Physics · Mechanics · Grade 7

Mass and density

Identical cubes: why is a gold cube about 240 times heavier than a cotton one?

The situation

At the market, try lifting a sack of cotton and a bucket of water: though the size is the same, the weight is very different. In the laboratory stand identical cubes of 1 dm³ volume: cotton, wood, ice, water, aluminium, iron, copper and gold. Which is how heavy, and how can we tell what a piece of unknown metal is made of?

Topic

Mass and density. Cotton, iron and water: the same volume — different masses

Key ideas

Density ρ = m/V
Measuring volume with a measuring cylinder
Identifying a substance by its density

Formulas

ρ = \frac{m}{V}

Density is mass per unit volume. A property for recognising a substance.

V = V_{2} - V_{1}

The volume of an irregular object: it equals the amount by which the water level in the measuring cylinder rises.

1 t = 1000 kg, 1 kg = 1000 g, 1 \frac{g}{{sm}^{3}} = 1000 \frac{kg}{m^{3}}

Units of mass and density. Volume: 1 mL = 1 cm³, 1 dm³ = 1 litre = 1000 cm³.

V = \frac{Δ m}{ρ_{w}}, ρ = ρ_{w} \frac{m}{Δ m}

Hydrostatic weighing: an object immersed in water makes the balance read an extra $Δ m = F_{A} / g$ , corresponding to the Archimedes force.

ε_{ρ} = ε_{m} + ε_{V} = \frac{Δ m}{m} + \frac{Δ V}{V}

The relative error of a quotient is the sum of the relative errors.

Theory

What is mass?

Mass is a measure of two properties of a body: its inertia (a heavy cart accelerates more slowly when pushed with the same force — “Newton's laws”) and its attraction to other bodies (a balance measures exactly the gravitational force and shows the mass). Experiments show that both methods give the same number. The unit of mass is the kilogram; $1$ t $= 1000$ kg, $1$ kg $= 1000$ g, $1$ g $= 1000$ mg. Mass does not depend on where the body is: on the Moon too a gold cube is 19.3 kg, only its weight is smaller.

Equal volume — different mass

All eight cubes have the same volume — 1 dm³: the volume of a regularly shaped body is calculated from its dimensions (cube $V = a^{3}$ , cuboid $V = a b c$ , cylinder $V = π r^{2} h$ ) — on the ruler beside the balance the edge of the cube is $a = 10$ cm. Yet the balance readings differ from 80 g to 19 kg. The ratio of mass to volume is called density: $ρ = m / V$ . It depends not on the size of the body but on the substance — a small gold coin and a large gold ingot have the same density. The table values are for room temperature; textbooks give them rounded (iron ≈ 7800, copper ≈ 8900 kg/m³).

Cotton and water

There is a lot of air between cotton fibres, so the density of loose cotton is very small — about 80 kg/m³ (it varies with how tightly it is packed). The water and cotton cubes are in thin containers: the mass of the container has been tared in advance, and the balance shows only the contents. Ice is lighter than water (917 < 1000), so it floats on water.

Measuring volume with a measuring cylinder

The shape of a stone or a ring is irregular — its volume cannot be found with a ruler. We lower it into the water in a measuring cylinder: the water level rises by the volume of the object, $V = V_{2} - V_{1}$ . Read the scale at the bottom of the meniscus, keeping your eye level with the surface. This measuring cylinder has a scale division of 2 mL and a reading error of ±1 mL.

Error and small objects

Density is calculated from two measurements, and their relative errors add. For a 45 cm³ stone the volume error is about $2 / 45 \approx 4$ %, but for a ring of 0.4 cm³ the level does not even rise by one division — the result is meaningless. A more accurate method for small objects is hydrostatic weighing: a beaker of water is tared on the balance, the object is lowered fully into the water on a thread, and the balance reads $Δ m = F_{A} / g = ρ_{w} V$ , corresponding to the Archimedes force. In the classical version the object hangs from a thread beneath the balance and is weighed in air ( $m$ ) and in water ( $m - Δ m$ ) — the result is the same: $ρ = ρ_{w} m / Δ m$ . With a balance accurate to 0.01 g the volume of a ring is found to within a few per cent (for the Archimedes force, see “Why doesn't a ship sink?”). According to legend, Archimedes checked that the king's crown was pure gold by a similar route — through the volume of water it displaced.

Did you know?

The densest chemical element is osmium: about 22.6 g/cm³. It is even heavier than gold: 1 dm³ of osmium has a mass of more than 22 kg.
A 1 dm³ gold cube has a mass of 19.3 kg — equal to the mass of more than 19 litres of water.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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