Welcome to A-Level Physics 9702: The Foundation!
Hi there! Starting A-Level Physics can feel like learning a new language, and this first chapter—Physical Quantities and Units—is your dictionary and grammar book. Everything we measure, calculate, and prove in the next two years relies on these fundamentals.
Don't worry! This chapter focuses on building good habits: ensuring your numbers make sense, your equations are balanced, and you correctly deal with direction. Let’s lay a rock-solid foundation!
1.1 Physical Quantities and Estimates
What is a Physical Quantity?
In physics, when we measure something, the result is a physical quantity. A physical quantity must always consist of two parts:
- Numerical Magnitude: The number itself (how much).
- Unit: The standard measure used (what kind).
Example: If you measure the mass of a textbook as 1.5 kg, 1.5 is the magnitude and kg (kilograms) is the unit.
A number without a unit is useless in Physics!
Making Reasonable Estimates
In exams, you often need to show that you have a basic intuition about the physical world. This means being able to make a reasonable estimate of a quantity.
Quick Review: Typical Estimates (Know these!)
- Speed of a fast car on a motorway: ~30 m/s (or 108 km/h).
- Mass of an apple: ~150 g (0.15 kg).
- Height of an average adult: ~1.8 m.
- Density of water: \(1000 \text{ kg m}^{-3}\).
Analogy: If you calculate the speed of a human runner to be \(500 \text{ m/s}\), your answer is technically correct based on your calculation, but it is *not* a reasonable estimate (that’s faster than the speed of sound!). Physics requires common sense alongside mathematics.
Key Takeaway 1.1: All measurements need a number (magnitude) and a standard (unit). Develop an intuition for whether your calculated answers are physically sensible!
1.2 SI Units: The Global Standard
The SI System
The International System of Units (SI) ensures that scientists worldwide use the same fundamental standards. This prevents confusion when sharing results.
SI Base Quantities and Units
All units in the universe of Physics are built from just a few Base Units. You must recall the following five base quantities and their corresponding SI units:
| Quantity | SI Base Unit | Symbol |
|---|---|---|
| Mass | kilogram | kg |
| Length | metre | m |
| Time | second | s |
| Electric Current | ampere | A |
| Thermodynamic Temperature | kelvin | K |
Did you know? Although the mole (mol) and candela (cd) are also SI base units, they are not typically assessed at the AS/A Level Physics 9702 syllabus depth.
Mnemonic Aid: To remember the five main base units:
Ma (Mass, kg)
Loves (Length, m)
To (Time, s)
Cuddle (Current, A)
Kittens (Kelvin, K)
Derived Units
Units that are formed by combining two or more base units are called Derived Units. You must be able to express any derived unit in terms of base units.
Example: Deriving the unit of Force
- We know the formula for Force is: \(F = ma\) (Force = mass \(\times\) acceleration).
- Base unit for Mass: kg.
- Base unit for Acceleration (change in velocity per unit time): Velocity is \((\text{m/s})\), so Acceleration is \((\text{m/s}) / \text{s}\) or \(\text{m s}^{-2}\).
- Therefore, the unit of Force (the Newton, N) is: \(\text{N} = \mathbf{kg \ m \ s^{-2}}\).
Similarly, the unit for Energy (the Joule, J) is \(\text{J} = \mathbf{kg \ m^2 \ s^{-2}}\).
Checking Homogeneity of Equations
Homogeneity means checking if the units on the left-hand side (LHS) of an equation are the same as the units on the right-hand side (RHS).
If an equation is correct, it must be homogeneous. If it is not homogeneous, it must be wrong.
Step-by-Step Check: \(E_k = \frac{1}{2} mv^2\)
- LHS (Energy, E): Base units are \(\text{kg m}^2 \text{ s}^{-2}\).
- RHS (Mass, m): Base unit is \(\text{kg}\).
- RHS (Velocity squared, \(v^2\)): Velocity is \(\text{m s}^{-1}\), so \(v^2\) is \((\text{m s}^{-1})^2 = \text{m}^2 \text{ s}^{-2}\).
- RHS Total: \(\text{kg} \times \text{m}^2 \text{ s}^{-2} = \text{kg m}^2 \text{ s}^{-2}\).
Since the LHS units match the RHS units (\(\text{kg m}^2 \text{ s}^{-2}\)), the equation is homogeneous.
Common Mistake: Remember, checking homogeneity only confirms the formula is *potentially* correct in structure. It does not check if the numerical constants (like the \(\frac{1}{2}\)) are correct!
SI Prefixes (Multiples and Submultiples)
We use prefixes to handle very large or very small numbers easily, avoiding writing too many zeros. You must recall and use the following prefixes:
| Prefix | Symbol | Power of 10 |
|---|---|---|
| Tera | T | \(10^{12}\) |
| Giga | G | \(10^9\) |
| Mega | M | \(10^6\) |
| Kilo | k | \(10^3\) |
| deci | d | \(10^{-1}\) |
| centi | c | \(10^{-2}\) |
| milli | m | \(10^{-3}\) |
| micro | \(\mu\) | \(10^{-6}\) |
| nano | n | \(10^{-9}\) |
| pico | p | \(10^{-12}\) |
Tip for using prefixes: When converting, always move towards the base unit (power of 0). If the prefix is large (like k or M), multiply by the positive power. If the prefix is small (like m or n), multiply by the negative power.
Example: Convert 5 \(\mu\text{m}\) to meters.
\(5 \ \mu\text{m} = 5 \times 10^{-6} \text{ m}\).
Key Takeaway 1.2: The five SI base units are your starting points. Derived units are combinations of these. Use homogeneity to check equations, and use prefixes correctly to handle scale.
1.3 Errors and Uncertainties
No measurement is perfect! Understanding errors is crucial for practical work (Paper 3 and Paper 5) and data analysis.
1. Precision vs. Accuracy
These terms are often confused, but they mean different things in physics:
- Accuracy: How close a measurement is to the true value.
- Precision: How close repeated measurements are to each other (how spread out they are). Precision is often related to the resolution of the measuring instrument.
Dartboard Analogy:
If all your shots (measurements) are clustered tightly together, you have high precision. If that tight cluster is near the bullseye (true value), you also have high accuracy.
2. Types of Errors
Random Errors
These cause measurements to be slightly spread out around the true value. They are unpredictable and usually result from human limitations or slight environmental variations.
- Effect: Reduces precision.
- Cause Examples: Parallax error when reading a scale, reaction time when timing, fluctuating air currents.
- Solution: Take many repeated readings and calculate the mean. Random errors tend to cancel themselves out.
Systematic Errors
These cause all measurements to shift consistently in one direction (always too high or always too low). This error is consistent and often instrumental.
- Effect: Reduces accuracy.
- Cause Examples: A ruler with a chipped end, a faulty ammeter, or, most commonly, a zero error (where the instrument reads a non-zero value when the true value is zero).
- Solution: Identify the fault (e.g., measuring the zero error) and calibrate or adjust the readings accordingly. You cannot eliminate systematic error by repeating readings.
3. Combining Uncertainties
When you use measured values (which have uncertainties) to calculate a final derived quantity, the uncertainty of the final value must be found by combining the uncertainties of the inputs.
We use two types of uncertainty: Absolute Uncertainty (\(\Delta x\)) and Percentage Uncertainty (\(\frac{\Delta x}{x} \times 100\%\)).
Rule 1: Addition or Subtraction (\(y = a \pm b\))
If you add or subtract quantities, you add their absolute uncertainties.
\[\Delta y = \Delta a + \Delta b\]
Example: Measuring the length of a rod (L) by taking two readings (\(x_1\) and \(x_2\)): \(L = x_2 - x_1\).
If \(x_1 = (1.0 \pm 0.1) \text{ cm}\) and \(x_2 = (5.0 \pm 0.1) \text{ cm}\).
\(\Delta L = 0.1 \text{ cm} + 0.1 \text{ cm} = 0.2 \text{ cm}\).
Thus, \(L = (4.0 \pm 0.2) \text{ cm}\).
Rule 2: Multiplication or Division (\(y = a \times b\) or \(y = a/b\))
If you multiply or divide quantities, you add their percentage uncertainties.
\[\frac{\Delta y}{y} \times 100\% = \frac{\Delta a}{a} \times 100\% + \frac{\Delta b}{b} \times 100\%\]
Example: Calculating area \(A = L \times W\). If L has 2% uncertainty and W has 3% uncertainty, the area A has \(2\% + 3\% = 5\%\) uncertainty.
Rule 3: Powers (\(y = a^n\))
If a quantity is raised to a power, multiply its percentage uncertainty by that power.
\[\frac{\Delta y}{y} \times 100\% = |n| \times \left(\frac{\Delta a}{a} \times 100\%\right)\]
Example: Kinetic Energy \(E_k = \frac{1}{2} mv^2\). The mass (m) has 1% uncertainty, and velocity (v) has 4% uncertainty.
The uncertainty in \(v^2\) is \(2 \times 4\% = 8\%\).
Total uncertainty in \(E_k = 1\% (\text{from m}) + 8\% (\text{from } v^2) = 9\%\).
Key Takeaway 1.3: Precision is about repeatability (affected by random errors, fixed by repetition). Accuracy is about closeness to the true value (affected by systematic errors, fixed by calibration). Use percentage uncertainties for multiplication/division and absolute uncertainties for addition/subtraction.
1.4 Scalars and Vectors
In physics, direction is sometimes everything. This distinction leads to two important types of physical quantities.
1. Understanding the Difference
Scalar Quantities
A scalar quantity is completely defined by its magnitude only (a number and a unit).
- Examples to recall: Distance, Speed, Mass, Time, Energy, Temperature.
Analogy: Telling a friend, "I have 50 J of energy." The direction of the energy doesn't matter.
Vector Quantities
A vector quantity is completely defined by both its magnitude and its direction.
- Examples to recall: Displacement, Velocity, Acceleration, Force, Momentum.
Analogy: Telling a friend, "A force of 50 N is pushing the door East." Direction is essential for understanding the effect.
2. Adding and Subtracting Coplanar Vectors
You cannot add vectors like simple numbers (scalars) unless they are acting along the exact same line. Since force and velocity often act at angles, we must use vector addition, usually represented graphically using the head-to-tail method or mathematically by resolving components (see below).
Resultant Vector: The single vector that has the same effect as all the original vectors combined.
Example: Adding Force Vectors
Imagine two forces, \(F_1\) and \(F_2\), acting on an object. The resultant force \(R\) is found by placing the tail of \(F_2\) at the head of \(F_1\). The resultant \(R\) connects the starting point (tail of \(F_1\)) to the end point (head of \(F_2\)).
If the vectors form a right-angled triangle, you can use Pythagoras' theorem and trigonometry to find the resultant magnitude and direction.
3. Representing a Vector as Two Perpendicular Components
Often, it is easier to deal with a vector by breaking it down (resolving it) into two component vectors that are perpendicular to each other (usually horizontal and vertical components).
If a vector \(F\) acts at an angle \(\theta\) to the horizontal:
- Horizontal Component (\(F_x\)): \(F_x = F \cos \theta\)
- Vertical Component (\(F_y\)): \(F_y = F \sin \theta\)
Tip: The Cosine Rule Trick!
If you resolve a vector, always use the angle \(\theta\) that is *adjacent* (next door) to the component you want to find. The component adjacent to the angle gets the cosine (\(\cos \theta\)).
Example: Resolving a Velocity
A projectile leaves the ground with a velocity \(v\) of \(20 \text{ m/s}\) at an angle of \(30^\circ\) to the horizontal.
- Horizontal velocity: \(v_x = 20 \cos (30^\circ) = 17.3 \text{ m/s}\)
- Vertical velocity: \(v_y = 20 \sin (30^\circ) = 10.0 \text{ m/s}\)
This technique is essential for tackling mechanics problems (like forces on inclined planes or projectile motion) later on.
Key Takeaway 1.4: Scalars have magnitude only; Vectors have magnitude and direction. Always resolve vectors into perpendicular components (\(F \cos \theta\) and \(F \sin \theta\)) before adding or subtracting them mathematically.
Chapter 1: Quick Review Checklist
Make sure you can:
- State the five AS level SI base units.
- Express derived units (like the Joule or Pascal) in terms of base units.
- Check if an equation is homogeneous (units balanced).
- Distinguish between random and systematic errors.
- Calculate uncertainty for addition, subtraction, multiplication, division, and powers.
- Define and give examples of scalar and vector quantities.
- Resolve a vector into its two perpendicular components using sine and cosine.
Congratulations on completing the first chapter! You now have the fundamental language and tools needed to measure the physical world.