Density
Section: Motion, Forces & Energy | Syllabus: Cambridge AS Level Physics 9702
Density Density Mass per unit volume of a substance. ρ = m / V ρ (rho) = density (kg/m³) | m = mass (kg) | V = volume (m³) The equation can be rearranged: m = ρ × V (mass = density × volume) V = m / ρ (volume = mass ÷ density) Units The SI unit of density is kg/m³ .
Density is also commonly expressed in g/cm³ . Conversion: 1 g/cm³ = 1000 kg/m³ (e.g. water: 1.0 g/cm³ = 1000 kg/m³) Figure: Density Formula Triangle A triangle divided into three sections. Top: "m" (mass, in kg).
Bottom-left: "ρ" (density, in kg/m³). Bottom-right: "V" (volume, in m³). A horizontal line separates m from ρ and V; a vertical line separates ρ and V. Instructions: Cover what you want to find. Cover m → m = ρ × V.
Cover ρ → ρ = m ÷ V. Cover V → V = m ÷ ρ. Measuring Density - Regularly Shaped Solid Measure the mass using a balance. Measure the dimensions using a ruler (e.g. length, width, height for a cuboid). Calculate the volume : V = length × width × height.
Calculate density: ρ = m / V . Figure: Measuring Density of a Regular Solid Two-step illustration. Left: A rectangular metal block on an electronic balance displaying "156.0 g". Right: The same block with a ruler alongside it, measuring length = 5 cm, width = 4 cm, height = 2 cm - each dimension labelled with an arrow.
Below: V = 5 × 4 × 2 = 40 cm³. ρ = 156 ÷ 40 = 3.9 g/cm³. Worked Example: Regular Solid A metal block has dimensions 5 cm × 4 cm × 2 cm and a mass of 200 g. Calculate its density. Volume = 5 × 4 × 2 = 40 cm³ Density = m / V = 200 / 40 = 5 g/cm³ (= 5000 kg/m³) Measuring Density - Irregularly Shaped Solid An irregular solid cannot have its volume measured with a ruler, so the displacement method is used.
Measure the mass using a balance. Part-fill a measuring cylinder with water. Record the initial volume V₁. Carefully lower the solid into the water (it must be fully submerged and must sink). Record the new water level V₂.
Volume of solid = V₂ − V₁. Calculate density: ρ = m / V . Key Point The solid must sink to be fully submerged. This method works because the volume of water displaced equals the volume of the solid. Figure: Displacement Method for an Irregular Solid Two measuring cylinders side by side.
Left: water only, level at 50 cm³ - labelled "V₁ = 50 cm³". Right: the same cylinder with an irregular stone fully submerged on a string, water level risen to 68 cm³ - labelled "V₂ = 68 cm³". Arrow points to calculation: Volume of stone = 68 − 50 = 18 cm³.
Note: "Alternatively, use a displacement (eureka) can - fill to the spout, lower the object in, collect the water that overflows into a measuring cylinder to find its volume directly." Worked Example: Irregular Solid A stone has a mass of 54 g.
When placed in a measuring cylinder, the water level rises from 20 cm³ to 38 cm³. Calculate the density of the stone. Volume = 38 − 20 = 18 cm³ Density = m / V = 54 / 18 = 3 g/cm³ Measuring Density - Liquid Measure the mass of an empty measuring cylinder (m₁).
Pour the liquid into the cylinder and read the volume V . Measure the mass of the cylinder + liquid (m₂). Mass of liquid = m₂ − m₁. Calculate density: ρ = m / V . Figure: Measuring Density of a Liquid Three steps.
Step 1: Empty measuring cylinder on a balance - display shows "45.2 g" (m₁). Step 2: Liquid poured into the cylinder - meniscus sits at the 80 cm³ mark (V = 80 cm³). Step 3: Cylinder with liquid on the balance - display shows "125.2 g" (m₂).
Calculation: mass of liquid = 125.2 − 45.2 = 80.0 g. Density = 80.0 / 80 = 1.0 g/cm³. Worked Example: Liquid An empty measuring cylinder has a mass of 45 g. After adding a liquid, the cylinder + liquid has a mass of 117 g, and the liquid volume reads 60 cm³.
Calculate the density of the liquid. Mass of liquid = 117 − 45 = 72 g Density = m / V = 72 / 60 = 1.2 g/cm³ Floating and Sinking An object floats in a liquid if its density is less than the density of the liquid.
An object sinks in a liquid if its density is greater than the density of the liquid. Worked Example: Float or Sink? Liquid X has a density of 1000 kg/m³. Object A has density 800 kg/m³. Object B has density 2700 kg/m³.
What happens to each object in liquid X? Object A: 800 < 1000 → density less than liquid → floats Object B: 2700 > 1000 → density greater than liquid → sinks One Liquid Floating on Another When two liquids that do not mix are placed together, the less dense liquid floats on top of the more dense liquid.
The liquid with the lower density forms the upper layer . The liquid with the higher density forms the lower layer . Worked Example: Liquids in Layers Three immiscible liquids are poured into a measuring cylinder: Liquid P (density 800 kg/m³), Liquid Q (density 1400 kg/m³), Liquid R (density 1000 kg/m³).
Describe the order of layers from bottom to top. Arrange by density (highest at bottom): Q (1400), R (1000), P (800) Bottom to top: Q - R - P Figure: Immiscible Liquids in Layers A tall measuring cylinder containing three distinct coloured layers.
Bottom layer: darkest colour, labelled "Liq…
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