Density of irregular solids: the core measurement idea
The density of irregular solids is determined from the definition of density: mass divided by volume. Mass is measured directly on a balance, while the volume of an irregular shape is commonly obtained by the water displacement method using a graduated cylinder.
\[ \rho = \frac{m}{V} \]Water displacement method (graduated cylinder)
For an irregular solid that fits in a cylinder and does not dissolve, the displaced water volume equals the solid’s volume. If the initial water reading is \(V_1\) and the final reading after fully submerging the object is \(V_2\), then:
\[ V_{\text{solid}} = V_2 - V_1 \]Experimental procedure (rigorous and practical):
- Measure the mass \(m\) of the dry irregular solid using a balance (record units, usually \(\mathrm{g}\)).
- Add water to a graduated cylinder and record the initial volume reading \(V_1\) at the bottom of the meniscus.
- Carefully lower the irregular solid into the water so it is completely submerged (avoid splashing; remove trapped bubbles).
- Record the final volume reading \(V_2\) at the bottom of the meniscus.
- Compute the solid’s volume \(V_{\text{solid}} = V_2 - V_1\), then compute density \(\rho = \frac{m}{V_{\text{solid}}}\).
Worked example (with units and conversions)
Suppose an irregular rock has mass \(m = 86.4\ \mathrm{g}\). A graduated cylinder reads \(V_1 = 42.0\ \mathrm{mL}\) before the rock is added and \(V_2 = 74.0\ \mathrm{mL}\) after full submersion.
Step 1. Find the volume from displacement.
\[ \begin{aligned} V_{\text{solid}} &= V_2 - V_1 \\ &= 74.0\ \mathrm{mL} - 42.0\ \mathrm{mL} \\ &= 32.0\ \mathrm{mL} \end{aligned} \]Step 2. Compute the density.
\[ \begin{aligned} \rho &= \frac{m}{V_{\text{solid}}} \\ &= \frac{86.4\ \mathrm{g}}{32.0\ \mathrm{mL}} \\ &= 2.70\ \mathrm{g/mL} \end{aligned} \]Step 3. Convert \(\mathrm{g/mL}\) to \(\mathrm{kg/m^3}\) if needed.
\[ \begin{aligned} 1\ \mathrm{g/mL} &= 1000\ \mathrm{kg/m^3} \\ \rho &= 2.70 \times 1000\ \mathrm{kg/m^3} \\ &= 2700\ \mathrm{kg/m^3} \end{aligned} \]Recording results and uncertainty (simple, lab-appropriate)
The dominant sources of uncertainty for the density of irregular solids are typically the cylinder reading (meniscus and parallax) and trapped air bubbles. If the volume reading uncertainty is \(\pm 0.5\ \mathrm{mL}\) for each reading, the displacement uncertainty is often approximated by:
\[ \Delta V_{\text{solid}} \approx \Delta V_2 + \Delta V_1 \]For the example, \(\Delta V_{\text{solid}} \approx 0.5\ \mathrm{mL} + 0.5\ \mathrm{mL} = 1.0\ \mathrm{mL}\), so \(V_{\text{solid}} = 32.0 \pm 1.0\ \mathrm{mL}\). If the mass uncertainty is small compared with the volume uncertainty, the density uncertainty is mainly controlled by \(V_{\text{solid}}\).
Common issues when measuring density of irregular solids:
| Issue | What goes wrong | Fix |
|---|---|---|
| Air bubbles on the surface | Volume appears larger, density is underestimated | Tap gently, wet the object first, rotate it to release bubbles |
| Not fully submerged | Volume appears smaller, density is overestimated | Use a sinker thread, ensure complete submersion without touching walls |
| Reading the top of the meniscus | Systematic volume error | Read the bottom of the meniscus at eye level |
| Object dissolves or absorbs water | Displacement no longer equals the object volume | Use another fluid or a geometric method if possible |
| Object floats (density less than water) | Cannot measure \(V_2\) by free submersion | Use a sinker and subtract the sinker’s displacement |
Special case: floating irregular solids (sinker correction)
If the irregular solid floats, a sinker (a dense weight) can be used to force submersion. Record cylinder readings for the sinker alone and for sinker + solid. If \(V_a\) is the reading with sinker alone and \(V_b\) is the reading with sinker plus the floating object held submerged, then:
\[ V_{\text{solid}} = V_b - V_a \]The density of irregular solids is then computed in the same way using \(\rho = \frac{m}{V_{\text{solid}}}\), where \(m\) is the mass of the solid alone.