Density

CORE VS EXTENDED GUIDE

• Core students study only the unlabelled sections.
• Extended students must study everything, including Extended Extended points.
• Extended = Core + Supplement.

Defining Density

Key Equations

Density equation:

$$\rho =\frac{m}{V}$$

Variables:

• $\rho $ = density, in kg m⁻³
• $m$ = mass, in kg
• $V$ = volume, in m³

SI unit of density: kg m⁻³ (g cm⁻³ is also commonly used in CIE problems)

Rearrangements:

Starting from $\rho =\frac{m}{V}$:

Rearranging for mass:

$$m=\rho \times V$$

Rearranging for volume:

$$V=\frac{m}{\rho }$$

ProportionalityDensity is directly proportional to mass when volume is constant, so doubling the mass doubles the density. Density is inversely proportional to volume when mass is constant, so doubling the volume halves the density.

Density is mass per unit volume.

Density tells you how much mass is packed into each unit of volume of a substance. A material with high density has a large mass concentrated in a small volume, whereas a material with low density has the same mass spread over a larger volume.

Because density depends on the substance itself — not on the size or shape of the object — it is a property of the material.

In the equation $\rho =\frac{m}{V}$, the Greek letter $\rho $ (rho) represents density. This symbol is standard in CIE Physics.

Representation note — reading the equation: The horizontal bar in $\frac{m}{V}$ means "divided by." The quantity on top (numerator) is divided by the quantity on the bottom (denominator). Subscripts and superscripts in units work the same way: kg m⁻³ means "kilograms per cubic metre," where the negative exponent (⁻³) signals "per" or "divided by."

Common unit conversions for density:

Worked Example

A metal block has a mass of 540 g and a volume of 200 cm³. Calculate its density in kg m⁻³.

Converting mass to SI units:

$$m=\frac{540}{1000}$$

$$m=0.540\text{ kg}$$

Converting volume to SI units:

$$V=\frac{200}{{10}^{6}}$$

$$V=2.00\times {10}^{-4}{\text{ m}}^{3}$$

Equation used — density equation

$$\rho =\frac{m}{V}$$

Substituting:

$$\rho =\frac{0.540}{2.00\times {10}^{-4}}$$

$$\rho =2700{\text{ kg m}}^{-3}$$

MisconceptionStudents often forget to convert both grams to kilograms and cm³ to m³ when a question asks for density in kg m⁻³. Converting only one of the two gives an answer that is wrong by a factor of 1000.

Exam TipCheck the unit the question requests, then convert all given values to match before substituting.

Exam TipDensity is a property of the material, not the object's size — never state that a larger object is automatically denser.

Determining the Density of a Liquid

Measuring the density of a liquid requires finding both the mass and the volume of a known sample of that liquid.

The method uses a measuring cylinder and a balance:

1. Place an empty measuring cylinder on the balance and record its mass, ${m}_{1}$.

2. Pour a known volume of the liquid into the measuring cylinder. Read the volume $V$ from the scale at the bottom of the meniscus [the curved surface of the liquid], keeping the eye level with the liquid surface to avoid parallax error.

3. Record the new mass of the cylinder plus liquid, ${m}_{2}$.

4. Calculate the mass of the liquid alone: $m={m}_{2}-{m}_{1}$.

5. Calculate density using $\rho =\frac{m}{V}$.

Examiner InsightCIE frequently asks why the cylinder is measured empty first. The answer is to find the mass of the liquid alone by subtraction, eliminating the mass of the container.

Exam TipAlways state "mass of liquid = mass of cylinder with liquid minus mass of empty cylinder."

Exam TipThe eye must be level with the bottom of the meniscus to avoid parallax error.

Determining the Density of a Regularly Shaped Solid

For a regularly shaped solid (such as a rectangular block, cylinder, or sphere), the volume is calculated directly from measured dimensions using standard mathematical formulae.

The method uses a ruler or vernier callipers and a balance:

1. Measure the mass of the solid on a balance and record $m$.

2. Measure the necessary dimensions with a ruler (or callipers for small objects):

• Rectangular block: length $l$, width $w$, height $h$ → $V=l\times w\times h$

• Cylinder: radius $r$, height $h$ → $V=\pi {r}^{2}h$

• Sphere: radius $r$ → $V=\frac{4}{3}\pi {r}^{3}$

3. Calculate density using $\rho =\frac{m}{V}$.

To improve accuracy, measure each dimension at several positions and calculate the mean.

Exam TipAlways measure each dimension in at least two places and take the mean to account for irregularities.

Determining the Density of an Irregularly Shaped Solid

An irregularly shaped solid has no simple mathematical formula for volume, so the volume is found by the displacement method instead.

The method uses a measuring cylinder (or eureka can), water, and a balance:

1. Measure the mass of the solid on a balance and record $m$.

2. Pour water into a measuring cylinder and record the initial water level, ${V}_{1}$, reading at the bottom of the meniscus at eye level.

3. Gently lower the solid into the water (it must sink completely). Record the new water level, ${V}_{2}$.

4. Calculate the volume of the solid: $V={V}_{2}-{V}_{1}$.

5. Calculate density using $\rho =\frac{m}{V}$.

If the solid is too large for a measuring cylinder, use a eureka can [a container with a spout at a set level]. Fill the can until water reaches the spout. Lower the solid in and collect the displaced water in a separate measuring cylinder. The volume of the collected water equals the volume of the solid.

MisconceptionStudents sometimes think the displacement method measures mass. It does not — it measures volume only. The mass must still be found separately on a balance.

Exam TipAlways state clearly that mass is measured on a balance and volume is found by displacement.

Exam TipThe solid must be fully submerged — if it floats, this method cannot be used directly.

Floating and Sinking Using Density Data

An object floats in a fluid when its density is less than the density of the fluid. An object sinks when its density is greater than the density of the fluid. If the densities are equal, the object remains at whatever position it is placed — it is said to be neutrally buoyant.

This principle applies because a denser object has a greater weight per unit volume than the fluid it displaces, so the upward buoyant force is not large enough to support it, and it sinks. A less dense object displaces a volume of fluid whose weight exceeds the object's own weight before it is fully submerged, so it floats with part of its volume above the surface.

Extended The same rule applies to immiscible liquids [liquids that do not mix]. When two immiscible liquids are placed together, the less dense liquid floats on top of the denser liquid. For example, if oil (ρ = 800 kg m⁻³) is poured into water (ρ = 1000 kg m⁻³), the oil floats above the water because its density is lower. With three or more immiscible liquids, they arrange themselves in layers in order of increasing density from top to bottom.

Examiner InsightCIE questions on floating often present a table of densities for several objects and a liquid, then ask which objects float. Compare each object's density individually against the fluid density — do not average the object densities together.

Exam TipState the comparison explicitly: "The object's density is less than the liquid's density, so it floats."

Exam TipAlways label the density of both the object and the fluid to justify the floating or sinking conclusion.

PRACTICAL: Determining the Density of a Regularly and Irregularly Shaped Solid

Aim & Principle: This investigation determines the density of solids by measuring their mass and volume, then applying the relationship ρ = m/V.

Variables

• Independent variable (IDV): type of solid (different materials).
• Dependent variable (DV): calculated density, in g cm⁻³ (from measured mass and volume).
• Control variables: same balance used for all mass measurements (to avoid systematic error); same measuring cylinder for displacement measurements; temperature of water kept constant (thermal expansion could change volume readings slightly).

Method

For a regularly shaped solid:

1. Measure the mass of the solid on a top-pan balance and record $m$.

2. Measure the length, width, and height of the block using a ruler (resolution ±0.1 cm). Measure each dimension at three different positions and calculate the mean.

3. Calculate the volume using $V=l\times w\times h$.

4. Calculate density using $\rho =\frac{m}{V}$.

For an irregularly shaped solid:

5. Measure the mass of the irregular solid on the same balance and record $m$.

6. Pour water into a measuring cylinder (resolution ±1 cm³) and record the initial level ${V}_{1}$ at the bottom of the meniscus at eye level.

7. Gently lower the solid into the water until fully submerged. Record the new level ${V}_{2}$.

8. Calculate volume: $V={V}_{2}-{V}_{1}$.

9. Calculate density using $\rho =\frac{m}{V}$.

Key Observation & Explanation: Different materials produce different density values. Metals such as steel or copper give densities significantly higher than water (above 1.0 g cm⁻³), which is consistent with their closely packed atomic structure giving a high mass per unit volume.

Measurement technique

The top-pan balance should be checked for zero error before use — press the tare/zero button with nothing on the pan. When reading the measuring cylinder, position the eye level with the bottom of the meniscus to avoid parallax error. For the regular solid, taking the mean of repeated dimension measurements reduces the effect of surface irregularities.

Recording & Processing:

Solid	Mass / g	Volume / cm³	Density / g cm⁻³
Regular block	—	—	—
Irregular stone	—	—	—

Record each measurement at least once. If time permits, repeat the displacement measurement for the irregular solid and take the mean volume. Check for anomalies — a density value far from the known density of that material suggests a measurement error.

Safety

SafetyAvoid dropping heavy solids into the measuring cylinder, which could crack the glass. Lower the solid gently using a string or tilt the cylinder slightly. Mop up any spilt water immediately to prevent slipping.

Exam TipAnnotate the eye-level reading position and label both water levels clearly.