Theory — Density, specific gravity, and buoyancy
Density measures how much mass is packed into a given volume. It is one of the most useful material properties in mechanics,
fluids, and basic engineering because it connects the size of an object to how heavy that object is. The standard formula is
very simple:
Density formula.
\[
\begin{aligned}
\rho &= \frac{m}{V}
\end{aligned}
\]
Here \(m\) is the mass and \(V\) is the volume. In SI units, mass is measured in kilograms and volume is measured in cubic
meters, so density is measured in kilograms per cubic meter \((\mathrm{kg/m^3})\). In laboratory and chemistry settings,
another very common unit is grams per cubic centimeter \((\mathrm{g/cm^3})\). These are closely related:
Common density unit conversion.
\[
\begin{aligned}
1\ \mathrm{g/cm^3} &= 1000\ \mathrm{kg/m^3}
\end{aligned}
\]
This conversion is important because water has a density close to \(1.0\ \mathrm{g/cm^3}\), which makes that unit especially
convenient for comparing materials to water. A material with a density smaller than water usually floats. A material with a
density larger than water usually sinks.
Specific gravity
Specific gravity is a dimensionless ratio that compares the density of a material to the density of water:
Specific gravity formula.
\[
\begin{aligned}
SG &= \frac{\rho}{\rho_{\text{water}}}
\end{aligned}
\]
Because it is a ratio of two densities, specific gravity has no unit. If the reference is water at about \(4^\circ\mathrm{C}\),
then \(\rho_{\text{water}} \approx 1000\ \mathrm{kg/m^3} \approx 1.0\ \mathrm{g/cm^3}\). That means a material with
\(SG = 2.5\) is two and a half times as dense as water, while a material with \(SG = 0.75\) is only three quarters as dense
as water.
Buoyancy and the float/sink idea
The connection between density and floating comes from Archimedes’ principle. A fluid pushes upward on an immersed object with
a buoyant force equal to the weight of the displaced fluid. If the object is less dense than the fluid, it can displace enough
fluid before becoming fully submerged, so it floats. If it is more dense than the fluid, then even full submersion does not
provide enough buoyant force, and it sinks.
Buoyant force.
\[
\begin{aligned}
F_b &= \rho_{\text{fluid}} g V_{\text{displaced}}
\end{aligned}
\]
For floating in water, the quick rule is:
Float or sink rule.
\[
\begin{aligned}
SG < 1 &\Rightarrow \text{usually floats} \\
SG \approx 1 &\Rightarrow \text{nearly neutral} \\
SG > 1 &\Rightarrow \text{usually sinks}
\end{aligned}
\]
If an object floats, the fraction of its volume that is submerged is approximately equal to its specific gravity when the fluid
is water. So an object with \(SG = 0.75\) will have about \(75\%\) of its volume below the water line.
Worked example
Use the sample values \(m = 250\ \mathrm{g}\) and \(V = 100\ \mathrm{cm^3}\). First apply the density formula directly in
the same unit system:
Direct density calculation.
\[
\begin{aligned}
\rho &= \frac{250\ \mathrm{g}}{100\ \mathrm{cm^3}} \\
&= 2.5\ \mathrm{g/cm^3}
\end{aligned}
\]
To find specific gravity, compare with water:
Specific gravity for the sample.
\[
\begin{aligned}
SG &= \frac{2.5}{1.0} \\
&= 2.5
\end{aligned}
\]
Since the specific gravity is greater than 1, the object is denser than water and is predicted to sink.
Why unit conversion matters
Density can be calculated in any consistent unit system, but the mass and volume must match. If mass is entered in grams and
volume in liters, for example, it is often safer to convert one or both quantities first. The calculator therefore converts the
input values to SI internally, computes the density in \(\mathrm{kg/m^3}\), and then converts to the requested display unit.
This avoids hidden mistakes from mixing incompatible units.
Typical reference values are helpful for interpretation. Ice has a density below \(1.0\ \mathrm{g/cm^3}\), which is why it floats.
Aluminum is around \(2.7\ \mathrm{g/cm^3}\), glass is around \(2.5\ \mathrm{g/cm^3}\), and lead is above
\(11\ \mathrm{g/cm^3}\). Those quick comparisons help identify whether an answer is reasonable.
Density and specific gravity are therefore not just bookkeeping quantities. They help identify unknown materials, compare
substances, predict buoyancy, and check whether a mechanics or fluids calculation is in the right range.