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

Density (\$\rho\$) is the mass of a substance per unit volume. It is a useful property for identifying materials and for solving many physics problems.

Objective

Describe how to determine the density of:

• a liquid,
• a regularly shaped solid,
• an irregularly shaped solid that sinks in a liquid (using volume displacement),

including the calculations required.

Key Formula

The fundamental relationship is

\$\rho = \frac{m}{V}\$

where

• \$\rho\$ = density (kg m⁻³ or g cm⁻³),
• \$m\$ = mass (kg or g),
• \$V\$ = volume (m³ or cm³).

1. Determining the Density of a Liquid

1. Measure a known volume of the liquid using a graduated cylinder or measuring jug. Record the volume \$V\$ in cm³ (or mL).
2. Weigh the container empty, then weigh it again with the liquid. The difference gives the mass \$m\$ of the liquid in grams.
3. Calculate density using \$\rho = m/V\$.

Example

Mass of liquid = 125 g, volume measured = 100 cm³.

\$\rho = \frac{125\ \text{g}}{100\ \text{cm}^3}=1.25\ \text{g cm}^{-3}\$

2. Determining the Density of a Regularly Shaped Solid

1. Measure the dimensions of the solid with a ruler or vernier calipers (to the nearest 0.1 mm).
2. Calculate the volume \$V\$ using the appropriate geometric formula (e.g., \$V = l \times w \times h\$ for a rectangular block, \$V = \frac{4}{3}\pi r^{3}\$ for a sphere).
3. Weigh the solid on a balance to obtain its mass \$m\$.
4. Use \$\rho = m/V\$ to find the density.

Example – Rectangular Block

Volume:

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

Density:

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

3. Determining the Density of an Irregularly Shaped Solid that Sinks (Displacement Method)

1. Weigh the solid to obtain its mass \$m\$.
2. Fill a graduated cylinder with a known volume of water \$V_{i}\$ (record the reading).
3. Gently submerge the solid completely without splashing. Record the new water level \$V_{f}\$.
4. The displaced volume \$V{\text{disp}}\$ equals \$V{f} - V_{i}\$, which is the volume of the solid.
5. Calculate density using \$\rho = m / V_{\text{disp}}\$.

Example

• Mass of irregular solid = 45 g.
• Initial water volume \$V_{i}\$ = 80.0 cm³.
• Final water volume \$V_{f}\$ = 115.0 cm³.

Displaced volume:

\$V{\text{disp}} = V{f} - V_{i} = 115.0 - 80.0 = 35.0\ \text{cm}^3\$

Density:

\$\rho = \frac{45\ \text{g}}{35.0\ \text{cm}^3}=1.29\ \text{g cm}^{-3}\$

Summary Table – Steps and Formulas

Common Pitfalls & Tips

• Always use the same unit system for mass and volume (g cm⁻³ or kg m⁻³).
• When measuring liquid volume, read the meniscus at eye level to avoid parallax error.
• For irregular solids that float, attach a small sinker so the object fully submerges; then subtract the sinker’s volume.
• Record all measurements with their uncertainties; propagate them if required for higher‑level work.