Density

• Density is a basic property of any material. It is the mass per unit volume, ρ = m / V, with SI unit kg/m³.
• Density differs from one material to another because of two factors: the difference in the atomic or molecular mass of the substance, and the difference in the spaces between its atoms or molecules.
• An object with lower density floats on a liquid with higher density.
• Density can be measured directly with one instrument, the hydrometer.
• It can also be measured indirectly: the mass is measured with a balance and the volume with a graduated cylinder, then the mass is divided by the volume.

Density in medicine

• Density is used in medical science to measure the density of blood and urine.
• Normal blood density ranges from 1040 kg/m3 to 1060 kg/m3.
• A higher density indicates a higher concentration of blood cells, while a lower density indicates a lower concentration, which may suggest anemia.
• Urine density is typically about 1020 kg/m3, and certain diseases that increase salt excretion raise it.

Pressure at a point

• Pressure at a point is the average force acting perpendicularly on a unit area surrounding that point.
• If a force F acts perpendicularly on a surface of area A, the pressure is P = F / A, measured in N/m2.
• Because pressure depends on area, a pointed heel can exert a greater pressure on the ground than an elephant’s foot, since the area of the pointed heel is extremely small.

Pressure inside a liquid

• Consider a horizontal plate of area A at depth h below the surface of a liquid of density ρ.
• The plate is the base of a column of liquid.
• The force on the plate equals the weight of this liquid column, because the liquid is incompressible.
• The volume of the column is Ah and its mass is Ahρ, so its weight is Fg = Ahρg, where g is the acceleration due to gravity.
• Dividing this force by the area A gives the pressure due to the liquid alone:

P = hρg (N/m2)

• If the free surface of the liquid is exposed to the atmospheric pressure P_a, the total pressure at a point inside the liquid is:

P = Pa + hρg

• Observations show that the pressure at a point increases as its depth increases, and that the pressure increases with the density of the liquid at the same depth.
• From this relation:

• All points on the same horizontal level inside a liquid have the same pressure.
• In connected vessels whose bases are at the same horizontal level, the liquid rises to the same level in every section, regardless of the geometric shape of each section. Thus, the sea level is the same for all interconnected seas.
• Dams are built with thicker bases to withstand the increasing pressure with depth.
• Divers wear suits and helmets to protect themselves from the pressure at great depths.

Immiscible liquids in a U-shaped tube

• Water and oil do not mix, and they form a separating level in a U-shaped tube.
• Let ho be the height of the oil above the separating level, and hw the height of the water above it.
• The pressures at the two points on the separating level are equal:

• P_a + ρ_ogh_o = P_a + ρ_wgh_w, so ρ_oh_o = ρ_wh_w, and ρ_o / ρ_w = h_w / h_o
• By measuring ho and hw, the relative density of oil can be found.
• Since the density of water is known, the density of the oil can then be found.

Pascal’s principle

• Take a liquid in a glass container with a piston on top.
• The pressure at a point at depth h is P = P1 + hρg, where P1 is the pressure at the liquid surface directly under the piston.
• If an extra pressure ΔP is applied by placing an additional weight on the piston, the piston does not move inward, because the liquid is incompressible.
• The pressure at the point A, at depth h, also increases by ΔP, so it becomes P = P1 + hρg + ΔP.

The hydraulic press

• A hydraulic press has a small piston of cross-sectional area a and a large piston of cross-sectional area A, with a suitable liquid between them.
• A force f on the small piston produces a force F on the large piston, and the pressures are equal:

f / a = F / A

• The mechanical advantage of the press is η = F / f = A / a.
• When the area ratio a : A is 1 : 100, a force from a 1 kg weight on the small piston balances a force from 100 kg on the large piston.

• The large piston moves a smaller distance than the small piston, since f y₁ = F y₂.
• Applications of Pascal’s principle