U-tube with two liquids.

Pour oil into one arm of a water U-tube; levels settle so pressures match at the interface level.

Measure the heights and find the unknown liquid's density to 2 significant figures.

Equating pressures at the same height in different liquids above the interface.

• Atmospheric pressure at a point is the weight of the air standing above it, per unit area: a column of 1 m² section running up to where the air ends. At sea level it is 1.013 × 10⁵ Pa (1 atm).
• Torricelli's mercury barometer: fill a long glass tube, sealed at its lower end, with mercury and turn it upside down into a dish of mercury. The space above the column holds only mercury vapour at negligible pressure, so P_A = 0.
• Point B inside the column and point C on the open trough surface are at the same level, so both are at P_a. Hence P_a = ρgh, with ρ the density of mercury and h the column height.
• At sea level the column stands about 76 cm high, which is 1 atm. Pressure is often quoted in cm or mm of mercury; 1 mm of Hg is called a torr (after Torricelli), and 1 torr = 133 Pa.
• Doctors and physiologists quote pressures in torr or mm of Hg. Weather science uses the bar and millibar: 1 bar = 10⁵ Pa.
• Open-tube manometer: a U-tube of liquid, one end open to air and the other joined to the system being tested. The level difference h gives the gauge pressure P − P_a = ρgh. Oil (low density) suits small pressure differences, mercury (high density) large ones.
• Pressure is equal at the same level in both arms of a U-tube holding one fluid at rest. Liquid density barely changes with pressure and temperature, so liquids are treated as incompressible; gas density changes a lot.
• If air kept its sea-level density 1.29 kg m⁻³ all the way up: ρgh = 1.01 × 10⁵ Pa with g = 9.8 m s⁻² gives h = 7989 m ≈ 8 km. In reality both air density and g fall with height and the atmosphere thins out over more than 100 km.
• Sea-level pressure is not always 760 mm of Hg: a fall of 10 mm or more in the barometer points to an approaching storm.
• 1000 m down in the sea (ρ = 1.03 × 10³ kg m⁻³, g = 10 m s⁻²): absolute pressure 104.01 × 10⁵ Pa ≈ 104 atm; gauge pressure 103 × 10⁵ Pa ≈ 103 atm. A submarine kept at 1 atm inside feels only the gauge pressure on a 20 cm × 20 cm window (0.04 m²): F = 4.12 × 10⁵ N.
• Tyre-pressure gauges and the blood-pressure gauge (sphygmomanometer) read gauge pressure.

Teaching this idea? Put the simulation on your own page or LMS with this code.