Density Revision Notes

Density is a physical property that describes how much mass is packed into a given volume of a substance. It tells us whether a material is heavy or light for its size. For example, a block of metal and a block of wood may have the same size, but the metal block usually has more mass, so it has a higher density.

• Density measures mass per unit volume.
• It is a characteristic property of materials.

Density (symbol ρ, pronounced 'rho') is calculated by dividing the mass (m) of an object by its volume (V). This is expressed as:

ρ = m ÷ V. Rearranging this formula gives: m = ρ × V and V = m ÷ ρ.

These formulas allow you to calculate any one of the three quantities if the other two are known.

• Density formula: ρ = m / V
• Mass formula: m = ρ × V
• Volume formula: V = m / ρ

Mass is measured in kilograms (kg) or grams (g), and volume in cubic metres (m³) or cubic centimetres (cm³). Density units depend on these and are typically kilograms per cubic metre (kg/m³) or grams per cubic centimetre (g/cm³).

For example, water has a density of about 1000 kg/m³ or 1 g/cm³. When calculating, ensure that mass and volume units are compatible to get the correct density units.

• Mass units: kg, g
• Volume units: m³, cm³
• Density units: kg/m³, g/cm³

Suppose you have a block of aluminium with a volume of 0.002 m³. The density of aluminium is approximately 2700 kg/m³. To find the mass, use the formula m = ρ × V.

Substitute the values: m = 2700 kg/m³ × 0.002 m³ = 5.4 kg.

Therefore, the mass of the aluminium block is 5.4 kilograms.

• Identify known values: ρ = 2700 kg/m³, V = 0.002 m³
• Calculate mass: m = 2700 × 0.002 = 5.4 kg

To measure density practically, you need to measure mass and volume accurately. Mass is measured using a balance. Volume measurement depends on the object's shape:

For regular shapes, use geometric formulas (e.g., length × width × height for a cube).

For irregular shapes, use water displacement in a graduated cylinder to find volume by the change in water level after submerging the object.

• Mass measured with a balance
• Volume measured geometrically or by displacement
• Calculate density using ρ = m / V

Density is defined as mass divided by volume (ρ = m/V). To measure the density of a liquid, you need to determine its mass and volume accurately.

First, use a clean, dry measuring cylinder to measure a known volume of the liquid, for example 50 cm³. Record this volume carefully, reading the bottom of the meniscus at eye level to avoid parallax errors.

Next, weigh an empty container (such as a beaker or a small flask) using a balance and record its mass. Then pour the measured volume of liquid into the container and weigh it again. The difference between the two masses gives the mass of the liquid alone (mass of container with liquid − mass of empty container). Alternatively, you can weigh the liquid directly if the measuring cylinder is on the balance and zeroed before pouring the liquid in, but this requires a balance with a tare function or careful subtraction of the cylinder’s mass, which is less precise due to the cylinder’s irregular shape and mass distribution.

• Use a graduated cylinder for volume measurement.
• Use a balance for mass measurement.
• Read volume at the meniscus to avoid parallax errors.
• Calculate mass by difference if using a container.

For solids with regular shapes (such as cubes, spheres, cylinders), volume can be calculated using geometric formulas based on measured dimensions.

First, measure the mass of the solid using a balance. Record this value in grams (g) or kilograms (kg).

Next, measure the dimensions needed to calculate volume. For example, for a cube measure the length of one side (l), for a cylinder measure the radius (r) and height (h), and for a sphere measure the diameter (d). Use a ruler or vernier callipers for precise measurements, recording values in centimetres (cm) or metres (m). Calculate the volume using the appropriate formula: cube: V = l³; cylinder: V = πr²h; sphere: V = (4/3)πr³ (where r = d/2). Ensure units are consistent, converting to cubic centimetres (cm³) or cubic metres (m³) as needed.

• Measure mass with a balance.
• Measure dimensions with a ruler or callipers.
• Calculate volume using geometric formulas.
• Ensure consistent units for volume and mass.

Irregular solids do not have simple geometric shapes, so their volume cannot be calculated directly. Instead, volume displacement is used.

First, measure the mass of the irregular solid using a balance.

Next, fill a measuring cylinder with a known volume of water and record this initial volume (V₁). Carefully submerge the solid completely in the water without spilling any water out of the cylinder. Record the new volume (V₂). The volume of the solid is the difference between these two readings: V = V₂ − V₁. This method works because the solid displaces an amount of water equal to its own volume.

• Measure mass of the solid with a balance.
• Record initial water volume in a measuring cylinder.
• Submerge solid fully and record new water volume.
• Calculate volume by subtracting initial from final volume.

Once mass and volume are known, calculate density using the formula:

ρ = m / V, where ρ is density, m is mass, and V is volume.

Ensure that mass and volume are in compatible units. For example, if mass is in grams and volume in cubic centimetres, density will be in grams per cubic centimetre (g/cm³). If mass is in kilograms and volume in cubic metres, density will be in kilograms per cubic metre (kg/m³). Always include units in your final answer and round appropriately based on measurement precision.

• Use ρ = m / V to calculate density.
• Match units of mass and volume for correct density units.
• Include units in final answers.
• Round results according to measurement precision.

Accuracy in measuring density depends on precise measurement of mass and volume. Use a balance that is calibrated and sensitive enough for the sample mass. For volume, use measuring cylinders or rulers with fine graduations.

Repeat measurements several times and calculate an average to reduce random errors.

When using water displacement, ensure no air bubbles cling to the solid, as this would reduce the apparent volume and cause errors. Also, ensure the solid sinks completely; if it floats, this method is invalid without adjustments. Use distilled water to avoid impurities affecting volume or mass. Dry the solid before weighing to avoid extra mass from water droplets.

• Use calibrated, sensitive balance.
• Take multiple measurements and average results.
• Avoid air bubbles during displacement.
• Ensure solid sinks fully in water.
• Dry solids before weighing.

Density is a physical property defined as the mass of an object divided by its volume (density, ρ = mass/volume). It indicates how much matter is packed into a given space. When an object is placed in a liquid, whether it floats or sinks depends on how its density compares to the density of the liquid.

For example, if a wooden block has a density of 600 kg/m³ and it is placed in water with a density of 1000 kg/m³, the block will float because it is less dense than water. Conversely, a metal object with a density of 7800 kg/m³ will sink in water because it is much denser.

• Density (ρ) = mass (m) / volume (V)
• Units of density are typically kilograms per cubic metre (kg/m³)
• An object’s density relative to the liquid determines floating or sinking

Buoyancy is the upward force exerted by a fluid that opposes the weight of an object immersed in it. This force arises because the pressure in a fluid increases with depth, creating a net upward force on the object.

If the buoyant force equals or exceeds the object's weight, the object floats or remains suspended. If the buoyant force is less than the object's weight, the object sinks.

The magnitude of the buoyant force equals the weight of the fluid displaced by the object (Archimedes’ principle).

• Buoyant force acts opposite to gravity
• Magnitude of buoyant force = weight of displaced fluid
• Floating occurs when buoyant force ≥ object weight

To assess whether an object will float or sink, compare its density to the density of the liquid. If the object’s density is less than the liquid’s density, it floats; if greater, it sinks.

This comparison works because the weight of the fluid displaced depends on the fluid’s density. A less dense object displaces a volume of fluid whose weight is enough to support it, causing it to float.

For example, a plastic bottle filled with air has a density less than water and floats, but if filled with water, its density increases and it sinks.

• Object density < liquid density → object floats
• Object density > liquid density → object sinks
• Density comparison predicts floating or sinking behaviour

Consider a cube with mass 2 kg and volume 0.002 m³ placed in water (density 1000 kg/m³). Calculate the cube’s density:

ρ = m / V = 2 kg / 0.002 m³ = 1000 kg/m³. Since the cube’s density equals the water’s density, it will be neutrally buoyant and remain suspended in the water without sinking or floating.

If the cube’s density were 900 kg/m³, it would float; if 1100 kg/m³, it would sink.

• Calculate density using ρ = m / V
• Compare calculated density with liquid density
• Equal densities result in neutral buoyancy

In real situations, factors such as shape, surface tension, and fluid viscosity can affect floating and sinking but density comparison remains the primary criterion.

Measuring density accurately requires precise mass and volume measurements. For irregular objects, volume can be found by water displacement.

Safety note: When testing floating and sinking, handle liquids and objects carefully to avoid spills or damage.

• Shape and surface effects can influence behaviour but density is key
• Use water displacement to find volume of irregular objects
• Ensure accurate measurements for reliable conclusions