Density sinking and floating

Overview

Density helps learners compare how much mass is packed into a given volume. Two objects may have the same size but different masses, or the same mass but different volumes. Density gives a fair comparison because it connects mass and volume in one physical quantity.

This chapter uses density to explain sinking and floating. A stone usually sinks in water because its density is greater than the density of water. A dry wooden block usually floats because its density is less than the density of water. A hollow object may float even if the solid material in it is dense, because the object's average density can be reduced by the space inside it.

The main idea is simple: floating and sinking depend on the relationship between the density of the object and the density of the fluid around it.

Prerequisites

• Concept of Physics: matter, mass, energy, observation, and measurement.
• Physical quantities and SI units: physical quantities, SI units, derived quantities, and unit checking.
• Measuring instruments in Physics: balances, measuring cylinders, rulers, and careful reading of scales.
• Basic arithmetic with multiplication and division.
• Familiarity with volume of regular shapes and volume readings in $\text{cm}^3$ or $\text{m}^3$.

Learning Scope

This page covers:

• Meaning of density.
• Relationship between density, mass, and volume.
• Units of density and simple unit conversion.
• Measuring density of regular solids, irregular solids, and liquids.
• Using density to explain floating, sinking, and suspension.
• Everyday examples involving water, oil, wood, metal, stones, and hollow objects.

This page does not teach the full theory of pressure, upthrust, or Archimedes' principle. Those ideas may support later explanations, but this Form I page keeps the main focus on density, sinking, and floating as stated in the official syllabus topic.

The 2022 examination format is not used here to define topic scope. It may later provide assessment signals only after review.

Subtopics

Density

Density is the mass per unit volume of a substance or object.

$$ \text{density} = \frac{\text{mass}}{\text{volume}} $$

Using symbols:

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

where $\rho$ is density, $m$ is mass, and $V$ is volume.

Key insight: density compares mass fairly by asking how much mass is found in each unit of volume. A large object is not automatically more dense than a small object. Density depends on both mass and volume.

For example, a small metal nut may be denser than a large piece of dry wood because the metal has more mass packed into each cubic centimetre.

Mass, Volume, And Density

Mass is the amount of matter in an object. Volume is the space occupied by an object. Density connects the two.

If two objects have the same volume, the one with greater mass has greater density.

If two objects have the same mass, the one with smaller volume has greater density.

Example:

• Object A has mass $100\ \text{g}$ and volume $50\ \text{cm}^3$.
• Object B has mass $100\ \text{g}$ and volume $200\ \text{cm}^3$.

Object A is denser because the same mass is packed into a smaller volume.

Units Of Density

The SI unit of density is kilogram per cubic metre, written as $\text{kg/m}^3$.

In school laboratory work, density is also often written in grams per cubic centimetre, $\text{g/cm}^3$, especially when mass is measured in grams and volume in cubic centimetres.

Examples:

$$ \begin{aligned} \rho &= \frac{600\ \text{kg}}{2\ \text{m}^3} \\ &= 300\ \text{kg/m}^3 \end{aligned} $$

and:

$$ \begin{aligned} \rho &= \frac{80\ \text{g}}{40\ \text{cm}^3} \\ &= 2\ \text{g/cm}^3 \end{aligned} $$

Key insight: use consistent units. If mass is in grams and volume is in $\text{cm}^3$, density will be in $\text{g/cm}^3$. If mass is in kilograms and volume is in $\text{m}^3$, density will be in $\text{kg/m}^3$.

Finding Density Of A Regular Solid

A regular solid has a shape whose volume can be found by measurement and calculation. Examples include a cube, cuboid, or cylinder.

For a rectangular block:

1. Measure the mass using a balance.
2. Measure length, width, and height using a ruler or metre rule.
3. Calculate volume:

$$ V = l \times w \times h $$

1. Calculate density:

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

Key insight: the mass and volume must describe the same object. Do not measure the mass of one block and the volume of another.

Finding Density Of An Irregular Solid

An irregular solid does not have a simple shape for direct volume calculation. Its volume can be found by water displacement if the object does not dissolve in water and does not absorb water quickly.

Method:

1. Measure the mass of the object using a balance.
2. Put some water in a measuring cylinder and record the initial volume.
3. Lower the object fully into the water.
4. Record the final volume.
5. Find the object's volume:

$$ \text{volume of object} = \text{final water reading} - \text{initial water reading} $$

1. Calculate density:

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

Key insight: water displacement measures the volume of space the object occupies. Read the measuring cylinder at eye level to reduce error.

Finding Density Of A Liquid

To find the density of a liquid:

1. Measure the mass of an empty container.
2. Add a known volume of liquid using a measuring cylinder.
3. Measure the mass of the container plus liquid.
4. Find the mass of the liquid:

$$ \text{mass of liquid} = \text{mass of container and liquid} - \text{mass of empty container} $$

1. Calculate density:

$$ \rho = \frac{\text{mass of liquid}}{\text{volume of liquid}} $$

Key insight: the container's mass must be subtracted. Density of the liquid should use only the mass of the liquid, not the mass of the container.

Floating

An object floats in a fluid when it remains at or near the surface without completely sinking.

For this topic, the density rule is:

• If the average density of an object is less than the density of the fluid, the object floats.

Dry wood often floats in water because its density is less than the density of water. A closed empty plastic bottle floats because the air inside reduces the average density of the bottle and air together.

Key insight: floating depends on average density of the whole object, not only the density of the material on the outside.

Sinking

An object sinks when it moves down through a fluid and settles below the surface.

For this topic, the density rule is:

• If the average density of an object is greater than the density of the fluid, the object sinks.

A stone usually sinks in water because its density is greater than the density of water. A solid metal nail also sinks in water for the same reason.

Key insight: a small object can sink and a large object can float. Size alone does not decide floating or sinking.

Suspension In A Fluid

An object may remain suspended in a fluid if its density is about equal to the density of the fluid. It neither rises strongly nor sinks strongly.

For this topic, the density rule is:

• If the average density of an object is nearly equal to the density of the fluid, the object may remain suspended.

This idea helps learners see that floating and sinking are not only about "heavy" and "light". They are about density comparison.

Comparing Fluids

An object may float in one liquid and sink in another if the liquids have different densities.

For example, suppose a small object has density between the density of cooking oil and the density of water. It may sink in oil but float in water.

Key insight: the same object can behave differently in different fluids because the comparison changes.

Hollow Objects And Average Density

A hollow object contains empty space or air. Its average density is found by considering the mass and total outer volume of the whole object.

A metal bowl may float if it traps air and has a large total volume compared with its mass. If water enters the bowl, its mass increases and its average density may become greater than the density of water, so it sinks.

Key insight: changing mass or changing volume can change density. A hollow object can float not because metal has become less dense, but because the object plus trapped air has a lower average density.

Density And Energy In Everyday Situations

Floating and sinking can involve energy changes when objects move in a fluid. A sinking stone loses height and can transfer energy to the water as it moves. A floating object may need work to be pushed fully under water. This page does not develop full energy calculations, but learners should notice that density, force, and energy are connected in practical situations.

For example, pushing a floating empty bottle underwater requires a force through a distance, so work is done. When released, the bottle rises because the fluid interaction pushes it upward.

Key Terms

• Density: mass per unit volume of a substance or object.
• Mass: amount of matter in an object.
• Volume: space occupied by an object.
• Fluid: a substance that can flow, such as a liquid or gas.
• Floating: remaining at or near the surface of a fluid without sinking completely.
• Sinking: moving downward in a fluid because the object's average density is greater than the fluid density.
• Suspension: remaining within a fluid when the object's density is nearly equal to the fluid density.
• Average density: density of a whole object found from its total mass and total volume.
• Water displacement: a method of finding volume by measuring how much water level rises when an object is immersed.

Worked Examples

Example 1: Calculate density of a regular block

A wooden block has mass $120\ \text{g}$, length $10\ \text{cm}$, width $4\ \text{cm}$, and height $3\ \text{cm}$. Find its density.

First find the volume:

$$ \begin{aligned} V &= l \times w \times h \\ &= 10\ \text{cm} \times 4\ \text{cm} \times 3\ \text{cm} \\ &= 120\ \text{cm}^3 \end{aligned} $$

Then find the density:

$$ \begin{aligned} \rho &= \frac{m}{V} \\ &= \frac{120\ \text{g}}{120\ \text{cm}^3} \\ &= 1.0\ \text{g/cm}^3 \end{aligned} $$

The density of the block is $1.0\ \text{g/cm}^3$.

Example 2: Find volume by displacement and calculate density

A stone has mass $75\ \text{g}$. A measuring cylinder reads $40\ \text{cm}^3$ before the stone is added and $65\ \text{cm}^3$ after the stone is fully immersed. Find the density of the stone.

Find the volume of the stone:

$$ \begin{aligned} V &= 65\ \text{cm}^3 - 40\ \text{cm}^3 \\ &= 25\ \text{cm}^3 \end{aligned} $$

Find the density:

$$ \begin{aligned} \rho &= \frac{75\ \text{g}}{25\ \text{cm}^3} \\ &= 3.0\ \text{g/cm}^3 \end{aligned} $$

The density of the stone is $3.0\ \text{g/cm}^3$.

Example 3: Predict floating or sinking

An object has density $0.80\ \text{g/cm}^3$. Water has density about $1.00\ \text{g/cm}^3$ in this school-level comparison. Predict whether the object floats or sinks in water.

Compare the densities:

$$ 0.80\ \text{g/cm}^3 < 1.00\ \text{g/cm}^3 $$

The object's density is less than the density of water, so the object floats.

Example 4: Find the density of a liquid

An empty cup has mass $50\ \text{g}$. When $100\ \text{cm}^3$ of a liquid is poured into it, the total mass becomes $130\ \text{g}$. Find the density of the liquid.

Find the mass of the liquid:

$$ \begin{aligned} m &= 130\ \text{g} - 50\ \text{g} \\ &= 80\ \text{g} \end{aligned} $$

Find the density:

$$ \begin{aligned} \rho &= \frac{80\ \text{g}}{100\ \text{cm}^3} \\ &= 0.80\ \text{g/cm}^3 \end{aligned} $$

The liquid has density $0.80\ \text{g/cm}^3$.

Common Mistakes

• Mistake: Saying that heavier objects always sink.

Correction: Floating and sinking depend on density, not mass alone. A large wooden log may float while a small metal nail sinks.

• Mistake: Forgetting to subtract the initial water reading in displacement.

Correction: The object's volume is the rise in water level, not the final reading alone.

• Mistake: Using mass of container plus liquid as the mass of the liquid.

Correction: Subtract the mass of the empty container first.

• Mistake: Mixing units without care.

Correction: Keep mass and volume units consistent before calculating density.

• Mistake: Thinking a hollow metal object floats because metal is less dense than water.

Correction: The hollow object's average density can be less than water because it includes trapped air.

Practice Tasks

1. Define density in words.
2. Write the formula connecting density, mass, and volume.
3. State the SI unit of density.
4. A block has mass $60\ \text{g}$ and volume $20\ \text{cm}^3$. Calculate its density.
5. A liquid has mass $150\ \text{g}$ and volume $200\ \text{cm}^3$. Calculate its density.
6. A stone raises water in a measuring cylinder from $30\ \text{cm}^3$ to $48\ \text{cm}^3$. What is the volume of the stone?
7. An object has density $1.5\ \text{g/cm}^3$. Water has density about $1.0\ \text{g/cm}^3$. Predict whether the object floats or sinks in water.
8. Explain why a sealed empty plastic bottle floats but may sink after it is filled with water.
9. Two objects have the same mass. Object A has volume $10\ \text{cm}^3$ and Object B has volume $50\ \text{cm}^3$. Which object has greater density? Explain.
10. Describe a simple method for finding the density of an irregular stone using a balance and measuring cylinder.

Generated Question Layer

Future generated practice can include:

• Direct recall questions on density, mass, volume, floating, sinking, and suspension.
• Calculation questions using $\rho = \frac{m}{V}$.
• Practical-method questions on measuring density of solids and liquids.
• Comparison questions using density of object and density of fluid.
• Error-spotting questions involving wrong units, missing subtraction, and confused mass-volume reasoning.
• Short explanation questions on hollow objects and average density.

Generated questions should remain original and should not be presented as official past-paper items unless separately reviewed.

Learner Aid Opportunities

• diagram: Show water displacement for an irregular solid in a measuring cylinder.
• diagram: Show floating, suspended, and sinking objects in the same fluid.
• chart: Compare mass, volume, density, and floating behaviour for simple objects.
• interactive: Let learners change mass and volume to see how density changes.
• LLM tutor: Give hints that distinguish mass, volume, density, and average density.

Exam-Derived Signals

• No past-paper mappings have been reviewed for this topic in this milestone.
• The 2022 CSEE Physics examination format is assessment guidance only. It may later support practice style, command words, and practical weighting, but it does not define this 2023 syllabus topic scope.
• Any future exam links should separate density calculation signals from broader pressure, force, and practical-work signals.

Source And Review Notes

• Official syllabus status: extracted from the 2023 Physics syllabus.
• Learner expansion status: original chapter text written from the official syllabus topic and existing repo context.
• Exam signal status: not mapped in this milestone.
• External enrichment status: no external web enrichment used.
• Textbook status: not used in this expansion.
• Review risk: density, floating, and sinking are kept at Form I level; later treatment of upthrust or Archimedes' principle should be linked only when the syllabus sequence supports it.