Theory Exercises

Density and How to Measure It

Density is a measure of how much mass is packed into a given volume. We use the Greek letter rho (ρ) for density and define it with a simple formula:

\[\rho = \frac{m}{V}\]

Where \(m\) is mass (in kilograms or grams) and \(V\) is volume (in cubic metres, litres or cubic centimetres). Common laboratory units are grams per cubic centimetre (\( \text{g/cm}^3\)) and kilograms per cubic metre (\( \text{kg/m}^3\)).

Important conversions: \(1 \text{ g/cm}^3 = 1000 \text{ kg/m}^3\) and \(1 \text{ cm}^3 = 1 \text{ mL}\).

Measuring mass

Mass is measured using a balance. Use an analytical or digital balance for good precision. Record units (g or kg) and convert if needed before calculations.

Measuring volume

There are two common ways to find the volume of an object:

1. Geometry (regular solids)

If you know the shape, use the appropriate volume formula:

Common volume formulas for geometric shapes

2. Displacement / Archimedes' method (irregular solids)

Submerge the object in water and measure the volume of displaced liquid. A measuring cylinder or overflow can works well. Since 1 mL = 1 cm³, the volume in mL equals cm³.

Archimedes & buoyancy (alternative using a balance)

If you can suspend the object from a balance and weigh it in air and then while submerged in water, you can use the apparent loss of weight to compute the volume:

\[ \text{Apparent loss of mass} = m_{ \text{air}} - m_{ \text{submerged}} = \text{mass of displaced water} = \rho_{ \text{water}} \times V\]

Therefore, if the fluid is water (\(\rho \approx 1 \text{ g/cm}^3\) at standard lab temperatures):

\[V = \frac{m_{ \text{air}} - m_{ \text{submerged}}}{\rho_{ \text{fluid}}}\]

When using grams and water, V in cm³ equals the numerical difference in grams.

Worked Examples

Example 1: A rectangular block measures 10 cm × 5 cm × 2 cm and has mass 200 g. Calculate its density in g/cm³ and kg/m³.
Step 1: Identify the given data
  • Length: \(l = 10 \text{ cm}\)
  • Width: \(w = 5 \text{ cm}\)
  • Height: \(h = 2 \text{ cm}\)
  • Mass: \(m = 200 \text{ g}\)
Step 2: Calculate the volume For a rectangular prism: \(V = l \times w \times h = 10 \times 5 \times 2 = 100 \text{ cm}^3\) Step 3: Calculate density in g/cm³
\[\rho = \frac{m}{V} = \frac{200 \text{ g}}{100 \text{ cm}^3} = 2 \text{ g/cm}^3\]
Step 4: Convert to kg/m³
\[\rho = 2 \text{ g/cm}^3 \times 1000 = 2000 \text{ kg/m}^3\]
Answer: The density is 2 g/cm³ or 2000 kg/m³.
Example 2: An irregular stone has mass 87 g and displaces 35 mL of water. Find its density.
Step 1: Identify the given data
  • Mass: \(m = 87 \text{ g}\)
  • Displaced water volume: \(35 \text{ mL}\)
Step 2: Convert volume units Since \(1 \text{ mL} = 1 \text{ cm}^3\): \(V = 35 \text{ mL} = 35 \text{ cm}^3\) Step 3: Calculate density
\[\rho = \frac{m}{V} = \frac{87 \text{ g}}{35 \text{ cm}^3} \approx 2.486 \text{ g/cm}^3\]
Answer: The density is approximately 2.49 g/cm³.
Example 3: A spherical metal ball has radius 5.0 cm and mass 600 g. Calculate its density.
Step 1: Identify the given data
  • Radius: \(r = 5.0 \text{ cm}\)
  • Mass: \(m = 600 \text{ g}\)
Step 2: Calculate the volume of the sphere Using the sphere volume formula: \(V = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi (5.0)^3 = \frac{4}{3}\pi \times 125 = \frac{500\pi}{3} \approx 523.60 \text{ cm}^3\) Step 3: Calculate density
\[\rho = \frac{m}{V} = \frac{600 \text{ g}}{523.60 \text{ cm}^3} \approx 1.146 \text{ g/cm}^3\]
Answer: The density is approximately 1.15 g/cm³.
Example 4: A cylindrical can has volume 785 cm³ and contains oil with density 0.92 g/cm³. Find the mass of oil and convert the density to kg/m³.
Step 1: Identify the given data
  • Volume: \(V = 785 \text{ cm}^3\)
  • Density: \(\rho = 0.92 \text{ g/cm}^3\)
Step 2: Calculate mass Rearranging \(\rho = \frac{m}{V}\) to solve for mass: \(m = \rho \times V = 0.92 \times 785 = 722.2 \text{ g}\) Step 3: Convert density to kg/m³
\[\rho = 0.92 \text{ g/cm}^3 \times 1000 = 920 \text{ kg/m}^3\]
Answer: The mass of oil is 722.2 g, and the density is 920 kg/m³.

Units & conversions

  • \(1 \text{ cm}^3 = 1 \text{ mL}\)
  • \(1 \text{ m}^3 = 1\,000\,000 \text{ cm}^3\)
  • \(1 \text{ g/cm}^3 = 1000 \text{ kg/m}^3\)
  • To convert \( \text{g/cm}^3 \rightarrow \text{kg/m}^3\): multiply by 1000
  • To convert \( \text{kg/m}^3 \rightarrow \text{g/cm}^3\): divide by 1000

Common pitfalls

  • Always use consistent units: convert mass and volume to compatible units before dividing.
  • Temperature can change fluid density; if you're using water for displacement, note approximate ρ_water = 1.00 g/cm³ but mention temperature if high precision is required.
  • Air bubbles when using displacement will make the measured volume too small — ensure the object is fully submerged and no trapped air remains.

Quick lab procedure (Archimedes displacement)

  1. Fill a measuring cylinder to a convenient mark and record the volume (V1).
  2. Carefully lower the object into the cylinder and record the new volume (V2).
  3. Displaced volume V = V2 − V1 (in mL = cm³).
  4. Measure mass m (g) on a balance. Density ρ = m / V (g/cm³).

Practice the conversions and methods with the short quiz on the right. Apply the formula ρ = m / V and check units at every step.