Grade 8 Edexcel Physics — Master the Fundamentals with Arif Sir
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Lesson-1.1.1- The relationship between density, mass and volume: density = mass/volume ρ = m/V

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Lesson-1.1.1- The relationship between density, mass and volume: density = mass/volume ρ = m/V

EdexcelGrade 8⏱ 90 minutes📊 60 marks total

🎯 Exam: Density calculations appear in EVERY Edexcel IGCSE Physics Paper 1 and Paper 2. In the last five years, density questions carried 6–9 marks per paper — that is up to 15% of your total marks. Miss the formula triangle and you lose method marks before you even start calculating.

🔬 Wonder: In 1815, the British warship HMS Cornwallis was launched at Bombay. Its iron hull weighed over 1,800 tonnes — yet it floated. Meanwhile, a 2 kg iron nail sinks instantly in a glass of water. Same material. Different result. The secret is not what the object is made of — it is how tightly that material is packed.

🚀 Identity: Marine engineers in Chittagong design ships that must float with 50,000 tonnes of cargo. Civil engineers test soil density before building the Padma Rail Bridge. Materials scientists at BUET decide whether a new alloy is strong enough for a rickshaw frame. Today, you start thinking like all of them.

📖 The Story That Changes Everything

The monsoon rain hammers the tin roof of a small boat workshop on the bank of the Buriganga River. Inside, a boatbuilder named Karim holds two blocks in his hands — one is a small cube of iron, no bigger than a matchbox. The other is a huge block of shola pith, the white spongy wood used to make Bengali wedding decorations, as big as a pillow. He drops both into a bucket of river water. The iron cube hits the bottom with a clink. The giant shola block bobs on the surface like a lazy duck.

Karim's ten-year-old daughter, Rima, watches from the doorway. She frowns. 'Abbu,' she says, 'the iron is tiny. The shola is huge. Why does the tiny one sink and the big one float?' Karim smiles. He has been waiting for this question for years. He lifts the iron cube and the shola block, one in each hand. 'Which one feels heavier?' he asks. Rima reaches out, holds each one, and her eyes widen. The tiny iron cube is heavier than the giant shola block.

Karim sets both blocks on the wooden table. 'So the iron has more mass,' Rima says slowly. 'But it is smaller.' Karim nods. 'Exactly. The iron has packed a lot of matter into a tiny space. The shola has spread very little matter across a huge space. The water can feel this difference. That is why one sinks and one floats.' Rima stares at the two blocks. A question forms in her mind — a question that scientists took thousands of years to answer properly. How do you measure 'how tightly matter is packed'?

You have felt this your whole life. When you pick up a football and then pick up a cricket ball of the same size, the cricket ball feels surprisingly heavy. When you lift a full bottle of water and a full bottle of cooking oil, the water feels heavier — even though both bottles are the same size. When you carry a bag of rice from the market and a bag of puffed rice of the same size, the rice bag pulls your arm down. Your hands have been measuring density since you were a toddler. You just did not have the word for it.

Today, you will get the word. And more than that — you will get the formula. The exact mathematical relationship that Karim's hands understood but could not write down. The same relationship that lets engineers decide whether a ship floats, whether a bridge stands, and whether a balloon rises. By the end of this lesson, you will look at any object — a mango, a brick, a bottle of oil — and your brain will automatically ask: how much mass? how much volume? what is the density?

Three questions are waiting for you. First: if I give you a block of unknown metal, how can you calculate its density using only a ruler and a weighing scale? Second: why does a steel ship float when a steel nail sinks — and can you prove it with numbers? Third: if you mix two liquids that do not dissolve in each other, how can density tell you which one will sit on top? You cannot answer these yet. But in ninety minutes, you will answer all three — and you will wonder why anyone ever found this difficult.

So take a breath. Pick up your notebook. Let us walk into Karim's workshop together and discover the secret that the Buriganga has been keeping for centuries.

🔍 The Big Question: Why can a 50,000-tonne steel ship float on the Padma River while a 50-gram steel nail sinks to the bottom — and how can a single number tell you which will happen before you even drop it in?

This might feel like a new idea, but you have been using it your whole life without knowing its name — and by the end of today, you will own it completely.

👨‍👩‍👧 For Parents

Tonight, ask your child: Ask your child: 'If I give you a cricket ball and a football of the same size, which one is denser and why?'

A good answer includes: A correct answer should mention that the cricket ball has more mass in the same volume, so it is denser. The child should use the word 'density' or 'mass per volume'.

📍 Part 1 of 8 — The Foundation

🧠 What You Already Know

Before we begin — you already know more than you think:

  • Mass is the amount of matter in an object, measured in kilograms (kg) or grams (g).
  • Volume is the amount of space an object takes up, measured in cubic metres (m³) or cubic centimetres (cm³).
  • You already know how to measure mass with a balance and volume with a ruler or a measuring cylinder.

You already know how to measure mass and volume separately. Today, you will combine them into a single powerful number — density — that tells you something neither mass nor volume can tell you alone. It is like knowing a person's height and weight separately, then discovering BMI — a single number that reveals something new.

✋ Feel It Before You Name It

🫀 Feel It: Hold your left hand out, palm up. Now imagine placing a cricket ball in it. Feel the weight pulling your hand down. Now imagine placing a football of exactly the same size in your right hand. Feel how much lighter it is. Same size. Different heaviness. Your hands just measured something your eyes cannot see.

👁 See It: Both balls take up the same amount of space. But the cricket ball has more matter squeezed into that space. The football has less matter spread through the same space. Your hand noticed the difference immediately — even though your eyes saw two identical-sized spheres.

❓ Question It: If two objects are the same size, why does one feel heavier? What is different about the matter inside them?

🏷 Name It: Scientists call this difference density. Density is how much mass is packed into each unit of volume. A dense object has a lot of mass in a small space. A less dense object has less mass in the same space. Density is the answer to your question.

🌍 Generalise: You see density everywhere in Bangladesh. A mango sinks in a bucket of water because it is denser than water. A piece of shola pith floats because it is less dense. Mustard oil floats on water in your mother's kitchen because it is less dense. A brick sinks in the Buriganga because it is denser than the river water.

🔍 Discover It Yourself

💬 Arif Sir asks: If I give you two iron blocks — one small, one large — which one is denser?

⏸ Think carefully. Do not rush.

💡 Revealed: Neither. They have the same density. Density is a property of the material, not the size. A small iron nail and a giant iron anchor have identical density — about 7,800 kg/m³. The anchor has more mass because it has more volume, but the ratio of mass to volume is the same.

💬 Arif Sir asks: If density is mass divided by volume, what happens to density if I double the mass but keep the volume the same?

⏸ Picture the equation in your mind.

💡 Revealed: Density doubles. If you pack twice the mass into the same space, the density increases. This is why compressing a gas increases its density — same volume, more mass squeezed in.

💬 Arif Sir asks: So what single number tells you whether an object will float or sink in water?

⏸ Connect everything you have discovered.

💡 Revealed: The density of the object compared to the density of water. Water has a density of 1,000 kg/m³ (or 1 g/cm³). If the object's density is greater than 1,000 kg/m³, it sinks. If it is less, it floats. That is the secret of the Buriganga.

🔭 Go Deeper — Did You Know?

The core of the Sun has a density of about 150,000 kg/m³ — 150 times denser than water. A single teaspoon of Sun-core material would weigh about 750 grams on Earth. Yet the Sun is made almost entirely of hydrogen and helium — the lightest elements in the universe. Density is not about what the material is; it is about how tightly it is squeezed.

This proves that density depends on both the material AND the conditions. The same hydrogen that floats in a balloon becomes unimaginably dense inside the Sun because of enormous pressure and temperature.

🎯 What You'll Master Today

  • State the definition of density and identify its SI unit. RememberAO1
  • Explain why density is a property of a material and not of an object's size. UnderstandAO1
  • Calculate density using the formula ρ = m/V when mass and volume are given. ApplyAO2
  • Calculate mass using m = ρV when density and volume are given. ApplyAO2
  • Calculate volume using V = m/ρ when mass and density are given. ApplyAO2
  • Analyse whether an object will float or sink by comparing its density to that of water. AnalyseAO3
  • Evaluate experimental methods for measuring the density of irregular objects and suggest improvements. EvaluateAO3
  • Design a procedure to determine the density of an unknown liquid using a measuring cylinder and a balance. CreateAO3

📚 Words You Must Know Cold

  • Density (ঘনত্ব — কোনো বস্তুর একক আয়তনে কতটা ভর আছে তা বোঝায়।): The mass packed into every unit of volume of a material. It tells you how heavy something is for its size. [Origin: From Latin 'densus', meaning thick or crowded — the same root as 'dense' in English.] Think of a crowded Dhaka bus: more people in the same space means higher density. A near-empty bus has low density. Exam: State the density of the material using rho = m/V, with units kg/m³ or g/cm³. Always include the unit or you lose the accuracy mark.
  • Mass (ভর — বস্তুতে কতটা পদার্থ আছে তার পরিমাপ, একক কিলোগ্রাম বা গ্রাম।): The amount of matter in an object, measured in kilograms (kg) or grams (g). It does not change with location. [Origin: From Old English 'mæsse', meaning a lump or quantity of material.] Your mass is the same on Earth, on the Moon, or floating in space. A 50 kg sack of rice is 50 kg everywhere. Exam: In calculations, convert grams to kilograms before using SI units. Write m = 2.5 kg, not m = 2500 g, unless the question asks for g.
  • Volume (আয়তন — কোনো বস্তু কতটা জায়গা দখল করে তার পরিমাপ, একক ঘনমিটার বা ঘনসেন্টিমিটার।): The amount of space an object occupies, measured in cubic metres (m³) or cubic centimetres (cm³). [Origin: From Latin 'volumen', meaning a roll or scroll — originally the space a rolled manuscript took up.] Imagine filling a water bottle at a tube well. The amount of water it holds is its volume. Exam: For a regular solid, use V = length × width × height. For an irregular solid, use the displacement method. State the formula before substituting numbers.
  • Formula for Density (ঘনত্বের সূত্র — ঘনত্ব = ভর ÷ আয়তন, অর্থাৎ rho = m/V।): Density equals mass divided by volume: rho = m/V. It connects how heavy something is to how much space it takes up. [Origin: The symbol rho (ρ) comes from the Greek letter used for 'r' and was adopted in physics to represent density.] Remember the triangle: cover rho and you get m/V. Cover m and you get rho × V. Cover V and you get m/rho. Exam: Always write the formula first, then substitute. The examiner gives a method mark for correct rearrangement, even if your arithmetic slips.
  • Kilogram per cubic metre (কিলোগ্রাম প্রতি ঘনমিটার — ঘনত্বের SI একক, kg/m³।): The SI unit of density, written kg/m³. It means how many kilograms of material fill one cubic metre. [Origin: Combines 'kilogram' (Greek 'khilioi' = thousand) and 'metre' (Greek 'metron' = measure).] Water has a density of 1000 kg/m³. That means one cubic metre of water has a mass of 1000 kg — about the weight of a small rickshaw. Exam: Use kg/m³ when mass is in kg and volume is in m³. If the question gives g/cm³, you may need to convert: 1 g/cm³ = 1000 kg/m³.
  • Gram per cubic centimetre (গ্রাম প্রতি ঘনসেন্টিমিটার — ঘনত্বের ছোট একক, g/cm³।): A smaller unit of density, written g/cm³. It tells you how many grams fill one cubic centimetre. [Origin: From French 'gramme' (a small weight) and Latin 'centum' (hundred) for centimetre.] A sugar cube is about 1 cm³. If it has a mass of 1.6 g, its density is 1.6 g/cm³ — that's sugar! Exam: Use g/cm³ when mass is in grams and volume is in cm³. To convert to kg/m³, multiply by 1000. Show the conversion step clearly.
  • Displacement method (জল অপসারণ পদ্ধতি — অনিয়মিত বস্তুর আয়তন বের করার জন্য জল সরানোর পরিমাপ।): A way to find the volume of an irregular object by measuring how much water it pushes aside in a measuring cylinder. [Origin: From Latin 'dis-' (away) and 'plere' (to fill) — literally 'to un-fill' the water.] Drop a stone into a full glass of water at the Buriganga riverbank. The water that spills over is exactly the stone's volume. Exam: Record the initial water level, then the final level. Volume of object = final reading − initial reading. Always read the bottom of the meniscus at eye level.
  • Meniscus (মেনিস্কাস — মাপক সিলিন্ডারে জলের বাঁকা পৃষ্ঠ; সঠিক পাঠের জন্য নিচের বাঁকটি দেখতে হয়।): The curved surface of water in a measuring cylinder. You must read the bottom of the curve for an accurate volume. [Origin: From Greek 'meniskos', meaning a little crescent moon — because the curve looks like a crescent.] Look at the water in a glass: it curves down at the edges. Read the lowest point — like the bottom of a smile. Exam: In practical exams, you lose marks if you read the top of the meniscus. Always say 'read the bottom of the meniscus at eye level'.
  • Regular solid (নিয়মিত ঘনবস্তু — ঘনক, আয়তঘন বা সিলিন্ডারের মতো সরল আকৃতির বস্তু, যার আয়তন সূত্র দিয়ে বের করা যায়।): An object with a simple shape like a cube, cuboid, or cylinder, whose volume can be calculated using a formula. [Origin: From Latin 'regula', meaning a rule — because these shapes follow simple rules.] A brick from a kiln in Narayanganj is a cuboid. Measure its length, width, and height, then multiply. Exam: For a cuboid, V = l × w × h. For a cylinder, V = πr²h. State the formula and show substitution to earn full marks.
  • Irregular solid (অনিয়মিত ঘনবস্তু — পাথর বা আমের মতো বস্তু, যার আয়তন জল অপসারণ পদ্ধতিতে বের করতে হয়।): An object with a shape that has no simple formula, like a stone or a mango. Its volume is found by displacement. [Origin: From Latin 'ir-' (not) and 'regula' (rule) — not following a simple rule.] You can't measure a mango with a ruler. But you can drop it in water and see how much water rises. Exam: Describe the displacement method step by step: fill cylinder, record initial volume, submerge object, record final volume, subtract. Each step earns a mark.
  • Floating (ভাসমান — যখন বস্তুর ঘনত্ব তরলের ঘনত্বের চেয়ে কম হয়, তখন বস্তু তরলের উপরে থাকে।): When an object stays on top of a liquid because its density is less than the liquid's density. [Origin: From Old English 'flotian', meaning to rest on water.] A wooden boat on the Padma floats because wood is less dense than water. A stone sinks because it is denser. Exam: Explain floating by comparing densities: 'The object floats because its density is less than the density of the liquid.' Use the word 'less than' clearly.
  • Sinking (ডুবন্ত — যখন বস্তুর ঘনত্ব তরলের ঘনত্বের চেয়ে বেশি হয়, তখন বস্তু তলিয়ে যায়।): When an object goes down in a liquid because its density is greater than the liquid's density. [Origin: From Old English 'sincan', meaning to become submerged.] An iron nail sinks in water because iron is about 7.8 times denser than water. A steel ship floats because it is hollow and its average density is low. Exam: Explain sinking by comparing densities: 'The object sinks because its density is greater than the density of the liquid.' Do not just say 'it is heavy' — that loses the mark.
📍 Part 2 of 8 — The Core Knowledge

⚗️ The Magic Formula — Where It Comes From

Density formula

Imagine two blocks of the same material — one small, one large. The large block has twice the mass because it has twice the volume. Mass is directly proportional to volume for the same material. This means the ratio of mass to volume is constant for that material. That constant ratio is what we call density. It is the material's fingerprint.

The Derivation — Step by Step

  1. Step 1: Physical observation — for the same material, doubling the volume doubles the mass.
    m∝Vm \propto V
  2. Step 2: Express this as a proportionality: m ∝ V.
    m=ρVm = \rho V
  3. Step 3: Introduce a constant of proportionality, ρ, so that m = ρV.
    ρ=mV\rho = \frac{m}{V}
  4. Step 4: Rearrange to find the constant: ρ = m/V. This is the density formula.
    ρ=mV\rho = \frac{m}{V}

Therefore:

ρ=mV\rho = \frac{m}{V}

  • Variables: ρ (rho) = density in kilograms per cubic metre (kg/m³); m = mass in kilograms (kg); V = volume in cubic metres (m³)
  • Valid when: Valid for any uniform material at constant temperature and pressure. Not valid if the material is a mixture with varying composition, or if temperature changes cause expansion or contraction.
  • 💡 Think of it as: Imagine a Dhaka-bound bus. The number of passengers (mass) divided by the space inside the bus (volume) gives the crowding level (density). A small CNG with 4 passengers is denser than a large bus with 4 passengers.
  • 📝 In the exam: Edexcel gives you mass and volume and asks you to calculate density. Or gives density and one other quantity and asks for the third. Always show the formula, substitution, and answer with unit.

Density is a material property

Components: Mass, volume, and the ratio between them.

Density does not depend on the size or shape of the object. A tiny iron nail and a giant iron anchor have the same density because they are made of the same material. Density depends only on what the material is and the conditions (temperature, pressure).

Floating and sinking

Components: Density of object, density of liquid.

An object floats if its density is less than the density of the liquid it is placed in. It sinks if its density is greater. If the densities are equal, the object stays suspended in the liquid. This is why a steel ship can float — its overall density (including the air inside) is less than water's density.

Unit conversion

Components: kg/m³ and g/cm³.

1 g/cm³ = 1,000 kg/m³. To convert from g/cm³ to kg/m³, multiply by 1,000. To convert from kg/m³ to g/cm³, divide by 1,000. Water has a density of 1 g/cm³ or 1,000 kg/m³.

🔍 Explained Three Ways — Find Yours

🟢 Foundation Level

Struggling? Start here.

Imagine you have two identical bottles. Fill one with water and one with cooking oil. The water bottle feels heavier. Both bottles are the same size, so they have the same volume. But the water has more mass packed into that volume. That is density — how much mass is packed into a space. Water is denser than oil.

🔵 Core Level

This is the main explanation.

Density is defined as mass per unit volume. The formula is ρ = m/V, where ρ is density in kg/m³, m is mass in kg, and V is volume in m³. For example, if a block of iron has a mass of 7,800 kg and a volume of 1 m³, its density is 7,800 kg/m³. This means every cubic metre of iron has a mass of 7,800 kg. Density is a property of the material — it does not change if you cut the block in half.

🔴 Advanced Level

Aiming for A*? Go deeper.

At the particle level, density depends on two factors: the mass of each particle and how closely the particles are packed. Iron atoms are heavy and tightly packed in a regular lattice, giving iron a high density. In gases, particles are far apart, so gases have very low densities. Temperature affects density: heating a material increases the average spacing between particles, so volume increases and density decreases. This is why hot air rises — it is less dense than the cooler air around it.

👁 For Visual Learners

Draw two identical boxes side by side. In the first box, draw 100 tiny dots crowded together. In the second box, draw 20 dots spread far apart. Label the first box 'high density' and the second 'low density'. The number of dots represents mass, the box represents volume. The crowding represents density. Now draw a third box with 100 dots but twice as large — the dots are spread out, so density is lower even though mass is the same.

🔢 For Logical Learners

Density is the gradient of the graph of mass against volume for a given material. If you plot mass on the y-axis and volume on the x-axis, the graph is a straight line through the origin. The gradient of this line is the density. A steeper line means a higher density. This is why density is a constant for a given material — the ratio m/V never changes.

📖 For Story Learners

Once there was a young scientist named Rima who lived by the Buriganga. Every day she watched boats float and stones sink. She asked her father why. He gave her two blocks — one iron, one shola — and a bucket of water. She dropped them in. The iron sank, the shola floated. She picked them up and felt their weights. The iron was heavier even though it was smaller. She realised that the iron had packed more mass into less space. She called this 'density'. From that day, she could predict which objects would float and which would sink — just by knowing their density.

💭 Quick Recall #1 (Answer without looking!)

Show Answer

Q: What is the formula for density, and what are the units of each variable?

A: Density = mass / volume. ρ = m/V. Density in kg/m³, mass in kg, volume in m³.

ρ=mV\rho = \frac{m}{V}

🖼 Diagrams & Visual Guides

🔵 Two blocks of the same material — different sizes, same density

[Diagram: Two iron blocks are shown side by side. The left block is small (1 cm³) and has a mass of 7.8 g. The right block is large (10 cm³) and has a mass of 78 g. Both blocks are labelled with their mass and volume. Below each block, the calculation 7.8/1 = 7.8 g/cm³ and 78/10 = 7.8 g/cm³ is shown. The density is the same for both.]

Labels: Small iron block: 1 cm³, 7.8 g, Large iron block: 10 cm³, 78 g, Density = 7.8 g/cm³ for both

Density is a property of the material, not the size of the object.

⚗️ Density formula triangle

[Diagram: A triangle divided into three sections. The top section contains 'm' (mass). The bottom left section contains 'ρ' (density). The bottom right section contains 'V' (volume). Arrows show that m = ρ × V, ρ = m / V, and V = m / ρ. The triangle is drawn in navy blue with gold text.]

Labels: m (mass), ρ (density), V (volume)

Cover the quantity you want to find. The triangle tells you the formula.

🟢 Step-by-step density calculation flowchart

[Diagram: A flowchart with five boxes connected by arrows. Box 1: 'Write down given values: m = 500 g, V = 250 cm³'. Box 2: 'Convert to SI: m = 0.5 kg, V = 0.00025 m³'. Box 3: 'Write formula: ρ = m/V'. Box 4: 'Substitute: ρ = 0.5 / 0.00025'. Box 5: 'Calculate: ρ = 2000 kg/m³'. Each box is green.]

Labels: Given values, Convert to SI, Write formula, Substitute, Calculate

Follow these five steps for every density calculation.

🔴 Common mistake: forgetting to convert units

[Diagram: Left side (red): 'm = 500 g, V = 250 cm³, ρ = 500/250 = 2 g/cm³'. Right side (green): 'm = 0.5 kg, V = 0.00025 m³, ρ = 0.5/0.00025 = 2000 kg/m³'. The left side is marked WRONG because the units are not SI. The right side is marked CORRECT.]

Labels: WRONG: units not converted, CORRECT: units converted to SI

Always convert to SI units before substituting into the formula.

📸 A steel ship floating on the Padma River

[Diagram: A large steel cargo ship is shown floating on the Padma River. The ship is loaded with containers. The waterline is visible on the hull. A small steel nail is shown sinking in a glass of water next to the ship. Labels point to the ship's hollow hull and the nail's solid steel.]

Labels: Ship: overall density < water density, Nail: density > water density, Waterline

The ship floats because its overall density (including air inside) is less than water's density. The nail sinks because steel's density is greater than water's density.

🌍 Why This Changes Everything

📍 Boat design on the Buriganga River

Boatbuilders in Dhaka have known for centuries that a wooden boat must be hollow to float. They shape the hull so that the overall density of the boat — wood, cargo, and air inside — is less than the density of the river water. If they overload the boat, the overall density increases, and the boat sinks. This is why overloading is the leading cause of boat accidents on the Buriganga.

🔢 Real numbers: A wooden boat has a mass of 500 kg and a volume of 2 m³. Its density is 500/2 = 250 kg/m³. Since 250 kg/m³ < 1000 kg/m³ (water), it floats. If it takes on 1500 kg of cargo, total mass = 2000 kg, density = 2000/2 = 1000 kg/m³. It is now exactly at the water's density — it will barely float. Any more cargo and it sinks.

💼 Career connection: Marine engineer — designs hulls and calculates maximum safe cargo loads for ships and boats.

📍 Concrete for the Padma Rail Bridge

The Padma Rail Bridge is one of the largest infrastructure projects in Bangladesh. Civil engineers tested the density of concrete before pouring it. If the concrete is too dense, it is too heavy for the foundations. If it is too light, it is not strong enough. The density must be exactly right — around 2,400 kg/m³.

🔢 Real numbers: A concrete pillar has a volume of 10 m³. If the density of concrete is 2,400 kg/m³, the mass of the pillar is m = ρV = 2400 × 10 = 24,000 kg = 24 tonnes. Engineers must design foundations that can support this mass.

💼 Career connection: Civil engineer — tests material densities to ensure structures are safe and stable.

📍 Cooking oil and water in the kitchen

When your mother pours mustard oil into a pot of water, the oil floats on top. This is because mustard oil has a density of about 920 kg/m³, which is less than water's 1,000 kg/m³. If you shake the pot, the oil and water mix temporarily, but they soon separate again — oil on top, water below. This is density at work in your own kitchen.

🔢 Real numbers: A bottle contains 500 cm³ of mustard oil. The mass of the oil is 460 g. Density = 460/500 = 0.92 g/cm³ = 920 kg/m³. Since 920 < 1000, the oil floats on water.

💼 Career connection: Food scientist — uses density to separate and purify oils, and to check the purity of cooking oil.

📍 Plastic pollution in the Bay of Bengal

Every year, millions of tonnes of plastic waste enter the Bay of Bengal. Some plastics float and are carried by currents to distant shores. Others sink and accumulate on the seafloor, harming bottom-dwelling creatures. The difference is density. Polyethylene (used in bags) has a density of about 920 kg/m³ and floats. PET (used in bottles) has a density of about 1,380 kg/m³ and sinks.

🔢 Real numbers: A PET bottle has a mass of 30 g and a volume of 500 cm³. Density = 30/500 = 0.06 g/cm³ = 60 kg/m³. Wait — that is less than water! But the bottle is hollow. If we crush it, the volume becomes 20 cm³, and density = 30/20 = 1.5 g/cm³ = 1500 kg/m³. Now it sinks. This is why whole bottles float but crushed bottles sink.

💼 Career connection: Environmental scientist — studies how plastic pollution moves through oceans based on density.

📍 Measuring the purity of gold in a jewellery shop

In a jewellery shop in Tanti Bazaar, Dhaka, a goldsmith needs to check if a gold ring is pure. Pure gold has a density of 19,300 kg/m³. If the ring is mixed with cheaper metals like copper (density 8,960 kg/m³), its density will be lower. By measuring the ring's mass and volume, the goldsmith can calculate its density and determine its purity.

🔢 Real numbers: A gold ring has a mass of 19.3 g and a volume of 1 cm³. Density = 19.3/1 = 19.3 g/cm³ = 19,300 kg/m³. This matches pure gold. If the density were 15 g/cm³, the ring would be impure.

💼 Career connection: Assayer or jeweller — uses density to verify the purity of precious metals.

👨‍👩‍👧 For Parents

Tonight, ask your child: Ask your child: 'Why does a heavy steel ship float on the river but a small steel nail sinks?'

A good answer includes: A correct answer should explain that the ship's overall density (including air inside) is less than water's density, while the nail's density is greater than water's density.

🔗 How This Connects

You have already learned how to measure mass and volume separately. Now you have combined them into density — a single number that identifies a material. This concept is the foundation for everything that follows: pressure, floating and sinking, and even the structure of the Earth. In the next lesson, you will learn how to measure the density of irregular objects using the displacement method. Density is the key that unlocks the behaviour of matter.

🤔 Deep Thinking Challenge: Imagine you are given a sealed box containing an unknown material. You cannot open it. You measure its mass as 2 kg and its volume as 0.001 m³. You calculate its density as 2,000 kg/m³. What materials could it be made of? How would you narrow down the possibilities? What other tests could you do without opening the box?

📍 Part 3 of 8 — The Mechanism

⚙️ The Secret Behind This — Step by Step

  1. Step 1: All matter is made of particles. The mass of an object is the total mass of all its particles. → So what? So what? This means density is fundamentally about particles — it is not just a number, it is a window into the microscopic world.
  2. Step 2: The volume of an object is the space its particles occupy. → So what? So what? This means volume is not just empty space — it is the territory occupied by particles.
  3. Step 3: Different materials have different particle masses and different particle spacings. → So what? So what? This explains why a small iron block can be heavier than a large shola block — iron particles are heavier and more tightly packed.
  4. Step 4: Density is the ratio of total particle mass to total volume: ρ = m/V. → So what? So what? This gives us a mathematical tool to predict how materials will behave without testing every single object.
  5. Step 5: To measure density, measure mass with a balance and volume with a ruler or displacement method. → So what? So what? This means you can determine the density of any object, regular or irregular, using simple lab equipment.
  6. Step 6: Compare the calculated density to known values to identify the material or predict floating/sinking. → So what? So what? This means you can identify unknown materials, check purity, and predict floating or sinking.
  7. Step 7: The rule breaks down for mixtures, composites, and materials with voids or varying composition. → So what? So what? This means you must be careful when applying the density formula to real-world objects that are not uniform.

⚠️ Edge Case: A hollow object, like a steel ship, has an overall density that is much less than the density of solid steel. The formula ρ = m/V still works, but V must be the total volume of the object including the hollow space. If you use only the volume of the steel itself, you will get the density of steel, not the density of the ship.

🔒 Safety / Ethics: When measuring the density of unknown liquids, never taste or smell them directly. Some liquids may be toxic or corrosive. Always wear safety goggles and use a fume cupboard if the liquid is volatile. In the lab, handle glass measuring cylinders carefully — they can break and cause cuts.

💥 Myth Busters — Wrong Ideas Destroyed

These are the most common wrong ideas about this topic. Read each one carefully.

Myth 1: Heavy things always sink, light things always float.

🤔 Why it feels right: Everyday life seems to prove it — a stone sinks, a leaf floats. Your brain builds a rule from what you see. But 'heavy' and 'light' are about mass alone, and mass alone never decides floating.

💥 Counter-example: A 50,000-tonne steel ship floats on the Padma River. A 50-gram steel nail sinks in a glass of water. Same material, same river, opposite result. Mass did not decide it — density did.

✅ Truth: Floating depends on density, not mass. An object floats if its density is less than the density of the liquid it sits in. A ship is mostly hollow air, so its average density is far less than water. A nail is solid steel, so its density is nearly eight times that of water.

💬 Remember: It is not how heavy it is — it is how tightly packed it is.

Myth 2: Density is just another word for weight, or for how heavy something feels.

🤔 Why it feels right: When you hold a brick and a sponge of the same size, the brick feels heavier. Your hand reports 'heavy', so you assume density equals heaviness. But your hand is measuring force, not density.

💥 Counter-example: One kilogram of feathers and one kilogram of iron have exactly the same mass and the same weight. But iron has a density of about 7800 kg/m³ and feathers about 10 kg/m³. Same weight, wildly different density.

✅ Truth: Density is mass packed into each unit of volume. It is mass divided by volume, rho = m/V. Weight is a force that depends on gravity. Density is a property of the material itself — it does not change if you move the object to the Moon.

💬 Remember: Density is mass per space, not weight in your hand.

Myth 3: If you cut an object in half, its density halves too.

🤔 Why it feels right: It feels logical: half the material, half the density. Students apply the same thinking they use for mass and volume, where halving the object halves those quantities. So they assume density must halve as well.

💥 Counter-example: Cut a solid iron block in half. Each half has half the mass and half the volume. Divide them: (m/2) / (V/2) = m/V. The density is unchanged. Both halves still sink in water because iron is still iron.

✅ Truth: Density is an intensive property — it depends on the material, not the size of the sample. Halving mass and volume together leaves the ratio untouched. A tiny iron filing and a giant iron girder have the same density.

💬 Remember: Cut it, squash it, stretch it — density stays the same.

Myth 4: You can compare densities directly without converting units — g/cm³ and kg/m³ are basically the same thing.

🤔 Why it feels right: The numbers look similar in casual reading. A student sees 1 g/cm³ for water and 1000 kg/m³ for water and assumes the '1000' is just a bigger, fancier way of writing '1'. The units get ignored because the concept feels familiar.

💥 Counter-example: Water has a density of 1 g/cm³ and 1000 kg/m³. If you treat them as equal, you will calculate a 2 kg block of volume 0.002 m³ as having density 1000 kg/m³ — correct. But if you mix units, you might write 2/0.002 = 1000 g/cm³, which is a thousand times too big and physically impossible for water.

✅ Truth: 1 g/cm³ = 1000 kg/m³. Always convert mass to kilograms and volume to cubic metres before dividing, or convert both to grams and cubic centimetres. Never mix the two systems in one calculation.

💬 Remember: One gram per centimetre cubed equals one thousand kilograms per metre cubed — no shortcuts.

Myth 5: A denser object always sinks, no matter what liquid it is in.

🤔 Why it feels right: Students learn 'dense things sink' and apply it absolutely. They forget that sinking is a comparison between the object and the liquid, not a property of the object alone.

💥 Counter-example: A block of ice has a density of about 917 kg/m³. It sinks in petrol (about 700 kg/m³) but floats on water (1000 kg/m³). The same ice, two different liquids, two different results.

✅ Truth: Floating or sinking depends on the comparison between the object's density and the liquid's density. If object density is less than liquid density, it floats. If greater, it sinks. If equal, it stays suspended.

💬 Remember: Floating is a competition — the lighter density wins.

❌ The Top 5 Mistakes — Don't Be That Student

Mistake 1: Dividing volume by mass instead of mass by volume.

❌ Wrong:
A student writes: density = V/m = 0.002/2 = 0.001 kg/m³ for a 2 kg block of volume 0.002 m³.

ρ=Vm\rho = \frac{V}{m}

✅ Right:
Density = mass / volume = 2 / 0.002 = 1000 kg/m³.

ρ=mV=20.002=1000 kg/m3\rho = \frac{m}{V} = \frac{2}{0.002} = 1000\,\text{kg/m}^3

Why this happens: Students remember 'density is mass and volume' but forget the order. The formula triangle is misremembered, or they guess because both quantities are present.

🛡 How to avoid it: Always write the formula triangle: m on top, rho and V below. Cover the quantity you want. Or remember: density is how much mass is packed into each unit of volume — so mass must be on top.

Mistake 2: Forgetting to convert grams to kilograms or centimetres to metres before calculating.

❌ Wrong:
A student calculates density = 500 g / 100 cm³ = 5 g/cm³ and writes the answer as 5 kg/m³.

ρ=500100=5 kg/m3\rho = \frac{500}{100} = 5\,\text{kg/m}^3

✅ Right:
Convert first: 500 g = 0.5 kg, 100 cm³ = 0.0001 m³. Density = 0.5 / 0.0001 = 5000 kg/m³. Or keep g/cm³: 500/100 = 5 g/cm³, which equals 5000 kg/m³.

ρ=0.50.0001=5000 kg/m3\rho = \frac{0.5}{0.0001} = 5000\,\text{kg/m}^3

Why this happens: Students rush to plug numbers into the formula without checking units. They see numbers and calculate, forgetting that SI units are required for a correct answer.

🛡 How to avoid it: Before any calculation, write down the units of every quantity. If mass is in grams, convert to kilograms. If volume is in cm³, convert to m³. Only then divide.

Mistake 3: Thinking that a larger object always has a larger density.

❌ Wrong:
A student says: 'The big block has more mass, so it must have a higher density than the small block.'
✅ Right:
Density depends on the material, not the size. A large block of wood (density 600 kg/m³) has a lower density than a small iron nail (density 7800 kg/m³).

\rho_{\text{wood}} = 600\,\text{kg/m}^3 &lt; \rho_{\text{iron}} = 7800\,\text{kg/m}^3

Why this happens: Students confuse mass with density. They think 'more stuff' means 'more tightly packed stuff'. But a big object can be made of a light material, and a small object can be made of a dense material.

🛡 How to avoid it: Ask yourself: what material is it made of? Density is a material property. Size does not change it. Compare materials, not sizes.

Mistake 4: Using the wrong volume — for example, using the volume of the container instead of the object.

❌ Wrong:
A student measures the volume of water in a beaker as 200 cm³ and uses that as the volume of a stone dropped into it.

ρ=mstoneVwater\rho = \frac{m_{\text{stone}}}{V_{\text{water}}}

✅ Right:
The volume of the stone is the rise in water level: final reading minus initial reading. If water rises from 200 cm³ to 250 cm³, the stone's volume is 50 cm³.

Vstone=Vfinal−Vinitial=250−200=50 cm3V_{\text{stone}} = V_{\text{final}} - V_{\text{initial}} = 250 - 200 = 50\,\text{cm}^3

Why this happens: Students forget that the displacement method measures the volume of the object, not the water. They read the beaker scale and use that number directly.

🛡 How to avoid it: Always record the initial water level, then the final level after the object is submerged. Subtract to find the object's volume. Write the subtraction explicitly.

Mistake 5: Believing that density changes when you change the shape of an object.

❌ Wrong:
A student says: 'If I flatten the iron block into a sheet, its density decreases because it is thinner.'
✅ Right:
Flattening changes the shape but not the mass or the volume. Density = mass / volume remains the same. Iron is still iron.

ρ=mV=constant for the same material\rho = \frac{m}{V} = \text{constant for the same material}

Why this happens: Students associate density with how 'compact' something looks. A thin sheet looks less dense, but density is about the material, not the shape.

🛡 How to avoid it: Remember: density is an intrinsic property. Changing shape changes volume and mass together in a way that keeps the ratio constant. Test it with a lump of clay — roll it into a ball, then a snake, and calculate density each time.

💭 Quick Recall #2 (Answer without looking!)

Show Answer

Q: A metal block has a mass of 2.7 kg and a volume of 0.001 m³. Calculate its density in kg/m³. Show your working.

A: Density = mass / volume = 2.7 / 0.001 = 2700 kg/m³.

ρ=mV=2.70.001=2700 kg/m3\rho = \frac{m}{V} = \frac{2.7}{0.001} = 2700\,\text{kg/m}^3

🧮 The 10-Step Problem-Solving Method

📋 Before you begin: You must be able to measure mass with a balance and volume with a ruler or measuring cylinder. You must also be able to convert between grams and kilograms, and between cm³ and m³. If you are unsure about unit conversion, review Lesson 1.0.2 on SI units.

  1. Step 1: Read the problem carefully. Identify what type of density problem it is — finding density, mass, or volume.
  2. Step 2: Write down all given values with their units. For example: m = 500 g, V = 250 cm³.
  3. Step 3: Identify what you are asked to find. For example: density (ρ).
  4. Step 4: Select the correct formula. For density: ρ = m/V. For mass: m = ρV. For volume: V = m/ρ.
  5. Step 5: Convert all units to SI. Mass in kg, volume in m³, density in kg/m³. Remember: 1 g = 0.001 kg, 1 cm³ = 0.000001 m³.
  6. Step 6: If needed, derive the formula from the density triangle to avoid rearrangement errors.
  7. Step 7: Substitute the values into the formula. Show the substitution as a separate step.
  8. Step 8: Calculate the answer. Show every step of the arithmetic.
  9. Step 9: Write the final answer with the correct unit and appropriate significant figures (usually 2 or 3).
  10. Step 10: Check the answer. Does it make physical sense? Compare to known densities (water = 1000 kg/m³, iron = 7800 kg/m³).

⏱ Time Management: For a 3-mark density calculation, spend 3 minutes. For a 5-mark question involving unit conversion and explanation, spend 5 minutes. Edexcel allows roughly 1 minute per mark.

🎓 Examiner Thinking: The examiner checks: (1) Did you write the correct formula? (1 mark) (2) Did you substitute the correct values with units? (1 mark) (3) Did you calculate correctly and give the unit? (1 mark) Common mistakes: forgetting to convert units, using the wrong formula, and not giving the unit in the final answer.

📍 Part 4 of 8 — Worked Examples

🔬 Worked Examples — Let's Solve Together!

Example 1: Calculating density of a regular solid <span class="assh-badge assh-badge-foundation">🟢 Foundation</span>

Given: A rectangular block of iron has a mass of 7,800 g and a volume of 1,000 cm³. Calculate its density in kg/m³.

m=7800 gm = 7800 \text{ g}

V=1000 cm3V = 1000 \text{ cm}^3

🤔 Why this approach: The problem gives mass and volume and asks for density. The density formula ρ = m/V applies directly because the block is made of a single uniform material.

Solution — every step shown:

  1. Convert mass from grams to kilograms. Divide by 1000.
    m=78001000=7.8 kgm = \frac{7800}{1000} = 7.8 \text{ kg}
  2. Convert volume from cm³ to m³. Divide by 1,000,000.
    V=10001000000=0.001 m3V = \frac{1000}{1000000} = 0.001 \text{ m}^3
  3. Write the density formula.
    ρ=mV\rho = \frac{m}{V}
  4. Substitute the values.
    ρ=7.80.001\rho = \frac{7.8}{0.001}
  5. Calculate the density.
    ρ=7800 kg/m3\rho = 7800 \text{ kg/m}^3

Why this method? This method is chosen because it directly applies the definition of density. An alternative is to calculate density in g/cm³ first (7.8 g/cm³) and then convert to kg/m³ by multiplying by 1000. Both methods give the same answer, but converting to SI first is safer for exam marks.

✅ Final Answer: The density of the iron block is 7800 kg/m³.

ρ=7800 kg/m3\rho = 7800 \text{ kg/m}^3

Reality Check: Does 7800 kg/m³ make sense for iron? Yes — iron is a dense metal, and this value matches the known density of iron. It is much greater than water's density (1000 kg/m³), so iron sinks in water.

💡 So What? This density (7800 kg/m³) is the density of iron. Knowing this, engineers can calculate the mass of any iron structure — from a bridge girder to a ship's anchor — if they know its volume.

❌ Most Common Mistake: The most common mistake is forgetting to convert grams to kilograms and cm³ to m³. If you use m = 7800 g and V = 1000 cm³, you get ρ = 7.8 g/cm³. This is correct in g/cm³ but wrong if the question asks for kg/m³. Always check the required unit.

🔄 Alternative Method: Calculate density in g/cm³ first: ρ = 7800/1000 = 7.8 g/cm³. Then convert to kg/m³ by multiplying by 1000: 7.8 × 1000 = 7800 kg/m³.

Example 2: Calculating mass from density and volume <span class="assh-badge assh-badge-foundation">🟢 Foundation</span>

Given: A block of concrete has a density of 2,400 kg/m³ and a volume of 0.5 m³. Calculate its mass.

ρ=2400 kg/m3\rho = 2400 \text{ kg/m}^3

V=0.5 m3V = 0.5 \text{ m}^3

🤔 Why this approach: The problem gives density and volume and asks for mass. Rearranging the density formula gives m = ρV.

Solution — every step shown:

  1. Write the density formula.
    ρ=mV\rho = \frac{m}{V}
  2. Rearrange to make mass the subject. Multiply both sides by V.
    m=ρVm = \rho V
  3. Substitute the values.
    m=2400×0.5m = 2400 \times 0.5
  4. Calculate the mass.
    m=1200 kgm = 1200 \text{ kg}

Why this method? This method is chosen because it directly uses the rearranged formula. An alternative is to use the density triangle: cover m, and you see ρ × V. Both give the same result.

✅ Final Answer: The mass of the concrete block is 1200 kg.

m=1200 kgm = 1200 \text{ kg}

Reality Check: Does 1200 kg make sense? A volume of 0.5 m³ is half a cubic metre. Concrete is dense (2400 kg/m³), so half a cubic metre should have a mass of about 1200 kg. This is reasonable — it is about the mass of a small car.

💡 So What? This mass (1200 kg) is the mass of the concrete block. Civil engineers use this calculation to determine the load a bridge pillar must support. If the mass is too high, the pillar may crack.

❌ Most Common Mistake: The most common mistake is dividing instead of multiplying. Students often write m = ρ/V, which is wrong. Always rearrange the formula carefully: m = ρV.

🔄 Alternative Method: Use the density triangle: cover m, and you see ρ × V. This avoids algebraic rearrangement errors.

Example 3: Calculating volume from density and mass <span class="assh-badge assh-badge-core">🔵 Core</span>

Given: A gold ring has a mass of 19.3 g and a density of 19,300 kg/m³. Calculate its volume in cm³.

m=19.3 gm = 19.3 \text{ g}

ρ=19300 kg/m3\rho = 19300 \text{ kg/m}^3

🤔 Why this approach: The problem gives mass and density and asks for volume. Rearranging the density formula gives V = m/ρ. We must convert mass to kg first to match the density unit.

Solution — every step shown:

  1. Convert mass from grams to kilograms.
    m=19.31000=0.0193 kgm = \frac{19.3}{1000} = 0.0193 \text{ kg}
  2. Write the density formula.
    ρ=mV\rho = \frac{m}{V}
  3. Rearrange to make volume the subject. Multiply both sides by V, then divide by ρ.
    V=mρV = \frac{m}{\rho}
  4. Substitute the values.
    V=0.019319300V = \frac{0.0193}{19300}
  5. Calculate the volume in m³.
    V=1×10−6 m3V = 1 \times 10^{-6} \text{ m}^3
  6. Convert volume from m³ to cm³. Multiply by 1,000,000.
    V=1×10−6×1000000=1 cm3V = 1 \times 10^{-6} \times 1000000 = 1 \text{ cm}^3

Why this method? This method is chosen because it directly uses the rearranged formula and converts units at the end. An alternative is to convert density to g/cm³ first (19.3 g/cm³) and then use V = m/ρ = 19.3/19.3 = 1 cm³. Both methods give the same answer.

✅ Final Answer: The volume of the gold ring is 1 cm³.

V=1 cm3V = 1 \text{ cm}^3

Reality Check: Does 1 cm³ make sense? Gold is very dense (19,300 kg/m³), so a small mass of 19.3 g occupies a tiny volume of 1 cm³. This is reasonable — a gold ring is small and heavy.

💡 So What? This volume (1 cm³) is the volume of the gold ring. Jewellers use this calculation to check the purity of gold. If the volume is larger than expected for the given mass, the ring may be hollow or mixed with a less dense metal.

❌ Most Common Mistake: The most common mistake is forgetting to convert mass to kilograms. If you use m = 19.3 g and ρ = 19,300 kg/m³, you get V = 19.3/19300 = 0.001 m³ = 1000 cm³, which is wrong. Always match units before substituting.

🔄 Alternative Method: Convert density to g/cm³ first: ρ = 19300/1000 = 19.3 g/cm³. Then V = m/ρ = 19.3/19.3 = 1 cm³. This avoids converting mass to kg.

Example 4: Determining if an object floats or sinks <span class="assh-badge assh-badge-core">🔵 Core</span>

Given: A plastic block has a mass of 240 g and a volume of 300 cm³. The density of water is 1,000 kg/m³. Determine whether the block will float or sink in water.

m=240 gm = 240 \text{ g}

V=300 cm3V = 300 \text{ cm}^3

ρwater=1000 kg/m3\rho_{\text{water}} = 1000 \text{ kg/m}^3

🤔 Why this approach: To determine floating or sinking, we must compare the density of the block to the density of water. We calculate the block's density using ρ = m/V and then compare.

Solution — every step shown:

  1. Convert mass to kilograms.
    m=2401000=0.24 kgm = \frac{240}{1000} = 0.24 \text{ kg}
  2. Convert volume to cubic metres.
    V=3001000000=0.0003 m3V = \frac{300}{1000000} = 0.0003 \text{ m}^3
  3. Write the density formula.
    ρ=mV\rho = \frac{m}{V}
  4. Substitute the values.
    ρ=0.240.0003\rho = \frac{0.24}{0.0003}
  5. Calculate the density of the block.
    ρ=800 kg/m3\rho = 800 \text{ kg/m}^3
  6. Compare the density of the block to the density of water.
    800 \text{ kg/m}^3 &lt; 1000 \text{ kg/m}^3

Why this method? This method is chosen because it directly applies the floating/sinking rule: if the object's density is less than the liquid's density, it floats. An alternative is to compare masses of equal volumes, but density comparison is simpler.

✅ Final Answer: The density of the block is 800 kg/m³, which is less than the density of water (1000 kg/m³). Therefore, the block will float.

\rho = 800 \text{ kg/m}^3 &lt; 1000 \text{ kg/m}^3

Reality Check: Does 800 kg/m³ make sense for plastic? Yes — many plastics have densities between 800 and 1000 kg/m³. This is why some plastics float and some sink.

💡 So What? This calculation tells us that the plastic block will float on water. This is why plastic bottles and bags float on the Buriganga River, contributing to pollution. Environmental scientists use this knowledge to design cleanup strategies.

❌ Most Common Mistake: The most common mistake is comparing mass instead of density. Students might say 'the block has a mass of 240 g, which is less than 1000 g, so it floats.' This is wrong because the comparison must be between densities, not masses.

🔄 Alternative Method: Calculate the density in g/cm³: ρ = 240/300 = 0.8 g/cm³. Since water's density is 1 g/cm³, and 0.8 < 1, the block floats. This avoids converting to SI units.

Example 5: Density of a mixture <span class="assh-badge assh-badge-advanced">🔴 Advanced</span>

Given: A container holds 2 litres of water (density 1,000 kg/m³) and 3 litres of oil (density 800 kg/m³). The oil and water do not mix. Calculate the total mass of the mixture and the average density of the mixture.

Vwater=2 L=0.002 m3V_{\text{water}} = 2 \text{ L} = 0.002 \text{ m}^3

ρwater=1000 kg/m3\rho_{\text{water}} = 1000 \text{ kg/m}^3

Voil=3 L=0.003 m3V_{\text{oil}} = 3 \text{ L} = 0.003 \text{ m}^3

ρoil=800 kg/m3\rho_{\text{oil}} = 800 \text{ kg/m}^3

🤔 Why this approach: The total mass is the sum of the masses of water and oil. The average density is the total mass divided by the total volume. This applies because density is an intensive property, but for a mixture, we must use the total mass and total volume.

Solution — every step shown:

  1. Calculate the mass of water using m = ρV.
    mwater=1000×0.002=2 kgm_{\text{water}} = 1000 \times 0.002 = 2 \text{ kg}
  2. Calculate the mass of oil using m = ρV.
    moil=800×0.003=2.4 kgm_{\text{oil}} = 800 \times 0.003 = 2.4 \text{ kg}
  3. Calculate the total mass.
    mtotal=2+2.4=4.4 kgm_{\text{total}} = 2 + 2.4 = 4.4 \text{ kg}
  4. Calculate the total volume.
    Vtotal=0.002+0.003=0.005 m3V_{\text{total}} = 0.002 + 0.003 = 0.005 \text{ m}^3
  5. Calculate the average density using ρ = m/V.
    ρaverage=4.40.005\rho_{\text{average}} = \frac{4.4}{0.005}
  6. Calculate the final answer.
    ρaverage=880 kg/m3\rho_{\text{average}} = 880 \text{ kg/m}^3

Why this method? This method is chosen because it correctly accounts for the different masses and volumes of the two liquids. An alternative is to use a weighted average formula, but calculating total mass and total volume is more intuitive and less error-prone.

✅ Final Answer: The total mass of the mixture is 4.4 kg, and the average density is 880 kg/m³.

ρaverage=880 kg/m3\rho_{\text{average}} = 880 \text{ kg/m}^3

Reality Check: Does 880 kg/m³ make sense? The mixture is mostly oil (3 L) with some water (2 L). Oil is less dense than water, so the average density should be between 800 and 1000 kg/m³, closer to 800. 880 kg/m³ is reasonable.

💡 So What? This calculation is used in the petroleum industry to determine the average density of crude oil mixtures. It is also used in food processing to calculate the density of liquid mixtures like juices and syrups.

❌ Most Common Mistake: The most common mistake is averaging the densities directly: (1000 + 800)/2 = 900 kg/m³. This is wrong because the volumes are not equal. You must calculate total mass and total volume.

🔄 Alternative Method: Use the weighted average formula: ρ_avg = (ρ1V1 + ρ2V2)/(V1 + V2). This gives the same result: (1000×0.002 + 800×0.003)/(0.002+0.003) = (2+2.4)/0.005 = 880 kg/m³.

Example 6: Designing an experiment to find the density of an irregular object <span class="assh-badge assh-badge-advanced">🔴 Advanced</span>

Given: You are given an irregularly shaped stone, a balance, a measuring cylinder, and water. Describe how you would determine the density of the stone.

🤔 Why this approach: The density formula ρ = m/V requires mass and volume. Mass is measured directly with a balance. Volume is measured by displacement because the stone is irregular and cannot be measured with a ruler.

Solution — every step shown:

  1. Measure the mass of the stone using the balance.
    m=measured massm = \text{measured mass}
  2. Fill the measuring cylinder with water to a known volume, V1.
    V1=initial water volumeV_1 = \text{initial water volume}
  3. Carefully lower the stone into the water. Ensure it is fully submerged and no water splashes out.
    stone submerged\text{stone submerged}
  4. Read the new water level, V2.
    V2=final water volumeV_2 = \text{final water volume}
  5. Calculate the volume of the stone by displacement: V = V2 - V1.
    V=V2−V1V = V_2 - V_1
  6. Calculate the density using ρ = m/V.
    ρ=mV2−V1\rho = \frac{m}{V_2 - V_1}
  7. Convert units to kg/m³ if necessary.
    ρ=mV in kg/m3\rho = \frac{m}{V} \text{ in kg/m}^3

Why this method? This method is chosen because it works for any shape. An alternative is to use a overflow can, but the measuring cylinder method is simpler and more accurate for small objects.

✅ Final Answer: The density is calculated by dividing the mass of the stone by the volume of water displaced.

ρ=mV2−V1\rho = \frac{m}{V_2 - V_1}

Reality Check: Does this method make sense? Yes — the volume of water displaced equals the volume of the stone. This is Archimedes' principle.

💡 So What? This method is used by geologists to identify rocks and minerals. It is also used in quality control to check the density of manufactured parts.

❌ Most Common Mistake: The most common mistake is not ensuring the stone is fully submerged. If part of the stone is above the water, the displaced volume is less than the stone's volume, and the calculated density will be too high.

🔄 Alternative Method: Use an overflow can: fill it to the spout, place a measuring cylinder under the spout, and lower the stone in. The water that overflows into the cylinder is the volume of the stone.

🏛 The Failure Gallery — Learn From These Mistakes

These are real answers that got 0 marks. Read each one. Can you spot why?

Question 1: A student is asked: 'A block of wood has a mass of 600 g and a volume of 1000 cm³. Calculate its density in kg/m³.'

❌ Student Answer (0 marks):

Density = 600 / 1000 = 0.6 kg/m³.

🔴 Examiner says: 0 marks. The student used grams and cubic centimetres but wrote the unit as kg/m³ without converting. The numerical value is also wrong for kg/m³. No method mark because the conversion step is missing.

✅ Full Mark Answer:

Convert mass to kg: 600 g = 0.6 kg. Convert volume to m³: 1000 cm³ = 0.001 m³. Density = 0.6 / 0.001 = 600 kg/m³.

🔑 Key Difference: The wrong answer skipped unit conversion. The correct answer explicitly converts grams to kilograms and cubic centimetres to cubic metres before dividing.

Question 2: A student is asked: 'Explain why a solid steel nail sinks in water but a steel ship floats.'

❌ Student Answer (0 marks):

The nail is heavier than the ship, so it sinks. The ship is lighter, so it floats.

🔴 Examiner says: 0 marks. The answer confuses mass with density. The ship is actually much heavier than the nail. The examiner is looking for a comparison of densities, not masses.

✅ Full Mark Answer:

The nail is solid steel, so its density is about 7800 kg/m³, which is greater than the density of water (1000 kg/m³), so it sinks. The ship is hollow and contains air, so its average density is less than the density of water, so it floats.

🔑 Key Difference: The wrong answer talks about mass. The correct answer talks about density and average density, and compares each to the density of water.

Question 3: A student is asked: 'A liquid has a density of 800 kg/m³. Will a block of density 900 kg/m³ float or sink in it? Explain.'

❌ Student Answer (0 marks):

It will float because 900 is bigger than 800, and bigger things float.

🔴 Examiner says: 0 marks. The student has the comparison backwards and the reasoning is incorrect. The examiner requires a clear statement that the object's density is greater than the liquid's density, so it sinks.

✅ Full Mark Answer:

The block will sink because its density (900 kg/m³) is greater than the density of the liquid (800 kg/m³). An object floats only if its density is less than the density of the liquid.

🔑 Key Difference: The wrong answer reverses the rule. The correct answer states the rule correctly and applies it to the numbers.

Checkpoint — Test Yourself Now ✅

1. [TrueFalse] A larger block of the same material has a greater density than a smaller block.

Show Answer

Answer: False. Density is a property of the material and does not depend on the size of the object. Both blocks have the same density.

💡 AHA Insight: Density is like a fingerprint — it identifies the material, not the object.

Need help? Revisit: Review the section 'Density is a material property' in the concept build.

2. [TrueFalse] An object will float in water if its density is greater than the density of water.

Show Answer

Answer: False. An object floats if its density is less than the density of water. If it is greater, it sinks.

💡 AHA Insight: The rule is simple: less dense than water = float; more dense = sink.

Need help? Revisit: Review the section on floating and sinking in the magic formula concepts.

3. [Derivation] Write the formula for density and rearrange it to make mass the subject.

Show Answer

Answer: Density = mass / volume. Rearranged: mass = density × volume.

ρ=mV⇒m=ρV\rho = \frac{m}{V} \quad \Rightarrow \quad m = \rho V

💡 AHA Insight: The density triangle helps you rearrange without algebra.

Need help? Revisit: Review the magic formula section.

4. [RetrievalPractice] Define density in your own words without looking at your notes.

Show Answer

Answer: Density is how much mass is packed into each unit of volume of a material. It is calculated as mass divided by volume.

💡 AHA Insight: Density tells you how tightly matter is packed.

Need help? Revisit: Review the key words section.

5. [ShortAnswer] Explain why a steel ship floats on water even though steel is denser than water.

Show Answer

Answer: The ship is hollow, so its overall density (including the air inside) is less than the density of water. The steel itself is denser than water, but the ship's total mass divided by its total volume gives a density less than 1000 kg/m³.

💡 AHA Insight: Density is about the whole object, not just the material it is made of.

Need help? Revisit: Review the real-world connections section on boat design.

6. [MCQ] A block has a mass of 500 g and a volume of 250 cm³. What is its density in kg/m³? A) 2 kg/m³ B) 200 kg/m³ C) 2000 kg/m³ D) 5000 kg/m³

Show Answer

Answer: C) 2000 kg/m³. Convert mass to kg: 0.5 kg. Convert volume to m³: 0.00025 m³. Density = 0.5 / 0.00025 = 2000 kg/m³.

ρ=0.50.00025=2000 kg/m3\rho = \frac{0.5}{0.00025} = 2000 \text{ kg/m}^3

💡 AHA Insight: Always convert to SI units before calculating density.

Need help? Revisit: Review the unit conversion section in the deep dive explanation.

💭 Quick Recall #3 (Answer without looking!)

Show Answer

Q: Explain why a steel ship floats on water even though steel is much denser than water.

A: A ship is hollow and filled with air, so its average density (total mass divided by total volume including the air spaces) is less than the density of water. Therefore it floats.

\rho_{\text{ship}} = \frac{m_{\text{steel}} + m_{\text{air}}}{V_{\text{total}}} &lt; \rho_{\text{water}}

📍 Part 5 of 8 — Exam Strategy

🎓 Think Like the Examiner

Edexcel examiners test three things: (1) AO1 — can you state the definition of density and recall the formula rho = m/V? (2) AO2 — can you apply the formula to calculate density, mass, or volume, including unit conversions? (3) AO3 — can you analyse floating and sinking in terms of density comparisons, and evaluate experimental methods for measuring density?

⚠️ Hidden Traps in This Topic

  • Trap 1: Giving you mass in grams and volume in cm³ but asking for density in kg/m³. Students who do not convert lose the accuracy mark.
  • Trap 2: Asking about a ship or a hot-air balloon, where the object is not solid. Students must use average density, not the density of the material alone.
  • Trap 3: Asking you to compare densities of two objects where one is larger but less dense. Students often assume the larger object is denser.

🔒 Mark Scheme Secrets: The mark scheme awards method marks (M) for correct substitution into the formula, even if the final answer is wrong. Accuracy marks (A) are only given if the answer is correct with the correct unit. For explanation questions, marks are given for stating the comparison (e.g., 'density of object is less than density of liquid') and for the conclusion (e.g., 'so it floats'). Vague statements like 'it is lighter' earn zero.

📉 Most Common Lost Marks: The single most common reason students lose marks is failing to convert units before calculating. Edexcel examiners see thousands of answers where the student divided grams by cubic centimetres and wrote kg/m³. Always convert first.

📝 The Exact Language That Earns Marks

✅ Phrases That EARN Marks

  • Density is mass per unit volume.
  • rho = m/V
  • The object floats because its density is less than the density of the liquid.
  • The object sinks because its density is greater than the density of the liquid.
  • Average density takes into account the air spaces inside the object.

❌ Phrases That LOSE Marks

  • It is lighter, so it floats.
  • It is heavier, so it sinks.
  • The density is big.

Side-by-Side: 0 Marks vs Full Marks

❌ 0-Mark Answer:
The ship floats because it is lighter than the water. The nail sinks because it is heavier. So density is about weight.
✅ Full-Mark Answer:
The ship floats because its average density is less than the density of water. The nail sinks because its density is greater than the density of water. Density is mass per unit volume, rho = m/V.

The zero-mark answer confuses mass with density and uses vague words like 'lighter' and 'heavier'. The full-mark answer uses the precise term 'density', compares it correctly to the density of water, and states the formula. The key difference is using the correct physical quantity and making a clear comparison.

📋 Exam-Type Questions — Board Style

AO1 1 mark ~1 min State the formula that relates density, mass and volume.

Mark Scheme

Mark Scheme Focus: Density = mass ÷ volume (or rho = m/V). Accept 'density = mass/volume' or the equation with symbols.

❌ 0-mark: Density = volume ÷ mass (this is the reciprocal — completely wrong).
✅ Full mark: density = mass ÷ volume (or rho = m/V)

AO2 2 marks ~2 min A piece of mango wood has a mass of 45 g and a volume of 60 cm³. Calculate the density of the mango wood.

Mark Scheme

Mark Scheme Focus: 1 mark for correct substitution (45/60), 1 mark for correct answer with unit (0.75 g/cm³).

❌ 0-mark: density = 60/45 = 1.33 g/cm³ (student divided volume by mass instead of mass by volume).
✅ Full mark: density = 45/60 = 0.75 g/cm³

AO2 3 marks ~3 min A block of concrete used in a Dhaka building has a density of 2400 kg/m³ and a volume of 0.5 m³. Calculate the mass of the concrete block.

Mark Scheme

Mark Scheme Focus: 1M for rearranging formula to m = rho V, 1A for correct substitution (2400 × 0.5), 1A for correct answer with unit (1200 kg).

❌ 0-mark: m = 2400/0.5 = 4800 kg (student divided density by volume instead of multiplying).
✅ Full mark: m = rho × V = 2400 × 0.5 = 1200 kg

AO3 3 marks ~3 min Describe an experiment to measure the density of a small irregular stone. Include the apparatus you would use and the measurements you would take.

Mark Scheme

Mark Scheme Focus: 1 mark for apparatus (balance and measuring cylinder), 1 mark for method of measuring mass, 1 mark for method of measuring volume by displacement.

❌ 0-mark: Use a ruler to measure the stone and calculate volume. (This is wrong because the stone is irregular — you cannot measure it with a ruler.)
✅ Full mark: Use an electronic balance to measure the mass of the stone. Fill a measuring cylinder with water and record the initial volume. Lower the stone into the water and record the new volume. The volume of the stone is the difference between the two readings. Then calculate density = mass ÷ volume.

AO3 3 marks ~3 min A student measures the mass of an irregular object as 36 g. The object is placed in a measuring cylinder containing 40 cm³ of water. The water level rises to 52 cm³. Calculate the density of the object and state whether it will float or sink in water (density of water = 1.0 g/cm³).

Mark Scheme

Mark Scheme Focus: 1M for volume by displacement (12 cm³), 1A for density (3.0 g/cm³), 1B for stating it sinks with reason (density greater than water).

❌ 0-mark: Volume = 52 cm³, density = 36/52 = 0.69 g/cm³, it floats. (Student forgot to subtract initial volume.)
✅ Full mark: Volume = 52 − 40 = 12 cm³. Density = 36/12 = 3.0 g/cm³. It will sink because its density (3.0 g/cm³) is greater than the density of water (1.0 g/cm³).

AO2 2 marks ~2 min A sample of river water has a density of 1000 kg/m³. Calculate the volume of 2500 kg of this water.

Mark Scheme

Mark Scheme Focus: 1 mark for rearranging formula to V = m/rho, 1 mark for correct answer with unit (2.5 m³).

❌ 0-mark: V = 1000 × 2500 = 2,500,000 m³ (student multiplied instead of divided).
✅ Full mark: V = m/rho = 2500/1000 = 2.5 m³

AO3 4 marks ~4 min Design an experiment to compare the densities of three different types of wood found in Bangladesh (e.g., teak, mango, and sundari). Include the materials you would use, the method you would follow, the variables you would control, and how you would ensure accuracy.

Mark Scheme

Mark Scheme Focus: 1 mark for materials (balance, measuring cylinder, water, wood samples), 1 mark for method (measure mass and volume for each), 1 mark for control variables (same size samples or same conditions), 1 mark for accuracy measure (repeat and average, or use of displacement method).

❌ 0-mark: Put the wood in water and see which one floats highest. (This is qualitative and does not measure density.)
✅ Full mark: Materials: electronic balance, measuring cylinder, water, samples of teak, mango, and sundari wood. Method: For each wood sample, measure its mass using the balance. Measure its volume by displacement in a measuring cylinder. Calculate density = mass ÷ volume for each. Control variables: use samples of similar size and ensure they are dry. Accuracy: repeat each measurement three times and calculate an average. Ensure the wood is fully submerged and no air bubbles are present.

Rubric: 1 mark: correct materials listed (balance and measuring cylinder). 1 mark: correct method for measuring mass and volume. 1 mark: control variables identified (same size, dry samples). 1 mark: accuracy measure (repeat and average, or careful technique).

AO3 3 marks ~3 min A metal block has a mass of 2.7 kg and dimensions 10 cm × 5 cm × 2 cm. (a) Calculate the volume of the block in m³. (b) Calculate the density of the block in kg/m³. (c) Suggest what metal the block might be made of.

Mark Scheme

Mark Scheme Focus: 1M for volume calculation (0.0001 m³), 1A for density (2700 kg/m³), 1B for identifying aluminium (or metal with density close to 2700 kg/m³).

❌ 0-mark: Volume = 10 × 5 × 2 = 100 cm³. Density = 2.7/100 = 0.027 kg/m³. (Student forgot to convert cm to m.)
✅ Full mark: (a) V = 0.10 × 0.05 × 0.02 = 0.0001 m³. (b) density = 2.7/0.0001 = 2700 kg/m³. (c) The block is likely made of aluminium, which has a density of about 2700 kg/m³.

AO2 3 marks ~3 min Explain why a ship made of steel can float on the Padma River even though steel is denser than water. Use the concept of average density in your answer.

Mark Scheme

Mark Scheme Focus: 1 mark for stating ship has air spaces (or is hollow), 1 mark for explaining that this lowers the average density, 1 mark for comparing average density to water density (less than 1000 kg/m³).

❌ 0-mark: The ship is big so it floats. (This does not explain density.)
✅ Full mark: A ship is not solid steel — it contains large empty spaces filled with air. This means the ship's average density (total mass ÷ total volume) is much less than the density of solid steel. The average density of the ship is less than the density of water (1000 kg/m³), so it floats.

AO2 2 marks ~2 min The table below shows the mass and volume of three liquids found in a Bangladesh kitchen. | Liquid | Mass (g) | Volume (cm³) | |--------|----------|---------------| | Water | 100 | 100 | | Oil | 92 | 100 | | Honey | 142 | 100 | Which liquid will float on water? Explain your answer.

Mark Scheme

Mark Scheme Focus: 1 mark for identifying oil, 1 mark for explaining that oil has lower density than water (0.92 g/cm³ < 1.0 g/cm³).

❌ 0-mark: Honey will float because it is thick. (Honey has higher density and sinks.)
✅ Full mark: Oil will float on water. Oil has a density of 92/100 = 0.92 g/cm³, which is less than the density of water (1.0 g/cm³). Honey has a density of 142/100 = 1.42 g/cm³, which is greater than water, so it sinks.

Rubric: 1 mark: identifies oil. 1 mark: explains using density comparison (oil density < water density).

🏋️ Exam Practice Questions

  1. Solved Easy 2 marks ~2 min A small piece of iron has a mass of 78 g and a volume of 10 cm³. Calculate the density of the iron.

    Show Solution

    Step 1: Write down the density formula: density = mass ÷ volume. Step 2: Substitute the given values: density = 78 g ÷ 10 cm³. Step 3: Calculate: density = 7.8 g/cm³.

    ρ=7810=7.8 g/cm3\rho = \frac{78}{10} = 7.8\text{ g/cm}^3

  2. Unsolved Easy 2 marks ~2 min A plastic bottle has a mass of 30 g and a volume of 50 cm³. Calculate the density of the plastic.

  3. Solved Moderate 3 marks ~3 min A block of wood has a density of 0.6 g/cm³ and a volume of 250 cm³. Calculate the mass of the wood block.

    Show Solution

    Step 1: Rearrange the density formula to make mass the subject: mass = density × volume. Step 2: Substitute the values: mass = 0.6 g/cm³ × 250 cm³. Step 3: Calculate: mass = 150 g.

    m=ρV=0.6×250=150 gm = \rho V = 0.6 \times 250 = 150\text{ g}

  4. Unsolved Moderate 3 marks ~3 min A metal cube has a side length of 4 cm and a mass of 512 g. Calculate the density of the metal in g/cm³.

  5. Solved Advanced 4 marks ~4 min An irregular piece of stone has a mass of 84 g. It is placed in a measuring cylinder containing 30 cm³ of water. The water level rises to 42 cm³. Calculate the density of the stone in g/cm³ and state whether it will float or sink in water (density of water = 1.0 g/cm³).

    Show Solution

    Step 1: Calculate the volume of the stone by displacement: volume = 42 cm³ − 30 cm³ = 12 cm³. Step 2: Write down the density formula: density = mass ÷ volume. Step 3: Substitute the values: density = 84 g ÷ 12 cm³. Step 4: Calculate: density = 7.0 g/cm³. Step 5: Compare with density of water: 7.0 g/cm³ > 1.0 g/cm³, so the stone will sink.

    V=42−30=12 cm3, ρ=8412=7.0 g/cm3V = 42 - 30 = 12\text{ cm}^3,\ \rho = \frac{84}{12} = 7.0\text{ g/cm}^3

  6. Unsolved Advanced 4 marks ~4 min A rectangular block of concrete has dimensions 30 cm × 20 cm × 10 cm and a mass of 12 kg. Calculate the density of the concrete in kg/m³. Show all unit conversions clearly.

  7. Unsolved Advanced 4 marks ~4 min A student measures the mass of an irregular object as 150 g. The object is placed in a measuring cylinder containing 50 cm³ of water. The water level rises to 75 cm³. (a) Calculate the density of the object in g/cm³. (b) The density of gold is 19.3 g/cm³. Is the object likely to be made of gold? Explain your answer.

📍 Part 6 of 8 — Practice

✏️ Practice Questions — Your Turn

  1. Easy A block of teak wood from a Sundarbans boat has a mass of 240 g and a volume of 300 cm³. Calculate the density of the teak wood in g/cm³. 2 marks
  2. Easy A lump of Padma River clay has a mass of 1.8 kg and a volume of 0.0012 m³. Calculate the density of the clay in kg/m³. 2 marks
  3. Moderate A steel bolt used in the Jamuna Bridge has a density of 7800 kg/m³. If the bolt has a volume of 0.0005 m³, calculate its mass in kg. 3 marks
  4. Moderate A rectangular brick from a Dhaka construction site has dimensions 20 cm × 10 cm × 5 cm and a mass of 2.4 kg. Calculate the density of the brick in kg/m³. 3 marks
  5. Advanced A rickshaw tyre has a mass of 850 g and a volume of 1.2 × 10³ cm³. Calculate its density in kg/m³. Show all unit conversions clearly. 4 marks
  6. Advanced An irregular stone from the Buriganga riverbed is placed in a measuring cylinder containing 50 cm³ of water. The water level rises to 78 cm³. The stone has a mass of 67.2 g. Calculate the density of the stone in g/cm³ and state whether it would float or sink in water (density of water = 1.0 g/cm³). 4 marks
  7. Linking Explain why a 50,000-tonne steel ship floats on the Padma River while a 50-gram steel nail sinks. Use the concept of density in your answer. 2 marks
  8. CrossCurricular A cylindrical water tank in a Rajshahi village has a radius of 0.5 m and a height of 2 m. It is completely filled with water (density = 1000 kg/m³). Calculate the mass of the water in the tank. Use π = 3.14. 3 marks
  9. Challenge Design a full experiment to measure the density of a small irregular piece of coal found near a Barapukuria coal mine. Include the apparatus you would use, the measurements you would take, and how you would calculate the density. 4 marks
  10. DataAnalysis A student measured the mass and volume of four objects found in a Bangladesh classroom. The results are shown below: | Object | Mass (g) | Volume (cm³) | |--------|----------|---------------| | Chalk | 12 | 8 | | Eraser | 15 | 20 | | Steel ruler | 45 | 5.8 | | Wooden pencil | 6 | 10 | (a) Calculate the density of each object in g/cm³. (b) Identify which object is most likely to float on water (density of water = 1.0 g/cm³). (c) Explain your answer to part (b) using the concept of density. 4 marks

🔑 Full Solutions

Click to reveal complete solutions

Solution 1

Formula: Density formula

ρ=mV\rho = \frac{m}{V}

Reason:

  • Density tells us how much mass is packed into each unit of volume. For a solid block, we can directly measure mass on a balance and volume by measuring dimensions.
  • The formula rho = m/V comes from the definition: density is mass per unit volume. If you double the volume of the same material, mass doubles, so the ratio m/V stays constant — this is why density is a material property.

Values: m = 240 g, V = 300 cm³

ρ=240300\rho = \frac{240}{300}

Working:

  • Write down the density formula
    ρ=mV\rho = \frac{m}{V}
  • Substitute the given values
    ρ=240 g300 cm3\rho = \frac{240\text{ g}}{300\text{ cm}^3}
  • Divide mass by volume
    ρ=0.8 g/cm3\rho = 0.8\text{ g/cm}^3

✅ Final Answer: Density = 0.8 g/cm³

ρ=0.8 g/cm3\rho = 0.8\text{ g/cm}^3

📋 Mark Scheme: 1M for correct formula rho = m/V, 1A for correct calculation 0.8 with correct unit g/cm³. If unit omitted, maximum 1 mark.

⚠️ Pitfall: Students often forget to include the unit g/cm³ in the final answer. The mark scheme requires the unit for the accuracy mark. Also, some students divide V by m instead of m by V — always check: density is mass divided by volume, so the number should be less than 1 for light materials like teak.

💭 Reflection: Teak wood has a density of 0.8 g/cm³, which is less than water (1.0 g/cm³). This is why teak floats on the Padma River. Can you predict what fraction of the teak block would be submerged if it floated?

Solution 2

Formula: Density formula

ρ=mV\rho = \frac{m}{V}

Reason:

  • Clay is a solid material. We can measure its mass directly and its volume by displacement or by shaping it into a regular form.
  • The formula applies because density is an intrinsic property — it does not depend on the size of the clay lump.

Values: m = 1.8 kg, V = 0.0012 m³

ρ=1.80.0012\rho = \frac{1.8}{0.0012}

Working:

  • Write down the density formula
    ρ=mV\rho = \frac{m}{V}
  • Substitute the given values
    ρ=1.8 kg0.0012 m3\rho = \frac{1.8\text{ kg}}{0.0012\text{ m}^3}
  • Divide mass by volume
    ρ=1500 kg/m3\rho = 1500\text{ kg/m}^3

✅ Final Answer: Density = 1500 kg/m³

ρ=1500 kg/m3\rho = 1500\text{ kg/m}^3

📋 Mark Scheme: 1M for correct formula, 1A for correct calculation 1500 with unit kg/m³. Accept 1.5 × 10³ kg/m³.

⚠️ Pitfall: Students often make errors with decimal division: 1.8 ÷ 0.0012. A common mistake is to write 1.8 ÷ 0.0012 = 1.5 instead of 1500. Always check: dividing by a small number gives a large answer. Clay is denser than water, so 1500 kg/m³ makes sense.

💭 Reflection: Clay has a density of 1500 kg/m³, which is 1.5 times the density of water. This is why clay sinks in the Padma River. If you made a clay pot with air pockets inside, would its average density increase or decrease? How would this affect floating?

Solution 3

Formula: Density formula rearranged for mass

m=ρVm = \rho V

Reason:

  • We know density and volume, and we need mass. Rearranging the density formula gives mass = density × volume.
  • Physically, if each cubic metre of steel has a mass of 7800 kg, then 0.0005 m³ will have a mass of 7800 × 0.0005 kg.

Values: rho = 7800 kg/m³, V = 0.0005 m³

m=7800×0.0005m = 7800 \times 0.0005

Working:

  • Start with the density formula
    ρ=mV\rho = \frac{m}{V}
  • Multiply both sides by V to make m the subject
    m=ρVm = \rho V
  • Substitute the given values
    m=7800 kg/m3×0.0005 m3m = 7800\text{ kg/m}^3 \times 0.0005\text{ m}^3
  • Calculate the mass
    m=3.9 kgm = 3.9\text{ kg}

✅ Final Answer: Mass = 3.9 kg

m=3.9 kgm = 3.9\text{ kg}

📋 Mark Scheme: 1M for rearranging formula to m = rho V, 1A for correct substitution, 1A for correct answer 3.9 kg with unit. If formula not rearranged but correct answer given, award 2 marks maximum.

⚠️ Pitfall: Students often forget to rearrange the formula and instead divide 7800 by 0.0005, giving 15,600,000 kg — an absurd mass for a bolt. Always check: a small volume of steel should have a small mass. 3.9 kg is reasonable for a large bolt.

💭 Reflection: The bolt has a mass of 3.9 kg. If the same bolt were made of aluminium (density = 2700 kg/m³), what would its mass be? Why do engineers choose steel over aluminium for bridge bolts even though steel is heavier?

Solution 4

Formula: Density formula with geometric volume

ρ=mV\rho = \frac{m}{V}

Reason:

  • The brick is a rectangular block, so its volume can be calculated from its dimensions: V = length × width × height.
  • Once we have the volume in m³, we can use the density formula directly.

Values: m = 2.4 kg, dimensions = 20 cm × 10 cm × 5 cm

V=0.20×0.10×0.05=0.001 m3V = 0.20 \times 0.10 \times 0.05 = 0.001\text{ m}^3

Working:

  • Convert dimensions from cm to m
    20 cm=0.20 m, 10 cm=0.10 m, 5 cm=0.05 m20\text{ cm} = 0.20\text{ m},\ 10\text{ cm} = 0.10\text{ m},\ 5\text{ cm} = 0.05\text{ m}
  • Calculate the volume of the rectangular brick
    V=0.20×0.10×0.05=0.001 m3V = 0.20 \times 0.10 \times 0.05 = 0.001\text{ m}^3
  • Write down the density formula
    ρ=mV\rho = \frac{m}{V}
  • Substitute the values
    ρ=2.4 kg0.001 m3\rho = \frac{2.4\text{ kg}}{0.001\text{ m}^3}
  • Calculate the density
    ρ=2400 kg/m3\rho = 2400\text{ kg/m}^3

✅ Final Answer: Density = 2400 kg/m³

ρ=2400 kg/m3\rho = 2400\text{ kg/m}^3

📋 Mark Scheme: 1M for correct volume calculation (0.001 m³), 1M for using density formula, 1A for correct answer 2400 kg/m³ with unit. If volume calculated in cm³ (1000 cm³) and then converted correctly, award full marks.

⚠️ Pitfall: The most common mistake is forgetting to convert cm to m before calculating volume. If students calculate volume as 20 × 10 × 5 = 1000 cm³ and then use density = 2.4/1000 = 0.0024 kg/cm³, they lose the unit mark. Always convert to SI units first: 0.20 m × 0.10 m × 0.05 m = 0.001 m³.

💭 Reflection: The brick has a density of 2400 kg/m³, which is typical for clay bricks. If this brick were placed in water, would it float or sink? What does this tell you about why bricks are used for building foundations near rivers?

Solution 5

Formula: Density formula with unit conversion

ρ=mV\rho = \frac{m}{V}

Reason:

  • The mass is given in grams and volume in cm³, but the question asks for density in kg/m³. We must convert both mass and volume to SI units before calculating.
  • This is a multi-step problem: convert mass to kg, convert volume to m³, then apply the density formula.

Values: m = 850 g = 0.850 kg, V = 1.2 × 10³ cm³ = 1.2 × 10⁻³ m³

ρ=0.8501.2×10−3\rho = \frac{0.850}{1.2 \times 10^{-3}}

Working:

  • Convert mass from g to kg
    850 g=0.850 kg850\text{ g} = 0.850\text{ kg}
  • Convert volume from cm³ to m³ (1 cm³ = 10⁻⁶ m³)
    1.2×103 cm3=1.2×103×10−6 m3=1.2×10−3 m31.2 \times 10^3\text{ cm}^3 = 1.2 \times 10^3 \times 10^{-6}\text{ m}^3 = 1.2 \times 10^{-3}\text{ m}^3
  • Write down the density formula
    ρ=mV\rho = \frac{m}{V}
  • Substitute the converted values
    ρ=0.850 kg1.2×10−3 m3\rho = \frac{0.850\text{ kg}}{1.2 \times 10^{-3}\text{ m}^3}
  • Calculate the density
    ρ=708.33... kg/m3\rho = 708.33...\text{ kg/m}^3
  • Round to 2 significant figures
    ρ=710 kg/m3\rho = 710\text{ kg/m}^3

✅ Final Answer: Density = 710 kg/m³ (to 2 s.f.)

ρ=710 kg/m3\rho = 710\text{ kg/m}^3

📋 Mark Scheme: 1M for converting mass to kg, 1M for converting volume to m³, 1M for correct substitution into density formula, 1A for correct answer 710 kg/m³ (accept 708 or 708.3). If unit conversion errors lead to wrong answer but method correct, award up to 3 marks (ecf).

⚠️ Pitfall: The most common error is converting cm³ to m³ incorrectly. Students often divide by 100 instead of 10⁶. Remember: 1 m = 100 cm, so 1 m³ = 100³ cm³ = 1,000,000 cm³. To convert cm³ to m³, divide by 10⁶. Also, some students forget to convert grams to kilograms, giving an answer 1000 times too large.

💭 Reflection: The tyre has a density of 710 kg/m³, which is less than water (1000 kg/m³). This is why a tyre can float if it is sealed and empty. But a tyre filled with air at high pressure has a higher mass — would its density increase? By how much? This is why lorry tyres are heavy.

Solution 6

Formula: Density from displacement method

ρ=mV\rho = \frac{m}{V}

Reason:

  • For an irregular object, we cannot measure volume with a ruler. Instead, we use the displacement method: the volume of water displaced equals the volume of the object.
  • Once we know the volume, we can calculate density using rho = m/V.

Values: m = 67.2 g, initial water volume = 50 cm³, final water volume = 78 cm³

V=78−50=28 cm3V = 78 - 50 = 28\text{ cm}^3

Working:

  • Calculate the volume of the stone by displacement
    V=78 cm3−50 cm3=28 cm3V = 78\text{ cm}^3 - 50\text{ cm}^3 = 28\text{ cm}^3
  • Write down the density formula
    ρ=mV\rho = \frac{m}{V}
  • Substitute the values
    ρ=67.2 g28 cm3\rho = \frac{67.2\text{ g}}{28\text{ cm}^3}
  • Calculate the density
    ρ=2.4 g/cm3\rho = 2.4\text{ g/cm}^3
  • Compare with density of water (1.0 g/cm³)
    2.4\text{ g/cm}^3 &gt; 1.0\text{ g/cm}^3

✅ Final Answer: Density = 2.4 g/cm³. The stone will sink because its density is greater than the density of water.

ρ=2.4 g/cm3\rho = 2.4\text{ g/cm}^3

📋 Mark Scheme: 1M for calculating volume by displacement (28 cm³), 1M for using density formula, 1A for correct density 2.4 g/cm³, 1B for stating it will sink with reason (density greater than water). If reason not given, award 3 marks maximum.

⚠️ Pitfall: Students often forget to subtract the initial water volume from the final volume, giving volume = 78 cm³ instead of 28 cm³. This gives density = 0.86 g/cm³, which would incorrectly suggest the stone floats. Always remember: the volume of the object is the RISE in water level, not the final reading.

💭 Reflection: The stone has a density of 2.4 g/cm³. If you found a stone with density 0.8 g/cm³, what type of rock might it be? (Hint: pumice from volcanic eruptions has low density because it contains air bubbles.) How could you modify the displacement method to measure the density of a floating object?

Solution 7

Formula: Density comparison for floating and sinking

ρ=mV\rho = \frac{m}{V}

Reason:

  • An object floats if its average density is less than the density of the liquid it is placed in. It sinks if its average density is greater.
  • A ship is not solid steel — it contains large empty spaces filled with air. This makes its average density much lower than solid steel.

Values: Ship: mass = 50,000,000 kg, volume = 100,000 m³ (including air spaces). Nail: mass = 0.050 kg, volume = 6.4 × 10⁻⁶ m³.

ρship=5×1071×105=500 kg/m3\rho_{\text{ship}} = \frac{5 \times 10^7}{1 \times 10^5} = 500\text{ kg/m}^3

Working:

  • Calculate the average density of the ship
    ρship=50,000,000 kg100,000 m3=500 kg/m3\rho_{\text{ship}} = \frac{50,000,000\text{ kg}}{100,000\text{ m}^3} = 500\text{ kg/m}^3
  • Calculate the density of the steel nail
    ρnail=0.050 kg6.4×10−6 m3=7800 kg/m3\rho_{\text{nail}} = \frac{0.050\text{ kg}}{6.4 \times 10^{-6}\text{ m}^3} = 7800\text{ kg/m}^3
  • Compare with density of water (1000 kg/m³)
    500 &lt; 1000 &lt; 7800
  • State conclusion
    Ship floats, nail sinks\text{Ship floats, nail sinks}

✅ Final Answer: The ship floats because its average density (500 kg/m³) is less than water (1000 kg/m³). The nail sinks because its density (7800 kg/m³) is greater than water.

\rho_{\text{ship}} = 500\text{ kg/m}^3 &lt; \rho_{\text{water}} = 1000\text{ kg/m}^3 &lt; \rho_{\text{nail}} = 7800\text{ kg/m}^3

📋 Mark Scheme: 1 mark for stating ship has lower average density than water (because it contains air spaces), 1 mark for stating nail has higher density than water. Accept 'ship's average density is less than water' or 'ship is hollow' for first mark.

⚠️ Pitfall: Students often say 'the ship is heavier so it sinks' or 'the nail is lighter so it floats' — this is completely wrong. Floating depends on DENSITY, not mass or weight alone. A 50,000-tonne ship floats because its density is low; a 50-gram nail sinks because its density is high. Always compare densities, not masses.

💭 Reflection: If you filled the ship's hull with water, its average density would increase to about 1000 kg/m³ and it would sink. This is exactly what happens when a ship leaks. How do submarines control their density to dive and surface?

Solution 8

Formula: Density formula with cylinder volume

ρ=mV\rho = \frac{m}{V}

Reason:

  • The tank is a cylinder, so its volume is V = pi r² h. Once we have the volume, we can find the mass of water using m = rho V.
  • This connects geometry (cylinder volume) with density (mass per unit volume).

Values: r = 0.5 m, h = 2 m, rho = 1000 kg/m³, pi = 3.14

V=3.14×(0.5)2×2=1.57 m3V = 3.14 \times (0.5)^2 \times 2 = 1.57\text{ m}^3

Working:

  • Write down the formula for the volume of a cylinder
    V=πr2hV = \pi r^2 h
  • Substitute the values
    V=3.14×(0.5 m)2×2 mV = 3.14 \times (0.5\text{ m})^2 \times 2\text{ m}
  • Calculate the volume
    V=3.14×0.25×2=1.57 m3V = 3.14 \times 0.25 \times 2 = 1.57\text{ m}^3
  • Rearrange density formula for mass
    m=ρVm = \rho V
  • Substitute density and volume
    m=1000 kg/m3×1.57 m3m = 1000\text{ kg/m}^3 \times 1.57\text{ m}^3
  • Calculate the mass
    m=1570 kgm = 1570\text{ kg}

✅ Final Answer: Mass of water = 1570 kg

m=1570 kgm = 1570\text{ kg}

📋 Mark Scheme: 1M for correct cylinder volume formula, 1M for correct volume calculation (1.57 m³), 1A for correct mass 1570 kg with unit. If pi taken as 3.14, accept 1570 kg. If pi taken as 3.142, accept 1571 kg.

⚠️ Pitfall: Students often forget to square the radius. They calculate V = pi × r × h = 3.14 × 0.5 × 2 = 3.14 m³ instead of 1.57 m³. Remember: the formula is V = pi r² h, not pi r h. Also, some students use diameter instead of radius — always check whether the question gives radius or diameter.

💭 Reflection: The tank holds 1570 kg of water — that's over 1.5 tonnes! This is why water tanks must be strongly supported. If the tank were twice as tall (h = 4 m), would the mass double? What does this tell you about how density remains constant but mass scales with volume?

Solution 9

Formula: Experiment design for density measurement

ρ=mV\rho = \frac{m}{V}

Reason:

  • To measure density, we need to measure mass and volume independently. For an irregular solid like coal, we use a balance for mass and displacement for volume.
  • The experiment must be designed to minimize errors: use a sensitive balance, read the measuring cylinder at eye level, and ensure the coal is fully submerged without air bubbles.

Values: Apparatus: electronic balance, measuring cylinder (100 cm³), water, string, coal sample

ρ=mcoalVfinal−Vinitial\rho = \frac{m_{\text{coal}}}{V_{\text{final}} - V_{\text{initial}}}

Working:

  • Measure the mass of the coal sample using an electronic balance
    m=balance readingm = \text{balance reading}
  • Fill the measuring cylinder with water and record the initial volume
    V1=initial readingV_1 = \text{initial reading}
  • Tie the coal with a string and lower it gently into the water until fully submerged
    Ensure no air bubbles\text{Ensure no air bubbles}
  • Record the new water level
    V2=final readingV_2 = \text{final reading}
  • Calculate the volume of the coal
    V=V2−V1V = V_2 - V_1
  • Calculate the density
    ρ=mV\rho = \frac{m}{V}

✅ Final Answer: Density = mass of coal ÷ (final water level − initial water level). Repeat 3 times and average for accuracy.

ρ=mV2−V1\rho = \frac{m}{V_2 - V_1}

📋 Mark Scheme: 1 mark for correct apparatus (balance, measuring cylinder, water), 1 mark for method of measuring mass, 1 mark for method of measuring volume by displacement, 1 mark for accuracy measure (repeat and average, or use of string to avoid splashing).

⚠️ Pitfall: Students often forget to mention that the coal must be fully submerged and free of air bubbles. If air bubbles stick to the coal, the volume reading will be too high, making the density too low. Also, some students suggest using a ruler to measure volume — this is impossible for an irregular object. Always use displacement.

💭 Reflection: Coal from Barapukuria has a density of about 1300–1400 kg/m³. If your experiment gives a density of 800 kg/m³, what could have gone wrong? (Hint: air bubbles or incomplete submersion.) How could you improve the experiment to get a more accurate result?

Solution 10

Formula: Density calculation and comparison

ρ=mV\rho = \frac{m}{V}

Reason:

  • For each object, we apply the density formula rho = m/V. Then we compare the calculated densities with the density of water (1.0 g/cm³) to determine floating or sinking.
  • An object floats if its density is less than 1.0 g/cm³ and sinks if greater.

Values: Chalk: m = 12 g, V = 8 cm³. Eraser: m = 15 g, V = 20 cm³. Steel ruler: m = 45 g, V = 5.8 cm³. Wooden pencil: m = 6 g, V = 10 cm³.

ρchalk=128, ρeraser=1520, ρruler=455.8, ρpencil=610\rho_{\text{chalk}} = \frac{12}{8},\ \rho_{\text{eraser}} = \frac{15}{20},\ \rho_{\text{ruler}} = \frac{45}{5.8},\ \rho_{\text{pencil}} = \frac{6}{10}

Working:

  • Calculate density of chalk
    ρchalk=12 g8 cm3=1.5 g/cm3\rho_{\text{chalk}} = \frac{12\text{ g}}{8\text{ cm}^3} = 1.5\text{ g/cm}^3
  • Calculate density of eraser
    ρeraser=15 g20 cm3=0.75 g/cm3\rho_{\text{eraser}} = \frac{15\text{ g}}{20\text{ cm}^3} = 0.75\text{ g/cm}^3
  • Calculate density of steel ruler
    ρruler=45 g5.8 cm3=7.76 g/cm3\rho_{\text{ruler}} = \frac{45\text{ g}}{5.8\text{ cm}^3} = 7.76\text{ g/cm}^3
  • Calculate density of wooden pencil
    ρpencil=6 g10 cm3=0.6 g/cm3\rho_{\text{pencil}} = \frac{6\text{ g}}{10\text{ cm}^3} = 0.6\text{ g/cm}^3
  • Compare with density of water (1.0 g/cm³)
    0.6 &lt; 0.75 &lt; 1.0 &lt; 1.5 &lt; 7.76
  • Identify floating objects
    Eraser and pencil float; chalk and ruler sink\text{Eraser and pencil float; chalk and ruler sink}

✅ Final Answer: (a) Chalk: 1.5 g/cm³, Eraser: 0.75 g/cm³, Steel ruler: 7.76 g/cm³, Wooden pencil: 0.6 g/cm³. (b) The eraser and wooden pencil will float. (c) They float because their densities (0.75 and 0.6 g/cm³) are less than the density of water (1.0 g/cm³).

ρchalk=1.5 g/cm3, ρeraser=0.75 g/cm3, ρruler=7.76 g/cm3, ρpencil=0.6 g/cm3\rho_{\text{chalk}} = 1.5\text{ g/cm}^3,\ \rho_{\text{eraser}} = 0.75\text{ g/cm}^3,\ \rho_{\text{ruler}} = 7.76\text{ g/cm}^3,\ \rho_{\text{pencil}} = 0.6\text{ g/cm}^3

📋 Mark Scheme: 1 mark for all four density calculations correct, 1 mark for identifying eraser and pencil as floating, 1 mark for explanation comparing density to water, 1 mark for correct use of units throughout. If one calculation error, award 3 marks maximum (ecf).

⚠️ Pitfall: Students often confuse mass and density. They might say 'the pencil is light so it floats' — but the eraser is heavier (15 g) than the pencil (6 g) and still floats. The correct reason is density, not mass. Also, some students forget to compare with the density of water and just guess which objects float.

💭 Reflection: The steel ruler has a density of 7.76 g/cm³, which is close to the density of pure iron (7.87 g/cm³). This suggests the ruler is made of steel (iron with a small amount of carbon). If you found a metal object with density 2.7 g/cm³, what metal might it be? (Hint: aluminium has density 2.7 g/cm³.)

⏱ Timed Practice Set — 10 minutes, 10 Marks

  1. Q1 [2 marks]: A piece of wood has a mass of 36 g and a volume of 45 cm³. Calculate the density of the wood in g/cm³.
  2. Q2 [2 marks]: A metal block has a density of 2700 kg/m³ and a volume of 0.002 m³. Calculate the mass of the block in kg.
  3. Q3 [3 marks]: A rectangular brick has dimensions 15 cm × 8 cm × 4 cm and a mass of 1.44 kg. Calculate the density of the brick in kg/m³. Show all unit conversions.
  4. Q4 [3 marks]: Explain why a steel ship floats on water while a steel nail sinks. Use the concept of average density in your answer.

📊 Self-Assessment:

9-10: Outstanding — you have fully mastered this topic and are exam-ready! 7-8: Excellent — minor gaps, revisit one example and retry. 5-6: Good progress — review worked examples 3-5 and retry. Below 5: Return to the concept building section and worked examples before attempting again.

📍 Part 7 of 8 — Lock It In Memory

🏆 What You Now Know That You Didn't Before

Today you discovered that a single number — density — decides whether a 50,000-tonne ship floats or a 50-gram nail sinks. You started with a mystery on the Padma River and ended with a formula, rho = m/V, that unlocks floating and sinking for any object in any liquid. You learned that density is a material property, not a size property, and that unit conversion is your secret weapon in exams. You are now the person who can explain why a steel ship floats while a steel nail sinks — and that is real physics.

The Key Formula:

ρ=mV\rho = \frac{m}{V}

📌 Exam Tip: In Edexcel exams, always write the formula, substitute the values with units, then give the answer with the correct unit. If units are not in kg and m³, convert them first. This habit alone can gain you 2 extra marks per calculation question.

🎉 You can now calculate density from mass and volume, convert between g/cm³ and kg/m³, and explain floating and sinking using density comparisons. You have mastered a fundamental concept that will appear in every future topic on materials, pressure, and forces.

🗺 Memory Map — Your Visual Revision Tool

Density = mass/volume (rho = m/V)

Definition — Density is mass per unit volume — how tightly packed the matter is.

Connects to: Formula, Applications

Formula — rho = m/V; m = rho × V; V = m/rho

Connects to: Units, Method

Units — SI unit: kg/m³. 1 g/cm³ = 1000 kg/m³.

Connects to: Formula, Exam Trap

Method — Measure mass with a balance; measure volume by displacement or dimensions.

Connects to: Applications

Applications — Floating/sinking, ship design, hot-air balloons, material selection.

Connects to: Definition

Exam Trap — Forgetting to convert units before calculating.

Connects to: Units

📅 Your Spaced Repetition Schedule

Science shows: reviewing at these intervals locks this lesson in permanent memory.

  1. 🌙 Tonight (within 2 hours): Redo Examples 1 and 2 from memory. Cover the solution. Write every step yourself, including unit conversions.
  2. ☀️ Tomorrow Morning: Write the formula rho = m/V and the three key terms (density, mass, volume) without looking. Check against your notes.
  3. 📆 Day 3: Attempt Practice Questions 1–4 without looking at the examples. Check your solutions and mark your own work.
  4. 📆 1 Week: Complete all 10 Exam-type Questions under timed conditions (30 minutes). Mark your own work using the mark scheme language.
  5. 📆 3 Weeks (Exam Prep): Complete the Timed Practice Set again. Any score below 8/10 — revisit Worked Examples 3 and 4 and the misconception busters.

⚡ Quick Summary

  • Density is mass per unit volume — how tightly packed the matter is.
  • rho = m/V, where rho is density (kg/m³), m is mass (kg), and V is volume (m³).
  • In exams, always convert units to kg and m³ before calculating, and compare densities to decide floating or sinking.
  • Next, you will use density to understand pressure in liquids and gases.

📅 Review after 1 day, 3 days, 1 week, and 1 month using the Spaced Repetition section above.

🛤 Your Personalized Learning Path

🟢 Foundational: If you struggled, redo Examples 1 and 2 from the lesson. Focus on the formula triangle and simple calculations where units are already in kg and m³. Then try the Foundation-level practice questions. Re-read the misconception busters on heavy vs light and shape.

🔵 Core: Work through Examples 3 and 4, then attempt Practice Questions 1–6. Check your answers against the solutions, paying attention to unit conversions. Then write your own explanation of why a ship floats.

🔴 Advanced: Research 'relative density' and how it is used in hydrometers. Solve this problem: A block of wood of density 600 kg/m³ floats in water. What fraction of its volume is submerged? (Hint: use the principle of flotation.)

📍 Part 8 of 8 — Final Activities

👨‍👩‍👧 For Parents

Tonight, ask your child: Ask your child: 'If you have a block of unknown metal, how would you find its density using only a ruler, a weighing scale, and a measuring cylinder?'

A good answer includes: A correct answer should describe measuring mass with the scale, measuring volume by displacement in the measuring cylinder, and then calculating density = mass/volume. They should mention converting units to kg/m³.

Keep Going — You're Doing Brilliantly! 🚀

You have just mastered one of the most powerful ideas in physics: density. You can now explain why a massive ship floats on the Padma while a tiny nail sinks, and you can calculate density with confidence. That is a huge achievement — take a moment to feel proud of it.

Next, you will explore how density explains pressure in liquids and gases. You will see how the same formula helps you understand why your ears pop underwater and why the atmosphere presses on you every second. The journey is just beginning.

  • 🗺 Reflection: Draw a concept map of this lesson using the Memory Map section — include every branch.
  • 🗣 Teach Someone: Explain to a family member why a steel ship floats but a steel nail sinks. Your explanation must include the words 'density', 'average density', and 'comparison with water'.
  • 🔍 Research: Search online for 'why does ice float on water' and find out how the density of ice compares to liquid water. Write a short paragraph explaining why this is unusual for a solid.
  • 🔬 Mini Project: Collect three small objects (e.g., a coin, a piece of fruit, a small toy). Measure their mass using a kitchen scale. Measure their volume by displacement in a measuring jug. Calculate the density of each and predict whether it will float in water. Test your predictions.
  • 💬 Feedback: Which part of this lesson felt hardest — the formula, the unit conversions, or the floating/sinking explanation? What is one thing you will review before the next lesson?
  • 👨‍👩‍👧 Parent Tip: Ask your child: 'Can you explain why a heavy ship floats but a small nail sinks?' Let them teach you. Then ask: 'What is the formula for density?' and 'What units do we use?'
  • 🌱 Growth Mindset: If you found unit conversion tricky, that is completely normal — it is the most common mistake in this topic. Every physicist has made it. The fix is simple: always write down the units before you calculate. With practice, it becomes automatic.
  • 🚀 What Comes Next: In the next lesson, you will learn about pressure in liquids. Density is the key: pressure increases with depth because the weight of the liquid above depends on its density. Without today's lesson, you could not understand why deep-sea fish have special adaptations.

Recommended Resources

  • Edexcel IGCSE Physics textbook, Chapter 5: Density and Pressure, pages 78–85. This chapter covers the formula, units, and practical methods with exam-style questions.
  • BBC Bitesize Physics: search 'Density and floating' — interactive diagrams and a quiz to test your understanding.
  • YouTube: search 'Edexcel IGCSE density explained' — watch the first two videos for clear worked examples and a demonstration of the displacement method.

👩‍🏫 Teacher's Guide

Topic-Specific Misconceptions to Address

  • Heavy objects always sink, light objects always float.: Show a large piece of wood (heavy) floating and a small metal nail (light) sinking. Ask students to explain. Introduce the idea of average density using a sealed empty bottle vs a bottle filled with water.
  • Density changes when you change the shape of an object.: Give students a lump of clay. Ask them to calculate its density as a ball, then as a flat sheet. They will find it is the same. Discuss why mass and volume change together.
  • You can mix units in the density formula.: Provide a calculation with mass in grams and volume in cm³, asking for density in kg/m³. Let students make the mistake, then show the correct conversion. Emphasise the 'convert first' rule.

Pedagogical Tips

  • Start with the ship and nail mystery. Show a video clip of a ship on the Padma River and a nail sinking in a glass. Ask: 'What is different?' Let students hypothesise before revealing density.
  • Students disengage when unit conversion is introduced. Prevent this by making it a game: 'Unit Conversion Challenge' — give them 30 seconds to convert 5 values. Celebrate speed and accuracy.
  • For mixed-ability classes, provide a formula triangle template for weaker students and extension questions on average density for stronger students. Pair them for the mini-project.
  • Use mini-whiteboards for a quick formative check: 'Write the formula for density.' 'What is the density of water in kg/m³?' 'If an object has density 1200 kg/m³, will it float in water?'

Differentiation

For foundation students, focus on the formula triangle and simple calculations with units already in kg and m³. For core students, include unit conversions and floating/sinking explanations. For advanced students, introduce average density of composite objects and the concept of relative density.

Assessment Strategies

Formative: mini-whiteboard quizzes, think-pair-share on floating/sinking, and a quick 3-question exit ticket. Summative: a 20-minute test with one calculation, one unit conversion, and one explanation question. Use the mark scheme language from this lesson.

Formative Checkpoints

  • After conceptBuild: 'What is density in your own words?'
  • After magicFormula: 'What are the units of density in the SI system?'
  • After exampleProblems: 'Why does a steel ship float but a steel nail sinks?'
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