If \(5 \mathrm{~kg}\) of glycerine occupies \(0.004 \mathrm{~m}^{3} .\) Determine the density of glycerine in \(\mathrm{g} \mathrm{cm}^{-3}\) and \(\mathrm{kg} \mathrm{m}^{-3}\).

Understanding the relationship between mass and volume is vital when we begin to talk about density. Mass refers to how much matter is contained in an object and is typically measured in kilograms (kg) or grams (g). Volume, on the other hand, measures the amount of space an object occupies and can be expressed in cubic meters (m³) or cubic centimeters (cm³).

When determining density, it's crucial to have accurate measurements for both mass and volume since any error could lead to incorrect calculations. To measure mass, one might use a scale, while volume can be measured with graduated cylinders or calculated based on geometric formulas for regular-shaped objects.

For instance, the given exercise states that 5 kg of glycerine occupies 0.004 m³. With these values, students can clearly visualize that the glycerine's mass is concentrated within a defined space, leading us to an understanding of its density.

The density formula is a key concept in physics and chemistry and is expressed as Density = Mass / Volume. This formula allows us to calculate how compacted or concentrated the matter within an object is. A high density means that there is a lot of mass in a given volume, while low density indicates the opposite.

Using the density formula, when we take the mass of glycerine which is 5 kg and divide it by its volume of 0.004 m³, we find the density to be 1250 kg/m³. Here, we see that glycerine is quite dense, packing a significant mass into a relatively small space. This kind of calculation is used in everyday applications, such as designing objects to float or sink and understanding the properties of various materials.

In science and mathematics, unit conversion is a necessary process when working with different measuring systems. It's not uncommon to have to convert mass from grams to kilograms, or volume from cubic centimeters to cubic meters, for uniformity in calculations or to adhere to specific scientific standards.

For density calculations, understanding how to convert units is essential. In our exercise, we first calculated the density in kg/m³, and to convert this to g/cm³, we utilized conversion factors: 1 kg equals 1000 g, and 1 m³ equals 1,000,000 cm³. This conversion led to the density of glycerine being expressed as 1.25 g/cm³.