Internal Structure and Density: Study Guide (Physics 1 8.1)

This guide is for the algebra-based Physics 1 course. Unit 8 applies the forces, energy and momentum ideas from earlier units to fluids. This first topic sets up the vocabulary: what a fluid is, and how to describe it with one number, its density.

Why solids, liquids and gases behave differently

All matter is made of particles (atoms and molecules) that interact with each other. Two things decide how a material behaves:

• how strongly neighbouring particles attract or repel each other, and
• how far apart the particles are, on average.

Compare the three everyday states:

State	Particle picture	Shape	Volume	Easy to squeeze?
Solid	Strong interactions hold particles in fixed places. They vibrate about those places.	fixed	fixed	no
Liquid	Particles are close together and still interact strongly, but they can slide past each other.	not fixed	fixed	hardly at all
Gas	Particles are far apart compared with their size. They interact only briefly, when they collide.	not fixed	not fixed	yes

This particle picture explains things you can see. A liquid keeps its volume because its particles are already touching: pushing them closer is very hard. A gas is easy to compress because most of its volume is empty space between particles.

What counts as a fluid

A fluid is a substance that has no fixed shape. If you push sideways on part of it, it flows. Pour it into a new container and it takes that container’s shape.

• Liquids are fluids. Water, oil and honey all flow.
• Gases are fluids too. Air flows round a moving car and fills any room it is in.
• Solids are not fluids. A steel block keeps its shape on any surface.

So “fluid” does not mean “liquid”. In this unit, everything you learn about fluids applies to air as well as to water, unless a model says otherwise.

Density: mass per unit volume

Two samples of the same fluid can have very different masses: a bucket of water has more mass than a cup of it. To describe the material rather than the sample, divide mass by volume. This ratio is the density:

ρ = m / V

• ρ (the Greek letter rho) is density, in kg/m³.
• m is mass, in kg.
• V is volume, in m³.

Density tells you how much mass is packed into each cubic metre. It is a property of the material. Cut a block in half and both m and V halve, so ρ stays the same.

Converting units

Labs often measure in grams and cubic centimetres. Since 1 g = 10⁻³ kg and 1 cm³ = 10⁻⁶ m³:

1 g/cm³ = 10⁻³ kg ÷ 10⁻⁶ m³ = 1000 kg/m³

Also useful: 1 mL = 1 cm³, and 1 L = 1000 cm³ = 10⁻³ m³.

Some real densities (approximate, at room conditions)

Material	Density (kg/m³)
air (at sea level)	about 1.2
cork	about 240
ice	about 920
fresh water	about 1000
aluminium	about 2700
iron	about 7900
gold	about 19,300

Gases are roughly a thousand times less dense than liquids and solids. That matches the particle picture: gas particles are far apart.

Mass, volume, weight, size and density are different words

Physics answers lose marks when these words are swapped. Keep them apart:

Word	Meaning	Unit
mass, m	how much matter; resistance to acceleration	kg
volume, V	how much space the object takes up	m³
weight, mg	the gravitational force on the object	N
density, ρ	mass per unit volume of the material	kg/m³
“size”	vague: say volume, length or area instead	—

A 2.0 kg block has weight 2.0 × 9.8 = 19.6 N on Earth. Its density depends on its volume too, so the mass alone cannot tell you the density.

Worked example 1: identifying a metal from its density

Question. A solid metal cylinder has diameter 2.00 cm and length 5.00 cm. Its mass is 42.4 g. Find its density in kg/m³ and suggest what metal it could be.

1. Radius: r = 2.00 cm ÷ 2 = 1.00 cm = 0.0100 m. Length: L = 0.0500 m.
2. Volume of a cylinder: V = πr²L = π × (0.0100 m)² × 0.0500 m = 1.57 × 10⁻⁵ m³.
3. Mass in kg: m = 42.4 g = 0.0424 kg.
4. Density: ρ = m / V = 0.0424 kg ÷ 1.57 × 10⁻⁵ m³ = 2.7 × 10³ kg/m³ (2699 kg/m³ before rounding).
5. Compare with the table: this matches aluminium (about 2700 kg/m³).

Check. Work in grams and cm³ instead: V = π × 1.00² × 5.00 = 15.7 cm³, and 42.4 g ÷ 15.7 cm³ = 2.70 g/cm³. Multiply by 1000 to get 2700 kg/m³. Both routes agree.

Mass–volume graphs

For pieces of one uniform material, m = ρV. Plot mass on the vertical axis and volume on the horizontal axis:

• The graph is a straight line through the origin (zero volume has zero mass).
• The slope of the line is Δm / ΔV, which is the density.
• A denser material gives a steeper line.

A graph is better than one measurement because the best-fit line averages out random errors in individual readings.

Worked example 2: density from a graph

Question. A student wants the density of a type of rock. She has five pebbles of it, an electronic balance and a measuring cylinder part-filled with water. Describe a method and use her results to find the density.

V (cm³)	4.0	7.5	11.0	15.5	19.0
m (g)	10.9	19.8	29.5	41.6	50.3

Method.

1. Measure each pebble’s mass on the balance.
2. Read the water level in the cylinder. Lower one pebble in gently so it is fully under water, and read the new level. The rise in level is the pebble’s volume (1 mL = 1 cm³). This works for irregular shapes that you cannot measure with a ruler.
3. Plot m (vertical) against V (horizontal) and draw a best-fit straight line.

Analysis.

1. The points lie close to a straight line that passes very near the origin (Figure 2). This supports the idea that all five pebbles are the same material.
2. Choose two points on the line, far apart: (2.0 cm³, 5.5 g) and (20.0 cm³, 53.2 g).
3. Slope = (53.2 − 5.5) g ÷ (20.0 − 2.0) cm³ = 47.7 g ÷ 18.0 cm³ = 2.65 g/cm³.
4. In SI units: ρ = 2.65 × 1000 = 2.65 × 10³ kg/m³ (granite, for comparison, is about 2700 kg/m³).

Check. Each pebble alone gives a ratio between 2.64 and 2.73 g/cm³. The slope sits inside that range but uses all five readings at once, so one bad reading has less effect.

Average density of an object made of parts

Many objects are not one uniform material. A boat, a football or a person contains several materials and often trapped air. For such an object, use the average density:

ρ_avg = total mass ÷ total volume

Worked example 3: a sealed hollow container

Question. A sealed box is made from aluminium sheet. The metal itself has a volume of 40 cm³. The box encloses 400 cm³ of air (density 0.0012 g/cm³). Find the box’s average density and compare it with water (1.0 g/cm³) and with solid aluminium.

1. Mass of aluminium: m = ρV = 2.70 g/cm³ × 40 cm³ = 108 g.
2. Mass of air: 0.0012 g/cm³ × 400 cm³ = 0.48 g.
3. Total mass: 108 + 0.48 = 108.48 g. Total volume: 40 + 400 = 440 cm³.
4. Average density: 108.48 g ÷ 440 cm³ = 0.25 g/cm³ = 2.5 × 10² kg/m³.

Compare. The box’s average density is about a quarter of water’s, even though aluminium is 2.7 times denser than water. Same mass, very different volume: a solid aluminium lump of 108 g would have a volume of only about 40 cm³.

Interpretation. The air adds almost nothing to the mass (ignoring it changes the answer by less than 0.5%) but a lot to the volume. In Topic 8.3 you will use average density to decide whether an object floats.

The ideal-fluid model

Real fluids are messy, so this course uses a simple model. An ideal fluid has two properties:

1. Incompressible. Squeezing it does not change its volume, so its density stays constant.
2. No viscosity. Viscosity is internal friction: layers of fluid dragging on each other as they slide. An ideal fluid flows with no such friction, so no kinetic energy is lost to thermal energy inside it.

How good is the model?

• Water fits well. It is very hard to compress, and it flows easily.
• Honey or thick oil has high viscosity. It flows slowly and loses energy as it flows, so the “no viscosity” part fails.
• Air is easy to compress. Halve the volume of air in a sealed syringe and its density doubles. The “incompressible” part fails when the pressure on a gas changes a lot.

Unless a question says otherwise, treat fluids as ideal.

Common misconceptions

• “Heavy means dense.” Weight depends on mass; density depends on mass and volume. A heavy object can have a low density if it is large.
• “A smaller piece has a smaller density.” Cutting a uniform object changes m and V by the same factor, so ρ does not change.
• “Fluid means liquid.” Gases are fluids too: they have no fixed shape and they flow.
• “Gases have no mass, so they have no density.” Air has a density of about 1.2 kg/m³. A room full of air has a mass of tens of kilograms.
• “A gas has a fixed density.” A gas fills whatever volume it is given, so the same mass of gas can have very different densities.
• “Things float because they are lighter than water.” A ship is far heavier than a bucket of water and still floats. What matters is comparing densities (developed in Topic 8.3).
• Unit slips. Mixing g with kg, or cm³ with m³, changes the answer by a factor of 1000 or more. Convert first, or work wholly in g and cm³ and multiply by 1000 at the end.
• Reading the mass–volume graph from one point only. Use the slope of the best-fit line, not m ÷ V for a single reading, when the question gives a graph.

Where this leads

Topic 8.2 (Pressure) asks what a fluid does to the surfaces it touches. Density appears straight away: the extra pressure under a column of fluid depends on ρ. Read the Topic 8.2 study guide next. First, try the practice questions, then use the revision notes and the checklist to consolidate. You can also return to the course roadmap.