An Introduction to Density

Density : The ratio between Mass and Volume

We often sometimes use terms like 'as heavy as lead' and 'as light as a cotton'. We often ask questions like Is lead heavier than cotton? or Is iron heavier than wood? These questions and terms often do not have one obvious meaning.

It is clear that a large log of wood can be considerably heavier than an iron hair pin and a grain of lead can be lighter than a mountain of cotton.
When such comparisons are made people often have density of material of the body in mind instead of mass of the body. Density of any material is determined by the mass of atoms of that material and how close or far these atoms are spaced in the material under consideration.
Density can be linked to the lightness and heaviness of different materials of the same size. For example if we take a iron cube and wooden cube of same dimensions then obviously iron cube would be heavier than wooden cube.
This is because atoms in iron are more tightly packed. They have greater mass in smaller volume than wood. So iron is denser than wood. Density of any material tells us about how much mass occupies a given space. Conceptually speaking density is a measurement that compares the amount of matter an object has to its volume.
The mass of a unit volume of a body is called its density.
It is clear that density of an iron block is same as the density of an iron hair clip.
Density is an important property of any material. Density is defined as mass per unit volume. We use Greek letter \( \rho \) (rho) for density. if m is the mass of homogeneous material of volume v then. The density \( \rho \) would be
\[ \bbox[aqua,5px,border:2px solid red]{ \mathbf{ \rho =\frac {m}{V}=\frac {mass}{volume} } } \, \color{navy}{\textbf{This is an important formula and is used frequently while solving various problems in physics}} \] Two objects of the same material have the same density even if they have different masses and different volumes. This happens because the ratio of mass to volume is the same for both objects. Density is a positive scalar quantity. As liquids are incompressible, their density remains constant at all pressures. The density of gases varies largely with pressure.

How to find density

We already know that density of any object is its mass per unit volume. In this section we will learn how we can find density using a worked example. We also studied about the units of density in previous section.
It is \(Kg/m^3 \) in SI units and \(gm/{cm}^3\) in cgs units. Now sometimes in problems measurement of quantities are given in different units. You must convert the given quantities in your preferred system of units.
Before going any further let us look at us look at conversion between different units of volume.
(a) \(m^3\) to \({cm}^3\) \begin{align*} \text{1m}^3&=\text{(100cm)}^3=\text{(}10^2\text{)}^3\text{cm}^3\\ &=10^6\text{cm}^3 \end{align*} (b) \({m}^3\) to \(Liter\) \[1m^3=10^3\text{L} \] (c) Liter to cubic cm \begin{align*} \text{1L}=10^3\text{cm}^3\\ \end{align*} Question : Calculate the density in kg per cubic meter, of a body that weighs \(500 gm\) and has a volume of \(52cm^3\).
Solution : Here in the question both volume and mass of the substance are given in cgs units but answer is asked in SI units. We can either convert the units of mass and volume before calculating density or we can first calculate density and then convert it into \(Kg/{cm}^3\)
Now from the formula of density \[d=\frac{m}{v}\] given that \(m=500gm\) and \(v=52{cm}^3\) density, \begin{align*} d&=\frac{500gm}{52{cm}^3}\\ &=9.6gm/{cm}^3 \end{align*} Now, \begin{align*} 1gm=10^{-3}kg\\ 1{cm}^3=10^{-6}{m}^3 \end{align*} density in \(kg/m^3\) \begin{align*} &=\frac{9.6 \times 10^{-3} kg} {10^{-6}m^3}\\ &=9.6 \times 10^3 kg/m^3 \end{align*}

density units

SI unit of density is \(kg/m^3\), CGS unit of density is \(g/cm^3\). We use these units in case of measuring densities of solids. However in case of liquids and gases we use \(g/ml\) for liquids and \(g/L\) for gases.
Dimensional formula of density is \([ML^{-3}]\)

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