Chapter 4.3: Density and Pressure – Why Do Things Float and Submarines Sink?
Welcome to this core section of AS Physics! Density and pressure are fundamental concepts that explain so much about the world around us, from why ships float to why deep-sea divers need special equipment.
Don't worry if you sometimes mix up mass and density, or force and pressure. We will break down these concepts step-by-step, using simple analogies to make sure everything clicks!
1. Understanding Density (\(\rho\))
Density is a measure of how much "stuff" (mass) is packed into a certain space (volume).
1.1 Definition and Formula
The density (\(\rho\)) of a substance is defined as its mass (\(m\)) per unit volume (\(V\)).
Formula:
\[
\rho = \frac{m}{V}
\]
SI Units:
- Mass (\(m\)) is in kilograms (\(\text{kg}\)).
- Volume (\(V\)) is in cubic metres (\(\text{m}^3\)).
- Therefore, the unit for density (\(\rho\)) is kilograms per cubic metre (\(\text{kg m}^{-3}\)).
Analogy: Imagine a party! If 10 people (mass) are crammed into a tiny elevator (volume), the density is high. If those 10 people are spread out in a huge ballroom, the density is low.
It’s not just about the total mass; it's about how concentrated that mass is.
Quick Tip for Calculations: Make sure all lengths are in metres (m) before calculating volume in \(\text{m}^3\). If you are given volume in \(\text{cm}^3\), you must convert! \(\text{1 m}^3 = 1,000,000 \text{ cm}^3\).
Key Takeaway: Density tells you how tightly packed the matter is. Higher density means more mass in the same space.
2. Understanding Pressure (\(P\))
While density is a property of a substance, pressure describes the effect of a force distributed over an area.
2.1 Definition and Formula
Pressure (\(P\)) is defined as the force (\(F\)) acting normally (perpendicularly) per unit area (\(A\)).
Formula:
\[
P = \frac{F}{A}
\]
SI Units:
- Force (\(F\)) is in Newtons (\(\text{N}\)).
- Area (\(A\)) is in square metres (\(\text{m}^2\)).
- Therefore, the unit for pressure (\(P\)) is Newtons per square metre (\(\text{N m}^{-2}\)). This unit is also called the Pascal (\(\text{Pa}\)).
Analogy: Sharp vs. Blunt. If you push a thumbtack with a force \(F\), it easily penetrates the wall because the force is concentrated over a tiny area \(A\) (high pressure). If you push the wall with your thumb (same force \(F\)) the area \(A\) is much larger, so the pressure is low, and nothing penetrates.
Important: Pressure depends on the area over which the force acts. A smaller area leads to a greater pressure for the same force.
Key Takeaway: Pressure is force concentrated on an area. Remember the force must be perpendicular to the surface.
3. Hydrostatic Pressure in Fluids
Fluids (liquids and gases) exert pressure. This pressure increases as you go deeper. This is why your ears pop when diving deep in a pool!
3.1 Derivation of Hydrostatic Pressure Difference (\(\Delta p = \rho g \Delta h\))
The syllabus requires you to derive the formula for the change in pressure with depth in a fluid. We look at a cylinder of fluid of height \(\Delta h\) and cross-sectional area \(A\).
Step-by-step derivation:
- Consider the mass (\(m\)) of the column of fluid acting on the area \(A\).
- Recall the definition of density: \(\rho = m/V\). So, mass \(m = \rho V\).
- The volume of the column is \(V = A \Delta h\).
- Substitute volume into the mass equation: \(m = \rho A \Delta h\).
- The force (\(F\)) exerted by this mass is its weight: \(F = mg\).
- Substitute the mass equation (step 4) into the force equation: \(F = (\rho A \Delta h) g\).
- The pressure difference (\(\Delta p\)) exerted by this column is \(P = F/A\).
- Substitute the force equation (step 6) into the pressure equation: \[ \Delta p = \frac{\rho A \Delta h g}{A} \]
- The area (\(A\)) cancels out!
Final Equation for Hydrostatic Pressure Difference: \[ \Delta p = \rho g \Delta h \]
Where:
- \(\Delta p\) is the pressure difference (Pa)
- \(\rho\) is the density of the fluid (\(\text{kg m}^{-3}\))
- \(g\) is the acceleration of free fall (\(\text{N kg}^{-1}\) or \(\text{m s}^{-2}\))
- \(\Delta h\) is the difference in vertical depth (m)
Did you know? This formula shows that the pressure change depends only on the density of the fluid and the vertical depth, not on the shape or width of the container!
Key Takeaway: Pressure in a fluid increases linearly with depth and is independent of the horizontal area.
4. Upthrust and Archimedes' Principle
Why does wood float, and why is it easier to lift your friend when they are submerged in a swimming pool? The answer is upthrust.
4.1 Origin of Upthrust
Upthrust is the net upward force exerted by a fluid on a submerged or partially submerged object.
The mechanism (Syllabus Item 5):
Since pressure increases with depth (\(\Delta p = \rho g \Delta h\)), an object submerged in a fluid will experience:
- A smaller downward pressure on its top surface.
- A greater upward pressure on its bottom surface.
4.2 Archimedes' Principle and Calculation
Archimedes' Principle is a rule that quantifies this upthrust force. It states that the upthrust acting on an object in a fluid is equal to the weight of the fluid displaced by that object.
Formula for Upthrust (Syllabus Item 6):
\[
F_{\text{upthrust}} = \rho g V
\]
Where:
- \(F_{\text{upthrust}}\) is the upthrust force (N).
- \(\rho\) is the density of the FLUID (\(\text{kg m}^{-3}\)). (This is a common mistake! Use the fluid's density, not the object's).
- \(g\) is the acceleration of free fall (\(\text{m s}^{-2}\)).
- \(V\) is the volume of the DISPLACED FLUID (i.e., the volume of the object that is submerged) (\(\text{m}^3\)).
Example: Floating vs. Sinking.
- An object floats if \(F_{\text{upthrust}}\) is equal to the weight of the object.
- An object sinks if the weight of the object is greater than \(F_{\text{upthrust}}\).
Memory Aid: Upthrust is related to the density of the *fluid* (\(\rho\)), whereas the weight of the object is related to the density of the *object*.
Common Mistake to Avoid: When calculating upthrust \(F = \rho g V\), always use the density of the liquid (\(\rho\)) and the volume of the object that is underwater (\(V\)). Do not accidentally use the object's density!
Key Takeaway: Upthrust exists because pressure increases with depth. The upthrust force is calculated using Archimedes' principle: \(F_{\text{upthrust}} = \text{Weight of displaced fluid}\).
5. Quick Review Box: Core Equations
You MUST know these definitions and equations for the exam:
- Density: \(\rho = m/V\)
- Pressure: \(P = F/A\) (where \(F\) is normal force)
- Hydrostatic Pressure (Difference): \(\Delta p = \rho g \Delta h\)
- Upthrust (Archimedes' Principle): \(F = \rho g V\) (where \(\rho\) is fluid density, \(V\) is submerged volume)