Describe how to determine the density of a liquid, of a regularly shaped solid and of an irregularly shaped solid which sinks in a liquid (volume by displacement), including appropriate calculations

1.4 Density

Learning objectives

• Define density and write the fundamental formula.
• Determine the density of a liquid.
• Determine the density of a regularly shaped solid using geometric formulas.
• Determine the density of an irregularly shaped solid that sinks by volume‑displacement.
• Predict whether an object will float or sink in a given fluid.
• Convert units, apply significant‑figure rules and propagate uncertainties correctly.

Key formula

\[

\rho = \frac{m}{V}

\]

• ρ = density (kg m⁻³ or g cm⁻³)
• m = mass (kg or g)
• V = volume (m³ or cm³)

Quick unit‑conversion box

Keep the same unit system throughout a calculation.

1. Determining the density of a liquid

1. Measure a known volume of the liquid with a graduated cylinder or measuring jug. Record the volume V (cm³ or mL). Read the bottom of the meniscus at eye level.
2. Weigh the empty container, then weigh it again with the liquid. The difference gives the mass m (g).
3. Calculate density using ρ = m / V.
4. Report the result with the correct number of significant figures (usually the same as the least‑precise measurement).
5. State the temperature, because liquid density varies with temperature.

Example – Water at 20 °C

• Mass of water = 125.0 g (± 0.1 g)
• Volume measured = 100.0 cm³ (± 0.1 cm³)

\[

\rho = \frac{125.0\ \text{g}}{100.0\ \text{cm}^3}=1.25\ \text{g cm}^{-3}

\]

Result to 3 sf (limited by the volume reading).

2. Determining the density of a regularly shaped solid

1. Measure all required dimensions with a ruler or vernier calipers (to the nearest 0.1 mm). Record uncertainties.
2. Calculate the volume V using the appropriate geometric formula (e.g. \(V=lwh\) for a rectangular block, \(V=\frac{4}{3}\pi r^{3}\) for a sphere).
3. Weigh the solid to obtain its mass m (g). Record the balance uncertainty.
4. Compute density with ρ = m / V, propagate uncertainties if required, and give the answer to the correct number of significant figures.

Example – Rectangular block

\[

V = l \times w \times h = 5.0 \times 3.0 \times 2.0 = 30.0\ \text{cm}^3

\]

\[

\rho = \frac{30.0\ \text{g}}{30.0\ \text{cm}^3}=1.00\ \text{g cm}^{-3}

\]

Both mass and volume are given to 3 sf, so the density is reported to 3 sf.

3. Determining the density of an irregular solid that sinks (displacement method)

1. Weigh the solid to obtain its mass m (g).
2. Fill a graduated cylinder with enough water to completely submerge the solid. Record the initial water volume V_i (cm³).
3. Gently lower the solid into the water, ensuring it is fully submerged and no water splashes out. Record the final water level V_f (cm³).
4. Displaced volume \(V{\text{disp}} = V{f} - V_{i}\). This equals the volume of the solid.
5. Calculate density \(\rho = \dfrac{m}{V_{\text{disp}}}\). Include uncertainties from the volume readings (typically ± 0.1 cm³).

Example – Irregular solid

• Mass = 45.0 g (± 0.1 g)
• Initial water volume \(V_{i}\) = 80.0 cm³ (± 0.1 cm³)
• Final water volume \(V_{f}\) = 115.0 cm³ (± 0.1 cm³)

\[

V_{\text{disp}} = 115.0 - 80.0 = 35.0\ \text{cm}^3

\]

\[

\rho = \frac{45.0\ \text{g}}{35.0\ \text{cm}^3}=1.29\ \text{g cm}^{-3}

\]

Result to 3 sf (limited by the volume measurement).

4. Determining the density of an irregular solid that floats (overflow‑can method)

1. Weigh the solid to obtain its mass m.
2. Fill an overflow can (or a graduated cylinder with a funnel) with water until it just begins to overflow. Record this initial volume V_i.
3. Place the floating solid gently on the water surface. The water level rises; collect the overflow water in a measuring container and record the volume collected, V_{\text{out}}\. This is the volume of water displaced, which equals the volume of the solid.
4. Calculate density \(\rho = m / V_{\text{out}}\).

Numeric example

• Mass of the wooden piece = 12.0 g (± 0.1 g)
• Overflow water collected = 15.0 cm³ (± 0.1 cm³)

\[

\rho = \frac{12.0\ \text{g}}{15.0\ \text{cm}^3}=0.80\ \text{g cm}^{-3}

\]

Since ρ = 0.80 g cm⁻³ < ρ_water = 1.00 g cm⁻³, the wood floats – the calculation confirms the observation.

5. Using density to predict flotation

An object will:

• Float if \(\rho{\text{object}} < \rho{\text{fluid}}\).
• Sink if \(\rho{\text{object}} > \rho{\text{fluid}}\).
• Remain suspended (neutral buoyancy) if the densities are equal.

Worked‑out decision

• Measured density of a metal sphere: \(\rho_{\text{sphere}} = 7.85\ \text{g cm}^{-3}\).
• Density of water at 20 °C: \(\rho_{\text{water}} = 1.00\ \text{g cm}^{-3}\).

Since \(7.85 > 1.00\), the sphere will sink in water.

6. Optional extensions (higher‑level)

• Relative density of liquids: If \(\rho{1} < \rho{2}\), liquid 1 will float on liquid 2 (e.g., oil on water).
• Density of an irregular solid that floats (alternative to overflow can): Attach a small sinker of known mass \(m{s}\) and volume \(V{s}\) to the object, submerge the combined system, measure the displaced volume, then subtract \(V_{s}\) to obtain the volume of the original object.

7. Summary table – methods, key measurements & formulas

8. Common pitfalls & tips

• Consistent units: Convert all measurements to the same system before using \(\rho = m/V\).
• Significant figures: Final answer should have the same number of significant figures as the least‑precise measurement (usually the volume).
• Uncertainty: Record the uncertainty of each reading (e.g., ± 0.1 cm³) and propagate it to the density if required.
• Meniscus reading: Always read the bottom of the meniscus at eye level.
• Floating objects: Use a sinker or the overflow‑can method to obtain the displaced volume.
• Temperature effect: State the temperature at which liquid measurements are made.