.001 m³. 1 cm³ = 1E−6 m³. 1 min = 60 s. L/min to m³/s: × 0.001 / 60.

1 and 2 = two spots in the same moving fluid. y = height (y = 0 at the lower spot). Open to air at both: type 0 for p₁ and p₂. Big tank surface or still fluid: v = 0. Hole in a tank: p₁ = p₂ = 0, v₁ = 0, y₁ = water height above hole, y₂ = 0, solve v₂ (= √(2gh)). Level pipe: y₁ = y₂ = 0. Know Q? Type Q/A in each v box, e.g. 0.002/(π*0.01²).