Density Revision Notes

Density is a physical property of matter that describes how much mass is contained in a given volume of a substance or object.

It is defined as the mass (m) of an object divided by its volume (V).

Mathematically, density (ρ, the Greek letter rho) is expressed as: ρ = m / V, where: m = mass in kilograms (kg), V = volume in cubic metres (m³), and ρ = density in kilograms per cubic metre (kg/m³).

• Density is a scalar quantity, having magnitude only.
• It characterises how compact or concentrated matter is in an object.
• Higher density means more mass packed into the same volume.

The fundamental formula relating mass, density, and volume is: m = ρ × V.

This formula can be rearranged to find any one of the three variables if the other two are known:

Density: ρ = m / V Volume: V = m / ρ Mass: m = ρ × V All variables must be in compatible units for correct calculation.

• Mass (m) is measured in kilograms (kg).
• Volume (V) is measured in cubic metres (m³) or cubic centimetres (cm³).
• Density (ρ) is measured in kilograms per cubic metre (kg/m³) or grams per cubic centimetre (g/cm³).
• 1 g/cm³ = 1000 kg/m³ (unit conversion).

Example: Calculate the mass of a block of metal with a volume of 0.02 m³ and a density of 7800 kg/m³ (typical for steel).

Step 1: Write down the formula: m = ρ × V Step 2: Substitute the known values: m = 7800 kg/m³ × 0.02 m³ Step 3: Calculate: m = 156 kg Step 4: State the answer with units: The mass of the block is 156 kilograms.

• Always check units before substituting values.
• Ensure volume and density units are compatible.
• Include units in every step to avoid errors.

A frequent mistake is mixing units, such as using volume in cm³ with density in kg/m³ without conversion.

Another error is forgetting to rearrange the formula correctly when solving for volume or density.

Sometimes students omit units or write incorrect units, leading to confusion or wrong answers.

• Convert all measurements to standard units before calculation.
• Use the correct formula rearrangement depending on the unknown.
• Write units clearly at every step to maintain accuracy.

Density can be investigated by measuring the mass and volume of an object and calculating its density.

For regular-shaped solids, volume can be calculated by measuring dimensions and using geometric formulas.

For irregular solids, volume can be found by water displacement in a measuring cylinder (Archimedes' principle).

• Measure mass using a balance (kg).
• Measure volume using rulers or displacement (m³ or cm³).
• Calculate density using ρ = m / V.
• Repeat measurements to improve accuracy and calculate an average.
• Handle apparatus carefully to avoid spills or damage.

Density (ρ) is a physical property defined as the mass (m) of a substance divided by its volume (V). It quantifies how much matter is packed into a given space.

The formula for density is: ρ = m / V, where: ρ is density in kilograms per cubic metre (kg/m³), m is mass in kilograms (kg), and V is volume in cubic metres (m³).

In practical laboratory measurements, mass is often measured in grams (g) and volume in cubic centimetres (cm³) or millilitres (ml), where 1 cm³ = 1 ml. The density unit in these cases is grams per cubic centimetre (g/cm³) or grams per millilitre (g/ml).

• Density = Mass ÷ Volume
• Units: kg/m³ (SI), or g/cm³ and g/ml (common)
• 1 cm³ = 1 ml

To measure the density of a liquid, you need to determine its mass and volume accurately.

Apparatus required: a balance (to measure mass), a clean graduated cylinder or measuring cylinder (to measure volume), a beaker, and a pipette or funnel if needed.

Procedure: 1. Place the empty graduated cylinder on the balance and zero (tare) the balance. 2. Pour a known volume of the liquid into the cylinder, for example, 50 ml. 3. Record the mass displayed on the balance; this is the mass of the liquid alone. 4. Use the volume reading from the graduated cylinder. 5. Calculate density using ρ = m / V, ensuring units are consistent (e.g., grams and millilitres).

• Use a balance to measure mass of liquid
• Use a graduated cylinder to measure volume
• Tare the balance with empty cylinder before adding liquid
• Calculate density with ρ = m / V
• Example: Mass = 40 g, Volume = 50 ml, Density = 40 g ÷ 50 ml = 0.8 g/ml

Regular solids have shapes such as cubes, cuboids, spheres, cylinders, or cones, for which volume can be calculated using geometric formulas.

Apparatus required: a balance to measure mass, a ruler or vernier callipers to measure dimensions (length, width, height, radius).

Procedure: 1. Measure the mass of the solid using the balance. 2. Measure the necessary dimensions accurately using the ruler or callipers. 3. Calculate the volume using the appropriate formula: for example, volume of a cube = side³; volume of a cylinder = π × radius² × height. 4. Calculate density using ρ = m / V, converting units as necessary (e.g., cm³ to m³ if using SI units).

• Measure mass with balance
• Measure dimensions with ruler or callipers
• Calculate volume using geometric formula
• Calculate density using ρ = m / V
• Example: Cube with side 4 cm, mass 64 g; Volume = 4³ = 64 cm³; Density = 64 g ÷ 64 cm³ = 1 g/cm³

Irregular solids have shapes that do not allow easy volume calculation. The volume can be found by measuring the volume of liquid displaced when the solid is submerged.

Apparatus required: a balance, a measuring cylinder or overflow can, water or another suitable liquid, and a beaker.

Procedure: 1. Measure the mass of the irregular solid using the balance. 2. Fill the measuring cylinder with a known volume of water and record this initial volume (V₁). 3. Carefully submerge the solid completely in the water without spilling, and record the new volume (V₂). 4. Calculate the volume of the solid as the difference: V = V₂ − V₁. 5. Calculate density using ρ = m / V, ensuring consistent units (e.g., grams and millilitres).

• Measure mass of solid with balance
• Record initial volume of water in measuring cylinder
• Submerge solid fully and record new volume
• Calculate volume displaced: V = V₂ − V₁
• Calculate density: ρ = m / V
• Example: Mass = 30 g, V₁ = 40 ml, V₂ = 55 ml; Volume = 15 ml; Density = 30 g ÷ 15 ml = 2 g/ml

Example: A metal block with regular shape has sides 5.0 cm, 3.0 cm and 2.0 cm. Its mass is 270 g. Calculate its density.

Step 1: Calculate volume: V = length × width × height = 5.0 cm × 3.0 cm × 2.0 cm = 30.0 cm³.

Step 2: Use density formula: ρ = m / V = 270 g ÷ 30.0 cm³ = 9.0 g/cm³ (or 9000 kg/m³). Note: 1 g/cm³ = 1000 kg/m³ if converting units to SI standard units (kg/m³).

• Calculate volume using correct formula
• Use consistent units for mass and volume
• Divide mass by volume to find density
• Common error: mixing units (e.g., grams with m³)
• Common error: incorrect volume calculation

Density (ρ) is defined as the mass (m) of an object divided by its volume (V). It is a measure of how much mass is contained in a given volume.

The formula for density is: ρ = m / V, where ρ is in kilograms per cubic metre (kg/m³), mass m is in kilograms (kg), and volume V is in cubic metres (m³).

Density is a scalar quantity, meaning it has magnitude only and no direction.

• Density formula: ρ = m / V
• Units: kilograms per cubic metre (kg/m³)
• Density is scalar

Archimedes' principle states that an object immersed in a fluid experiences an upward buoyant force equal in magnitude to the weight of the fluid displaced by the object.

The buoyant force (upthrust) acts opposite to gravity and can support the object against sinking.

If the buoyant force equals the object's weight, the object floats; if less, the object sinks; if more, the object rises to the surface and floats.

• Buoyant force = weight of displaced fluid
• Upthrust opposes weight
• Floating occurs when buoyant force ≥ weight

To assess if an object will float or sink, compare its density with the density of the fluid it is placed in.

If the object's density is less than the fluid's density, the object will float because it displaces a volume of fluid whose weight is equal to the object's weight before being fully submerged.

If the object's density is greater than the fluid's density, the object will sink because it cannot displace enough fluid to balance its weight before being fully submerged.

• Object density < fluid density → object floats
• Object density > fluid density → object sinks
• Equal densities → object remains suspended

Example: A block has a mass of 2.0 kg and a volume of 0.0025 m³. It is placed in water with density 1000 kg/m³. Will it float or sink?

Step 1: Calculate the block's density: ρ = m / V = 2.0 kg / 0.0025 m³ = 800 kg/m³.

Step 2: Compare with water's density: 800 kg/m³ < 1000 kg/m³, so the block is less dense than water and will float.

• Calculate object density using ρ = m / V
• Compare densities to fluid density
• Determine floatation based on comparison

In practice, objects less dense than the fluid float partially submerged, displacing a volume of fluid equal in weight to the object.

Objects denser than the fluid sink to the bottom, fully submerged.

Surface tension and shape can affect floatation slightly but density comparison is the primary factor for floating or sinking in fluids like water or oil.

• Floating objects partially submerged
• Sinking objects fully submerged
• Density comparison is primary factor

Density (ρ) is defined as the mass (m) of a substance per unit volume (V). It is a scalar quantity and is expressed mathematically as:

\[ \rho = \frac{m}{V} \] where ρ is density in kilograms per cubic metre (kg/m³), m is mass in kilograms (kg), and V is volume in cubic metres (m³).

Density is an intrinsic property of a substance and varies between different liquids depending on their molecular composition and temperature.

• Density formula: ρ = m ÷ V
• Units of density: kilograms per cubic metre (kg/m³)

Buoyancy is the upward force exerted by a fluid that opposes the weight of an object or another fluid placed in it. When two immiscible liquids are placed together, the liquid with the lower density experiences a net upward buoyant force relative to the denser liquid and thus floats on top.

This occurs because the denser liquid has more mass per unit volume, exerting a greater downward force. The less dense liquid displaces a volume of the denser liquid equal to its own weight, allowing it to float.

The condition for floating is: the density of the floating liquid is less than the density of the liquid below it.

• Buoyant force acts upward against gravity
• Lower density liquid floats on higher density liquid
• Floating occurs when ρ₁ < ρ₂, where ρ₁ is the density of the upper liquid

To determine if one liquid will float on another, compare their densities:

If \( \rho_1 < \rho_2 \), liquid 1 will float on liquid 2.

If \( \rho_1 > \rho_2 \), liquid 1 will sink below liquid 2, assuming they do not mix (immiscible).

• Compare densities of the two liquids
• Lower density liquid floats on top
• Assumes liquids are immiscible (do not mix)

Suppose you have two liquids: Liquid A with density 800 kg/m³ and Liquid B with density 1000 kg/m³. To evaluate which floats:

Since 800 kg/m³ < 1000 kg/m³, Liquid A will float on Liquid B.

Conversely, Liquid B will sink below Liquid A if placed together without mixing.

• Example: Liquid A (800 kg/m³) floats on Liquid B (1000 kg/m³)
• Use density comparison to predict layering

To draw a diagram showing two immiscible liquids: 1. Draw a vertical container (e.g., a beaker) with a horizontal boundary separating two layers.

2. Label the top layer with the liquid of lower density and write its density value (e.g., 800 kg/m³). Label the bottom layer with the higher density liquid (e.g., 1000 kg/m³).

• Draw container with two horizontal layers
• Label each layer with liquid name and density
• Top layer: lower density liquid; bottom layer: higher density liquid