Definition 6.1.1. Mass Density.
The mass density of an object, \(\rho\text{,}\) can be found by dividing the object’s mass by its volume:
\begin{equation*}
\rho = \frac{m}{V}
\end{equation*}
Section 6.1 Density
In The Law of Inertia (Newton’s First Law), you learned about mass as a property of an object. In some situations, you might care about density, which describes how mass is distributed in space.
Note 6.1.2. 1D and 2D Density.
The symbol \(\rho\) represents the volume (three-dimensional) mass density. Physics also commonly makes use of surface (two-dimensional) mass density, symbolized by \(\sigma\text{,}\) and linear (one-dimensional) mass density, symbolized by \(\lambda\text{.}\)
Exercises Activities
1. Sensemaking: Units.
Determine the units for mass density \((\rho)\text{.}\)
Answer.
\(\mathrm{kg/m^3}\)
2. Sensemaking: Numbers.
One cup of water has a mass of approximately \(240 \mathrm{~g}\text{.}\) Use this to estimate the density of water. Make sure to convert your final answer to SI units, and comment on whether or not this number makes sense.
Answer.
The density of water is approximately \(1000 \mathrm{~kg/m^3}\) or \(1 \mathrm{~g/mL}\text{.}\)
3. Sensemaking: Covariation.
Density depends on two quantities: mass and volume. Use the covariational reasoning sensemaking strategy to check that the definition for density makes sense given your everyday experience.
4. Calculation.
Suppose you find a rock that breaks into two parts, each of which has the same volume \(V_o\text{.}\) You are able to determine that one part has density \(\rho_1\text{,}\) while the other has density \(\rho_2\text{.}\) Write an equation for the mass of the entire rock.
