11.2 Density

Density is a key property that describes how much mass is packed into a given volume. It's calculated by dividing mass by volume and helps explain why some objects float while others sink.

Understanding density is crucial for grasping buoyancy and Archimedes' principle. These concepts explain why ships float, hot air balloons rise, and how we can measure the volume of irregularly shaped objects.

Density

Definition of Density

Density tells you how tightly matter is concentrated in a space. Two objects can be the same size but have very different masses, and density captures that difference.

• Mathematically defined as mass divided by volume: $\rho = \frac{m}{V}$, where $\rho$ (the Greek letter "rho") is density, $m$ is mass, and $V$ is volume
• Higher mass for a given volume means higher density. A block of lead is much denser than an equal-sized block of aluminum because lead atoms are heavier and more tightly packed.
• Larger volume for a given mass means lower density. A chunk of styrofoam takes up far more space than the same mass of steel.
• Density is an intensive property, meaning it doesn't depend on how much of the substance you have. A small gold ring and a large gold bar have the same density.

Density Calculations

You can rearrange the density equation to solve for any of the three variables. The SI unit for density is $\text{kg/m}^3$, but you'll often see $\text{g/cm}^3$ in problems. Water's density is $1.00 \text{ g/cm}^3$ (or $1000 \text{ kg/m}^3$), which is a handy reference value.

• Finding mass from density and volume: $m = \rho V$
  • Example: A substance with density $2.5 \text{ g/cm}^3$ and volume $10 \text{ cm}^3$ has mass: $m = (2.5 \text{ g/cm}^3)(10 \text{ cm}^3) = 25 \text{ g}$
• Finding volume from mass and density: $V = \frac{m}{\rho}$
  • Example: A substance with mass $50 \text{ g}$ and density $5 \text{ g/cm}^3$ has volume: $V = \frac{50 \text{ g}}{5 \text{ g/cm}^3} = 10 \text{ cm}^3$

Watch your units carefully. If density is given in $\text{kg/m}^3$ but volume is in $\text{cm}^3$, you need to convert before plugging in.

Density Comparisons and Buoyancy

Different materials have different densities because of differences in atomic mass and how tightly their atoms or molecules are arranged.

• Gases generally have much lower densities than liquids, which in turn have lower densities than solids. Air at sea level has a density of about $1.29 \text{ kg/m}^3$, while water is $1000 \text{ kg/m}^3$.
• Among solids, metals tend to be denser than materials like wood or plastic. Steel has a density of about $7800 \text{ kg/m}^3$, while balsa wood is only around $160 \text{ kg/m}^3$.

Density is what determines whether an object floats or sinks in a fluid:

• An object floats if its density is less than the fluid's density. Wood (roughly $500\text{–}800 \text{ kg/m}^3$) floats in water.
• An object sinks if its density is greater than the fluid's density. A rock (typically $2500\text{–}3000 \text{ kg/m}^3$) sinks in water.

Archimedes' principle connects density to buoyancy: the buoyant force on a submerged object equals the weight of the fluid it displaces. The buoyant force acts upward, opposing gravity.

• If the buoyant force equals or exceeds the object's weight, the object floats. Ice ($917 \text{ kg/m}^3$) is less dense than liquid water, so it floats with about 90% of its volume submerged.
• If the buoyant force is less than the object's weight, the object sinks.

One practical application: you can find the volume of an irregularly shaped object by submerging it in water and measuring how much fluid it displaces. Once you know the volume, you can calculate density from the object's mass.

Density Measurement Techniques

• Relative density (also called specific gravity): the ratio of a substance's density to the density of a reference substance, usually water at $4°\text{C}$. It's dimensionless, so a relative density of 2.7 means the substance is 2.7 times as dense as water.
• Pycnometer: a small glass flask used to measure liquid density precisely. You compare the mass of the flask filled with the unknown liquid to its mass filled with water.
• Hydrometer: a sealed glass tube with a weighted bottom that floats upright in a liquid. It sinks deeper in less dense liquids and rides higher in denser ones. The density is read directly from a scale on the tube's stem.