Volume and density

A small metal part can be heavier than a large wooden one. Are we comparing size or material? To separate these effects, we will first learn to measure volume, then compare the masses of equal volumes.

How many identical small cubes fit inside a rectangular block? Its length and width determine the number in one layer; its height determines how many layers fit. This gives volume V, a measure of occupied space: V = abc, where a, b, c are the three dimensions. Double only the height and volume doubles. Double every dimension and volume becomes eight times as large. If dimensions are in metres, the product is in m³, cubic metres.

A stone has no three dimensions whose product gives its exact volume. But we can let water ‘measure’ its shape. A fully submerged object occupies space previously filled by water, so the graduated cylinder reading rises by the object's volume. Subtract the initial reading from the final reading. The object must not dissolve or absorb water, and no bubbles should cling to it. Open cavities that fill with water are not included in the displaced volume measured this way.

Why is multiplying by 100 not enough when converting cubic metres? Imagine cutting a cube of side 1 m into centimetre cubes. There are 100 along each of the three directions, giving 100·100·100 = 1 000 000 in total. Hence 1 м³ = 1 000 000 см³. One millilitre equals 1 cm³, and 1 л = 1000 см³ = 0,001 м³. A cubic unit includes all three dimensions. Sketching the cube is a useful check on a conversion.

Take aluminium and iron cubes with equal volumes. Their different masses can no longer be explained by size: the material differs. The mass per unit volume of a uniform substance is its density ρ. A greater mass in the same volume means a greater density. The SI unit is kg/m³: the value tells us the mass of one cubic metre. We can also use g/cm³, provided mass and volume are expressed in compatible units.

We do not need a whole cubic metre of material. Equal volumes within a uniform sample have equal masses, so divide the mass of any sample by its volume: ρ = m/V. Cut it in half: both mass and volume halve, leaving the ratio unchanged. Density describes the substance, whereas mass describes the particular object. But compress the same mass into a smaller volume and its density rises. Constancy assumes unchanged conditions and state of matter.

We measure a density and find a similar value in a table. Does that identify the material for certain? No: different materials can have similar densities, and a cavity lowers an object's average density. Table values refer to particular temperatures and states of matter. First match the units and conditions, then use the agreement as evidence for a possible material. Ice and liquid water have different densities even though they are the same substance.

If each cubic metre has a mass of ρ kilograms, what mass is contained in V cubic metres? Adding equal portions gives m = ρV. At the same density, doubling volume doubles mass; half the volume has half the mass. The units agree too: (кг/м³)·м³ = кг. For a composite or hollow object, use its average density or calculate the masses of the parts separately.

We now know the mass to be accommodated and the mass per cubic metre. The required volume is V = m/ρ. For the same mass, a denser substance occupies less volume because each cubic metre holds more mass. For a hollow or composite object, total mass divided by external volume gives average density. This describes the whole chosen object, including its cavities, and need not equal the density of its wall material.

A graduated cylinder reads 40 ml before an insoluble part is fully submerged, and 70 ml afterwards. The part does not absorb water and has no bubbles attached; its mass is 81 g. Find its average density.

The part's volume is the difference between readings: 70 − 40 = 30 мл = 30 см³.

Density is mass divided by volume: 81/30 = 2,7 г/см³.

In SI units this is 2700 kg/m³: 1 г/см³ = 1000 кг/м³. To identify the material, we must also know that the part is uniform and has no cavities.

Average density: 2,7 г/см³ = 2700 кг/м³.

Density lets us compare substances independently of the size of a uniform sample: cutting it leaves m/V unchanged.

The formulas ρ = m/V, m = ρV and V = m/ρ answer different questions about the same relationship between mass and volume.

Displacement measurements and comparisons with tables require the stated conditions; average density matters for hollow objects.

Mass also affects Earth's gravitational pull on an object. Next we will distinguish mass, gravitational force and apparent weight.