Density, mass and volume
Density is mass per unit volume: rho = m / V. Hence m = rho V and V = m / rho. In SI density is measured in kg/m3; problems often give grams and centimetres: 1 g/cm3 = 1000 kg/m3. An aluminium bar of 540 g and 200 cm3 has the density 540 / 200 = 2.7 g/cm3 = 2700 kg/m3.
| Substance | kg/m3 | Substance | kg/m3 |
|---|---|---|---|
| Water | 1000 | Ice | 900 |
| Sea water | 1030 | Aluminium | 2700 |
| Kerosene | 800 | Steel, iron | 7800 |
| Ethanol | 790 | Copper | 8900 |
| Mercury | 13600 | Lead | 11300 |
Pressure of a solid
Pressure is the force per unit area of support: p = F / S. For a body lying on a horizontal surface the force equals the weight, F = mg. The unit is the pascal: 1 Pa = 1 N/m2. A boy of 48 kg on skis of 0.3 m2 presses on snow with 48 × 9.8 / 0.3 = 1568 Pa, but on boots of 0.032 m2 with 14.7 kPa. That is why a larger area lowers the pressure and a sharp blade raises it.
Pressure of a liquid
The pressure of a liquid column at depth h is p = rho g h. It does not depend on the shape of the vessel and grows by 9.8 kPa for each metre of water. The total pressure adds the pressure at the surface: p = p0 + rho g h. Engineering problems (pressure on walls and gates, manometers, communicating vessels) are in the hydrostatics calculator.
Buoyancy and floating
A body in a liquid is pushed up by F_A = rho_l g V_sub, where V_sub is the submerged volume. Compare it with the weight P = rho_b V g: if rho_b < rho_l the body floats and in equilibrium F_A = P, so the submerged share is rho_b / rho_l. Ice (900) floats in water (1000) with 90 % of its volume below the surface. If rho_b > rho_l the body sinks and its weight in the liquid P - F_A is less than in air. For equal densities the body can hang at any depth.
Typing units
- Any field accepts a unit: 540 g, 200 cm3, 2 l, 300 cm2, 76 cm.
- A number without a unit is read in SI: kg, m3, m2, N, Pa.
- The solution shows Given, the conversion to SI and the formula with the numbers.