Chapter 1: Physical Quantities and Units (9702 AS Level)
Welcome to the foundational chapter of AS Physics! Everything we do in Physics involves measuring and describing the world, and this chapter gives us the essential toolkit to do that accurately and clearly. Don't worry if measuring errors or vectors seem tricky; by the end of these notes, you'll have a strong grasp of the language of Physics.
1.1 Physical Quantities and Estimates
A Physical Quantity is anything that can be measured. To fully define a quantity, you need two things:
- A Numerical Magnitude (how much).
- A Unit (what kind).
Example: If a mass is 5 kg, 5 is the numerical magnitude and kg is the unit.
Making Reasonable Estimates
Physics often requires you to judge whether an answer makes sense. This means being able to make a reasonable estimate of quantities, especially those related to everyday life.
- Height of a human: About 1.7 m. (Not 17 m or 0.17 m).
- Mass of a car: About 1000 kg (1 tonne).
- Time for a heartbeat: About 1 second.
Quick Tip: Always check your calculated answers against a realistic value. If you calculate a car's speed as \(10^{8}\) m/s, you know you have made an error, as this is faster than light!
Key Takeaway 1.1: All physical measurements need both a number and a unit. Practice estimating common quantities to check your results.
1.2 SI Units (The International System)
The SI system (Système International d'Unités) is the worldwide standard for measurement. It ensures that scientists and engineers globally are using the exact same basis for their calculations.
Base Quantities and Base Units
These are the fundamental building blocks of all other quantities. They cannot be defined in terms of other units.
The syllabus requires you to recall the following five base quantities and their units:
| Quantity | SI Base Unit | Symbol |
|---|---|---|
| Mass | kilogram | kg |
| Length | metre | m |
| Time | second | s |
| Electric Current | ampere | A |
| Thermodynamic Temperature | kelvin | K |
Derived Units
Derived units are built up by multiplying or dividing the base units. They are used for derived quantities (like speed, force, or energy).
Example 1: Force (Newtons, N)
From \(F = ma\):
Force = mass \(\times\) acceleration
Unit of Force = unit of mass \(\times\) unit of acceleration
\(1 \, \text{N} = 1 \, \text{kg} \times (1 \, \text{m/s}^2) = \mathbf{\text{kg m s}^{-2}}\)
Example 2: Charge (Coulombs, C)
From \(Q = It\):
Charge = current \(\times\) time
Unit of Charge = unit of current \(\times\) unit of time
\(1 \, \text{C} = 1 \, \text{A} \times 1 \, \text{s} = \mathbf{\text{A s}}\)
Checking Homogeneity of Equations
A key skill is using base units to check if an equation is dimensionally correct. This is called checking for homogeneity.
If an equation is homogeneous, the overall SI base units on the Left-Hand Side (LHS) must equal the overall SI base units on the Right-Hand Side (RHS).
Important Note: A homogeneous equation isn't necessarily physically correct (it might be missing a constant, like the factor \(\frac{1}{2}\) in \(E_k = \frac{1}{2}mv^2\)), but a non-homogeneous equation is definitely wrong!
Step-by-Step Example: Check if \(E_k = mv^2\) is homogeneous.
- LHS Unit (Energy, E): Energy is Work Done, \(W = Fd\). Unit of \(W = (\text{kg m s}^{-2}) \times \text{m} = \mathbf{\text{kg m}^{2} \text{s}^{-2}}\)
- RHS Unit (\(mv^2\)): Unit of mass (\(m\)) \(\times\) Unit of velocity squared (\(v^2\))
Unit of \(mv^2 = \text{kg} \times (\text{m/s})^2 = \text{kg} \times (\text{m}^2 \text{s}^{-2}) = \mathbf{\text{kg m}^{2} \text{s}^{-2}}\) - Comparison: LHS unit (\(\text{kg m}^{2} \text{s}^{-2}\)) = RHS unit (\(\text{kg m}^{2} \text{s}^{-2}\)). The equation is homogeneous.
SI Prefixes
Prefixes are essential shortcuts to handle very large or very small numbers.
| Prefix | Symbol | Multiplication Factor | Power |
|---|---|---|---|
| Tera | T | 1,000,000,000,000 | \(10^{12}\) |
| Giga | G | 1,000,000,000 | \(10^{9}\) |
| Mega | M | 1,000,000 | \(10^{6}\) |
| Kilo | k | 1,000 | \(10^{3}\) |
| (Base Unit) | |||
| Deci | d | 0.1 | \(10^{-1}\) |
| Centi | c | 0.01 | \(10^{-2}\) |
| Milli | m | 0.001 | \(10^{-3}\) |
| Micro | \(\mu\) | 0.000001 | \(10^{-6}\) |
| Nano | n | 0.000000001 | \(10^{-9}\) |
| Pico | p | 0.000000000001 | \(10^{-12}\) |
Did you know? Using the correct case is vital! M (Mega, \(10^6\)) is very different from m (milli, \(10^{-3}\)).
Key Takeaway 1.2: Memorize the 5 base units. Use homogeneity checks to catch dimensional errors in formulae. Be fluent with the standard SI prefixes.
1.3 Errors and Uncertainties
No measurement is perfect. Errors and uncertainties are intrinsic to experimental physics.
Random Errors vs. Systematic Errors
You need to understand the difference between these two main types of errors:
1. Random Errors
- Definition: Errors that cause results to scatter randomly around the true value. They are unpredictable and vary from one reading to the next.
- Causes: Human error (e.g., poor reaction time when using a stopwatch), fluctuating conditions (e.g., small vibrations), difficulty reading a scale (e.g., parallax error).
- Effect: Reduces precision.
- Solution: Taking many repeat readings and calculating an average.
2. Systematic Errors
- Definition: Errors that cause all readings to shift consistently in one direction (always too high or always too low).
- Causes: Faulty calibration, environmental effects, or a zero error (where the instrument reads a non-zero value when the true value should be zero, e.g., a balance reading 0.5 g before placing an object on it).
- Effect: Reduces accuracy.
- Solution: Identifying and adjusting the instrument (e.g., correcting the zero error) or using a completely different method.
Precision and Accuracy
These terms are often confused! Think of them using a dartboard analogy:
- Accuracy: How close your measured value is to the true or accepted value (hitting the bullseye). Affected by systematic errors.
- Precision: How close repeated measurements are to one another (darts landing tightly clustered together). Affected by random errors.
A result can be precise (all readings are close) but inaccurate (all readings are far from the true value due to a large zero error).
Combining Uncertainties
When you use measurements (each having an uncertainty) to calculate a derived quantity, you must combine their uncertainties. We use two simple rules for the syllabus:
Rule 1: Addition and Subtraction (\(Z = A + B\) or \(Z = A - B\))
If you add or subtract two quantities, you add their absolute uncertainties.
Absolute uncertainty in \(Z = (\text{Absolute uncertainty in } A) + (\text{Absolute uncertainty in } B)\)
Example: Measuring a length L by finding the difference between two readings: \(L_2 = 10.0 \pm 0.1 \text{ cm}\) and \(L_1 = 2.0 \pm 0.1 \text{ cm}\). The length L is \(8.0 \pm (0.1 + 0.1) \text{ cm} = 8.0 \pm 0.2 \text{ cm}\).
Rule 2: Multiplication, Division, and Powers (\(Z = A \times B\) or \(Z = A / B\) or \(Z = A^n\))
If you multiply or divide quantities, you add their percentage uncertainties.
Calculation Steps:
- Calculate the percentage uncertainty for each initial quantity: \[\text{Percentage Uncertainty} = \left( \frac{\text{Absolute Uncertainty}}{\text{Value}} \right) \times 100\%\]
- Add these percentages together to find the percentage uncertainty in the final result (\(Z\)).
- (Optional but often required) Convert the final percentage uncertainty back into an absolute uncertainty.
Example: If density (\(\rho\)) is calculated using mass (\(m\)) and volume (\(V\)), \(\rho = m/V\).
\[\frac{\Delta\rho}{\rho} = \frac{\Delta m}{m} + \frac{\Delta V}{V}\]
For Powers (\(Z = A^n\)): If a value A is raised to the power of \(n\), multiply its percentage uncertainty by \(n\).
\[\frac{\Delta Z}{Z} = n \times \frac{\Delta A}{A}\]Quick Review: Uncertainty Rules
- ADD/SUBTRACT quantities $\implies$ ADD Absolute Uncertainties.
- MULTIPLY/DIVIDE quantities $\implies$ ADD Percentage Uncertainties.
- POWER quantities ($A^n$) $\implies$ MULTIPLY Percentage Uncertainty by $n$.
Key Takeaway 1.3: Systematic errors reduce accuracy; random errors reduce precision. When combining variables in calculations, you always add their uncertainties, either as absolute values or percentages, depending on the calculation type.
1.4 Scalars and Vectors
Quantities in physics are classified based on whether direction matters.
Distinction Between Scalars and Vectors
1. Scalar Quantities
- Definition: A quantity defined completely by its magnitude (size) only. Direction is irrelevant.
- Examples: Mass, Time, Energy, Temperature, Distance, Speed.
2. Vector Quantities
- Definition: A quantity defined by both its magnitude and its direction.
- Examples: Force, Velocity, Acceleration, Momentum, Displacement.
Memory Aid: A vector tells you Vhere to go (it needs direction!).
Adding and Subtracting Coplanar Vectors
Since vectors have direction, they must be added using geometric (graphical) or trigonometric methods, not simple arithmetic.
Addition (Head-to-Tail Rule):
- Draw the first vector.
- Draw the second vector starting from the arrowhead (head) of the first vector.
- The resultant vector (sum) is drawn from the tail of the first vector to the head of the second vector.
Subtraction:
Subtracting vector \(\mathbf{B}\) from vector \(\mathbf{A}\) (\(\mathbf{A} - \mathbf{B}\)) is the same as adding the negative of \(\mathbf{B}\) (\(\mathbf{A} + (-\mathbf{B})\)). The negative vector \(-\mathbf{B}\) has the same magnitude as \(\mathbf{B}\) but points in the exact opposite direction.
Resolving a Vector into Perpendicular Components
Often, forces or velocities act at angles. It's much easier to analyze motion and forces by breaking (or resolving) a single vector (\(\mathbf{F}\)) into two perpendicular components (usually horizontal, \(F_x\), and vertical, \(F_y\)).
If a vector \(\mathbf{F}\) acts at an angle \(\theta\) to the horizontal:
1. Component Parallel (Horizontal, Adjacent to \(\theta\)):
\[F_x = F \cos \theta\]
2. Component Perpendicular (Vertical, Opposite \(\theta\)):
\[F_y = F \sin \theta\]
Analogy: Imagine pushing a lawnmower. If you push diagonally (the main Force, \(\mathbf{F}\)), part of your effort moves the mower forward (\(F_x\)) and part pushes it down into the ground (\(F_y\)).
Don't worry if this seems tricky at first: Resolving vectors is one of the most important skills in AS Physics. Practice drawing the right-angled triangles to identify which side is $\sin \theta$ and which is $\cos \theta$. The component adjacent (next to) the angle \(\theta\) usually gets the cosine ($\cos$).
Key Takeaway 1.4: Scalars need only magnitude (like speed), while vectors need magnitude and direction (like velocity). We add vectors geometrically, and break them down using trigonometry into perpendicular components for easy calculation.