AP® Physics 1: Algebra-Based review sheet from Aim for Five (aimforfive.com/physics/units/8/8-1)
Unit 8 · Topic 8.1
8.1 Internal Structure and Density
Solids, liquids and gases behave differently because their particles attract each other with different strengths. Liquids and gases are fluids: they have no fixed shape and they flow. You describe a fluid by its density, ρ = m/V, and in AP Physics 1 you treat it as ideal: incompressible and with no viscosity.
Key terms
- fluid
- density
- ideal fluid
- incompressible
- viscosity
Solids, liquids and gases
Every material is made of atoms and molecules that pull on each other. How strong those attractions are, compared with how much the particles jiggle around, decides the state of matter.
| State | Particle picture | Shape and volume |
|---|---|---|
| Solid | strong attractions lock particles into fixed positions; they vibrate in place | fixed shape, fixed volume |
| Liquid | particles stay close but can slide past each other | takes the container's shape, fixed volume |
| Gas | particles are far apart and interact only in brief collisions | fills the whole container |
What counts as a fluid
A fluid is anything that can flow. It has no shape of its own, so it takes the shape of whatever holds it. Liquids and gases are both fluids. Water, air, oil and honey all count.
The rest of this unit treats a fluid as a huge number of tiny particles that obey Newton's laws. Pressure, buoyancy and flow all come from those particles pushing on each other and on the surfaces they touch.
Density
Density is mass per unit volume: ρ = m/V. The symbol ρ is the Greek letter rho. The SI unit is kg/m³. Fresh water is 1000 kg/m³, which is the same as 1.0 g/cm³; to convert g/cm³ to kg/m³, multiply by 1000.
Density belongs to the material, not to the object. Cut a block of aluminum in half and each half has half the mass and half the volume, so the density stays the same.
| Material | Approximate density (kg/m³) |
|---|---|
| Air at room conditions | 1.2 |
| Ice | 917 |
| Fresh water | 1000 |
| Seawater | 1025 |
| Aluminum | 2700 |
| Iron | 7870 |
| Mercury | 13,600 |
Ideal fluids
AP Physics 1 assumes every fluid is ideal unless a problem says otherwise. An ideal fluid has two properties.
It's incompressible: squeezing it doesn't shrink its volume, so its density never changes. Liquids come close to this in real life. Gases are easy to compress, but you'll still treat them as ideal in this course.
It has no viscosity. Viscosity is a fluid's internal friction, its resistance to flowing; honey has a lot and water has a little. With no viscosity, a flowing fluid loses no energy to internal friction, which is what lets you use energy conservation in 8.4.
Measuring density
Find mass with a balance. Find volume from an object's dimensions, or for an irregular object by lowering it into a graduated cylinder and reading how much the water level rises. For better results, measure several samples and graph mass against volume: the slope of the best-fit line is the density. A graph also handles a constant offset, such as the mass of a container, which ends up in the intercept instead of spoiling your answer.
Worked examples
Try each one yourself first, then open the solution.
- Example 1Calculator allowed
Identify the metal
A rectangular metal block measures 10 cm × 5.0 cm × 2.0 cm and has a mass of 0.27 kg. Find its density in kg/m³ and use the table to identify the metal.
Show the solutionHide the solution
- Step 1: Convert to meters first: 0.10 m × 0.050 m × 0.020 m = 1.0 × 10⁻⁴ m³.
- Step 2: ρ = m/V = 0.27 kg / (1.0 × 10⁻⁴ m³) = 2700 kg/m³.
- Step 3: That matches aluminum. (In cm: V = 100 cm³ and ρ = 270 g / 100 cm³ = 2.7 g/cm³, the same thing.)
Answer: 2700 kg/m³, so aluminum
- Example 2Calculator allowed
How heavy is the air in a room?
A classroom is 5.0 m long, 4.0 m wide and 3.0 m high. The air has density 1.2 kg/m³. What is the mass of the air in the room?
Show the solutionHide the solution
- Step 1: V = 5.0 × 4.0 × 3.0 = 60 m³.
- Step 2: m = ρV = (1.2 kg/m³)(60 m³) = 72 kg.
- Step 3: Air is not weightless: the air in a classroom has about the mass of an adult.
Answer: About 72 kg
- Example 3Calculator allowed
Density from a graph (classic trap)
A student pours different volumes of a liquid into the same beaker and measures the total mass each time: V = 20, 40, 60 and 80 cm³ give m = 66, 82, 98 and 114 g. The student calculates 66 g / 20 cm³ = 3.3 g/cm³. What's wrong, and what is the liquid's density?
Show the solutionHide the solution
- Step 1: The measured masses include the beaker, so dividing one total mass by one volume gives a value that's far too big.
- Step 2: Graph mass against volume. Each extra 20 cm³ adds 16 g, so the slope is 16 g / 20 cm³ = 0.80 g/cm³. That slope is the liquid's density.
- Step 3: The line's intercept (the mass at zero volume) is 66 − 16 = 50 g, which is the beaker's mass.
- Step 4: In SI units: 0.80 g/cm³ × 1000 = 800 kg/m³.
Answer: The beaker's mass was included. The slope gives 0.80 g/cm³ = 800 kg/m³ (the beaker is 50 g).
Common mistakes
- Mixing units: using a volume in cm³ with a mass in kg. Convert to kg and m³ (1 cm³ = 10⁻⁶ m³) before using density in other equations.
- Thinking a bigger piece of a material has a bigger density. Density is the same for any amount of the same material.
- Thinking gases aren't fluids. A fluid is anything that flows, so gases count.
- Forgetting that ideal means incompressible: in this course a liquid's density doesn't change with depth or pressure.
On the exam
- Experimental questions may ask how to measure density, or give mass and volume data to graph. The slope of mass against volume is the density, and a nonzero intercept points to a container or other constant mass.
- When asked why fluids behave as they do, explain in terms of particles: their spacing and how strongly they interact.
Connected topics
Videos
Check yourself
4 questions on 8.1 Internal Structure and Density. Pick an answer to see if you got it, and why.
A small metal block has a mass of 0.27 kg and a volume of 1.0 × 10⁻⁴ m³. What is the density of the metal?
A room measures 5.0 m by 4.0 m by 3.0 m. The density of the air in it is 1.2 kg/m³. What is the mass of the air in the room?
A uniform block of wood is cut into two pieces. One piece has one-third of the original volume. How does the density of that piece compare with the density of the original block?
Cube X and cube Y have the same mass. Each edge of cube X is twice as long as each edge of cube Y. What is the ratio of the density of cube X to the density of cube Y, ρ_X/ρ_Y?
0 of 4 answered