Measurement

NZ Level 6 (NZC) Level 1 (NCEA)

Use density to determine volume or mass

Lesson

The density of a substance is its mass for each unit of volume.

The Greek letter $\rho$`ρ` is often used for density. Given a quantity of a substance, we write

$\rho=\frac{m}{V}$`ρ`=`m``V`

where $m$`m` is the mass of the sample and $V$`V` is its volume.

If the mass is measured in kilograms and volume in cubic metres, the density measure is in the units $kg.m^{-3}$`k``g`.`m`−3, (kilograms per cubic metre).

If the density is known and either the mass or the volume, then the remaining unknown quantity can be calculated. That is, if $\rho=\frac{m}{V}$`ρ`=`m``V` is given, then $m=\rho V$`m`=`ρ``V` and $V=\frac{m}{\rho}$`V`=`m``ρ`.

The density of most substances varies with temperature. This is because the volume of a given mass changes with temperature. Water, for example, is at its most dense when its temperature is $4^\circ$4°C.

When a mass of a solid substance is immersed in a fluid and weighed while immersed, it is found to have lost weight equal to the mass of the fluid displaced. This is the principle of buoyancy supposed to have been discovered by Archimedes.

At $4^\circ$4°C water has a density of $1.0$1.0 g/cm$^3$3.

What is the mass of $800$800 cm$^3$3 of water at this temperature?

Using $\rho=\frac{m}{V}$`ρ`=`m``V`, we have $1=\frac{m}{800}$1=`m`800. Therefore the mass is $800$800 g.

An instrument consisting of a sealed glass tube with weights at the lower end is used to measure the density of the wort in a brewing process, which in turn depends on the concentration of sugar in the brew.

Suppose the instrument weighs $250$250 g and its total volume is $275$275 cm$^3$3. If the instrument were to be immersed in water at its maximum density of $1.0$1.0 g.cm$^{-3}$−3, what fraction of the tube would be below the surface of the water?

The tube sinks until $250$250 g of water has been displaced. So, a volume $V=\frac{250}{1}$`V`=2501 cm$^3$3 is displaced and this is $\frac{250}{275}=\frac{6}{7}$250275=67 of the tube length.

A calibration mark could be made on the tube to indicate the depth to which it sinks when immersed in a liquid with density $1.0$1.0.

When used to test the density of a brewing wort, the tube would not sink down to the mark if the density of the wort were greater than $1$1. In effect, this procedure compares the density of a wort with the density of water and in this situation, the result is called the specific gravity, which is the ratio of the density being tested to that of water.

Apply the relationships between units in the metric system, including the units for measuring different attributes and derived measures

Apply measurement in solving problems