Density and pressure

Density

Density is a measure of how much mass is contained in a given volume. It tells us how compact or concentrated a substance is.

Definition

Density (ρ\rho) is defined as mass per unit volume.

Formula
ρ=mV\rho = \frac{m}{V}

where ρ\rho = density (kg m−3^{-3}), mm = mass (kg), VV = volume (m3^3)

A minimalist diagram showing two cubes of equal size: one labeled 'metal' and filled densely with small dots, the other labeled 'liquid' with fewer dots. This visually compares high and low density in a clean, modern style.

Different materials have different densities. For example, metals are usually much denser than liquids or gases.

Pressure

Pressure is the force applied per unit area. It describes how concentrated a force is on a surface.

Definition

Pressure (pp) is defined as force per unit area.

Formula
p=FAp = \frac{F}{A}

where pp = pressure (Pa), FF = force (N), AA = area (m2^2)

A simple, clean diagram of a flat horizontal surface with a downward arrow labeled 'F' and a shaded rectangle labeled 'A' beneath it, illustrating force applied over an area. Use thin lines and minimalist labels.

The SI unit of pressure is the pascal (Pa), where 1 Pa = 1 N m−2^{-2}.

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at rest, due to the weight of the fluid above a certain point.

Derivation of Hydrostatic Pressure Equation

Consider a column of liquid of height Δh\Delta h, density ρ\rho, and cross-sectional area AA.

  • Volume of liquid: V=AΔhV = A\Delta h
  • Mass of liquid: m=ρV=ρAΔhm = \rho V = \rho A \Delta h
  • Weight of liquid: W=mg=ρAΔh gW = mg = \rho A \Delta h \, g

Pressure at the base due to this liquid column:

Δp=WA=ρAΔh gA=ρgΔh\Delta p = \frac{W}{A} = \frac{\rho A \Delta h \, g}{A} = \rho g \Delta h
Formula
Δp=ρgΔh\Delta p = \rho g \Delta h

where Δp\Delta p = pressure difference (Pa), ρ\rho = density (kg m−3^{-3}), gg = gravitational field strength (N kg−1^{-1}), Δh\Delta h = height difference (m)

A clean, vertical cross-section of a container filled with liquid. Show a shaded column of height 'Δh', area 'A', and label the top and bottom of the column. Use simple arrows to indicate pressure at the bottom. Minimalist, modern style.

This equation is used to calculate the pressure difference between two points at different depths in a fluid.

Exam Tip

Always use the density of the fluid, not the object, when applying Δp=ρgΔh\Delta p = \rho g \Delta h.

Upthrust and Archimedes’ Principle

When an object is submerged in a fluid, it experiences an upward force called upthrust (or buoyant force). This is due to the difference in hydrostatic pressure between the top and bottom surfaces of the object.

  • The pressure at the bottom of the object is greater than at the top.
  • The difference in pressure creates a net upward force.

A minimalist diagram of a rectangular block fully submerged in water. Show arrows at the top (smaller) and bottom (larger) faces labeled with pressure, and a net upward arrow labeled 'Upthrust'. Keep the design clean and modern.

Archimedes’ principle states that the upthrust is equal to the weight of fluid displaced by the object.

Formula
F=ρgVF = \rho g V

where FF = upthrust (N), ρ\rho = density of fluid (kg m−3^{-3}), gg = gravitational field strength (N kg−1^{-1}), VV = volume of fluid displaced (m3^3)

1

Example

A block of volume 0.020.02 m3^3 is fully submerged in water (ρ=1000\rho = 1000 kg m−3^{-3}). Find the upthrust.

Summary

  • Density: ρ=mV\rho = \frac{m}{V}
  • Pressure: p=FAp = \frac{F}{A}
  • Hydrostatic pressure: Δp=ρgΔh\Delta p = \rho g \Delta h
  • Upthrust (Archimedes’ principle): F=ρgVF = \rho g V

Understanding these concepts is essential for solving problems involving fluids, floating and sinking, and pressure in liquids.

← Equilibrium of forces

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