Study Notes: Density (Chapter 1.4)
Hello future Physicists! This chapter on Density is one of the foundational concepts in IGCSE Physics. It helps us understand the fundamental property of matter: how much 'stuff' is packed into a given space. Mastering density is crucial, especially as we move on to topics like pressure and forces later in the course!
Let’s dive into what density is, how we measure it, and why some things float while others sink.
1. Defining Density: Mass per Unit Volume (Core)
What is Density?
Imagine you have two boxes of the exact same size (the same Volume). One is filled with feathers, and the other is filled with bricks. Which box is heavier? The box of bricks, obviously!
The box of bricks has a higher Density because it has more mass packed into the same volume compared to the feathers.
- Definition: Density ($\rho$) is defined as the mass ($m$) of a substance per unit volume ($V$).
- It tells us how 'compact' a substance is.
The Density Equation
The relationship between density, mass, and volume is simple and must be memorised:
$$ \rho = \frac{m}{V} $$
Where:
- $\rho$ (rho) is the symbol for Density.
- $m$ is the Mass of the object (or substance).
- $V$ is the Volume it occupies.
🧠 Memory Aid: The Density Triangle
If you find rearranging formulas tricky, use the 'Density Triangle'. Cover the quantity you want to find, and the triangle shows you the equation:
(Imagine a triangle with M on top, and $\rho$ and V on the bottom row.)
To find Mass ($m$): Cover M. You get $\rho \times V$.
To find Density ($\rho$): Cover $\rho$. You get $m / V$.
To find Volume ($V$): Cover V. You get $m / \rho$.
Units of Density
Density units depend on the units used for mass and volume. The standard SI unit is:
- Kilograms per cubic metre (kg/m³)
However, it is very common (especially when dealing with liquids or small solids) to use:
- Grams per cubic centimetre (g/cm³)
Quick Conversion Fact:
The density of water is approximately $1000 \text{ kg/m}^3$, which is the same as $1 \text{ g/cm}^3$.
Key Takeaway 1:
Density is the ratio of mass to volume ($\rho = m/V$). The higher the density, the more concentrated the mass is in that material.
2. Determining Density Practically (Core)
To find the density of any object, we need two measurements: Mass and Volume. The way we measure the volume depends on the object's shape and state (solid or liquid).
2.1 Determining the Density of a Regular Solid
A regular solid is something like a cube, cuboid, or cylinder. Its volume can be calculated using geometry.
- Measure the Mass ($m$): Use a suitable balance (like a digital top-pan balance) to measure the mass of the solid in grams (g) or kilograms (kg).
- Measure the Volume ($V$):
- Use a ruler or Vernier calipers to measure its length, width, and height (or radius and height for a cylinder).
- Calculate the volume. (e.g., Volume of a cuboid $V = l \times w \times h$).
- Calculate Density ($\rho$): Divide the mass by the calculated volume ($\rho = m/V$).
Common Mistake to Avoid: Always ensure the units you measure are consistent. If you measure length in cm, your volume will be in cm³, and your density unit will be g/cm³.
2.2 Determining the Density of a Liquid
- Measure the Mass of the Container ($m_{empty}$): Use a balance to find the mass of an empty measuring cylinder or beaker.
- Measure the Mass of the Container + Liquid ($m_{total}$): Pour the liquid into the container and measure the total mass.
- Calculate the Mass of the Liquid ($m$): Subtract the mass of the empty container from the total mass:
$$ m_{liquid} = m_{total} - m_{empty} $$ - Measure the Volume of the Liquid ($V$): Read the volume directly from the markings on the side of the measuring cylinder. (Remember to read the meniscus correctly!).
- Calculate Density ($\rho$): Use the formula $\rho = m/V$.
2.3 Determining the Density of an Irregular Solid (Volume by Displacement)
If the solid is irregularly shaped (like a rock or a key) and sinks in water, we cannot use a ruler. We find its volume using the technique of volume by displacement.
The principle is simple: When an object is fully submerged in a fluid, the volume of the fluid displaced (pushed out of the way) is equal to the volume of the object.
Method using a Measuring Cylinder:
- Measure Mass ($m$): Find the mass of the irregular solid using a balance.
- Initial Volume ($V_1$): Pour some water into a measuring cylinder and record the initial volume $V_1$.
- Final Volume ($V_2$): Gently lower the solid into the water (it must be completely submerged). Record the new, higher water level $V_2$.
- Calculate Volume ($V$): The volume of the solid is the difference in the water levels:
$$ V = V_2 - V_1 $$ - Calculate Density ($\rho$): $\rho = m/V$.
If the object is very large, you might use a Eureka Can (or displacement can). Water is filled right up to the spout. When the object is lowered in, the displaced water flows out of the spout and is collected in a separate measuring cylinder. The volume of collected water is the volume of the object.
Key Takeaway 2:
The hardest part of measuring density is finding the volume. For irregular solids that sink, we must use the displacement method.
3. Floating and Sinking (Core & Supplement)
Why do some things float and others sink? The answer is density!
3.1 The Principle of Flotation (Core)
The rule is based on comparing the density of the object with the density of the fluid (liquid or gas) it is placed in:
- If $\rho_{object}$ > $\rho_{fluid}$, the object sinks. (Example: A rock sinks in water).
- If $\rho_{object}$ < $\rho_{fluid}$, the object floats. (Example: A piece of wood floats on water).
- If $\rho_{object}$ = $\rho_{fluid}$, the object is suspended (it floats within the liquid, neither sinking nor rising).
Did you know? Even though a large steel ship is heavy, it floats because its shape creates a huge overall volume. The ship contains a massive amount of air, making the average density of the ship (steel + air) much lower than the density of water.
3.2 Layering of Immiscible Liquids (Supplement - Extended Content)
Liquids that do not mix (like oil and water) are called immiscible liquids. If you pour two or more immiscible liquids into the same container, they will settle into distinct layers based on their densities.
Rule: The liquid with the lowest density will float on top of the liquid with the highest density.
Example: If you mix cooking oil (density $\approx 0.92 \text{ g/cm}^3$) and water (density $\approx 1.00 \text{ g/cm}^3$):
- Since $\rho_{oil}$ is less than $\rho_{water}$, the oil will form a layer on the surface of the water.
If you were given three unmixing liquids (A: $1.2 \text{ g/cm}^3$, B: $0.8 \text{ g/cm}^3$, C: $1.0 \text{ g/cm}^3$), the layers would stack from bottom to top as A, C, B (Highest density A sinks to the bottom, lowest density B floats on top).
Key Takeaway 3:
Density dictates flotation. Less dense objects/liquids float on top of more dense objects/liquids.