Internal structure and density

Solids hold their atoms in place. Liquids and gases let them flow, so both take the shape of whatever holds them; both are fluids. Density, $\rho = m/V$, says how tightly mass is packed. The trap: density belongs to the substance, not the sample.

§1

What makes something a fluid.

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A fluid takes the shape of whatever holds it. Liquids do. Gases do. Solids don't. Pour water into a tall glass and it fills the bottom; pour it into a wide bowl and it spreads. Air does the same thing in 3D, just invisibly.

The picture comes from how the atoms are arranged:

Fig. 8.1.1 In a solid, particles sit in fixed positions; in a liquid they're close but can slide past each other; in a gas, they fly apart with large gaps between them. Liquids and gases both flow, so both are fluids.

Unit 8 works with the ideal-fluid model, defined in §3.

§2

Density: $\rho = m/V$.

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Density $\rho$ is mass per unit volume:

$$\rho \;=\; \dfrac{m}{V}$$

SI units are kg/m$^3$. The AP equation sheet also uses g/cm$^3$, where $1$ g/cm$^3$ = $1000$ kg/m$^3$. Water is about $1000$ kg/m$^3$ ($1.00$ g/cm$^3$).

Here's the part students miss: $\rho$ is the same number for any sample of the same substance. Cut a block in half: both $m$ and $V$ drop by half, so the ratio holds. Stack two identical blocks: both double, ratio still holds. The substance picks the ratio.

Worked example

Aluminum has $\rho = 2700$ kg/m$^3$. A piece of aluminum has mass $m = 5.4$ kg. What is its volume?

Rearrange

$\rho = \dfrac{m}{V}$ gives $V = \dfrac{m}{\rho}$.

Plug in

$V = \dfrac{5.4 \text{ kg}}{2700 \text{ kg/m}^3} = 0.0020 \text{ m}^3$.

Check the units

kg ÷ (kg/m$^3$) = m$^3$. Volume. The answer carries the right unit.

$\rho$ is a scalar: magnitude in kg/m$^3$, no direction. You won't see $\vec{\rho}$ in this course.

Density is a property of the substance, not the sample. $m$ and $V$ scale with how much you have; their ratio doesn't.

§3

Ideal fluids.

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For the ideal-fluid models in this unit, assume incompressible and inviscid unless a problem says otherwise. That label is shorthand for two assumptions:

Incompressible. A given mass of fluid keeps the same volume no matter the pressure, so $\rho$ stays fixed. Liquids are nearly incompressible; gases are not, so this model fits liquids best and applies to a gas only when a problem says to assume it.

Inviscid (no viscosity). The fluid slides past itself with no resistance. Honey is viscous; water mostly isn't. Same move as ignoring friction in mechanics: wrong in detail, fine for the big picture.

These two are what make pressure, buoyancy, continuity, and Bernoulli (in 8.2 through 8.4) work cleanly. Assume ideal unless told otherwise.

§4

Where students go wrong.

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Pitfall · 01

"A bigger block is denser than a smaller one."

The bigger block weighs more on the scale. "More" sounds like "more dense." Easy slide.

Fix. Bigger block, more $m$ and more $V$, in the same ratio. The top and the bottom of $\rho = m/V$ scale together, so the ratio doesn't move. Same substance, same density.

Pitfall · 02

"Heavy things sink, light things float."

Everyday experience backs this up. Most heavy things you handle are denser than water; most light things aren't. The rule looks like it works.

Fix. What sinks or floats is set by density, not mass. A tiny lead pellet sinks; a massive log floats. The full buoyancy story waits for 8.3; the language matters now.

Pitfall · 03

"Density and mass are the same thing, just different units."

Both describe "how much stuff," both involve kg in their units, and $\rho = m/V$ looks like a rearrangement of $m$.

Fix. The units carry the meaning. Mass (kg) says how much. Density (kg/m$^3$) says how tightly packed. Weight ($mg$, in newtons) is a third quantity. Three quantities, three jobs.

§5

Skill check.

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Ten scenarios: three conceptual, four symbolic, three numeric. Progress saves as you go. Each wrong answer is sorted by the misconception behind it.