Unit AS 1: Forces, Energy and Electricity

Chapter 1.1: Physical Quantities

Welcome to AS Level Physics! Whether you love maths or feel a little nervous about physics calculations, this chapter is the foundation for everything you will study in Unit AS 1 and beyond. In physics, we describe the universe using measurements. To make sense of those measurements and communicate them across the world, physicists use a standardised language called the SI System (International System of Units). In this guide, we will break down what physical quantities are, how base and derived units work, how to handle unit prefixes, and how to verify physical equations using homogeneity.

Key Takeaway: Every measurement in physics requires two parts: a numerical magnitude and a unit.


1. The Nature of Physical Quantities

A physical quantity is any feature of an object or phenomenon that can be measured.

Every physical quantity must be stated with both:

1. A numerical magnitude (the size, number, or value)
2. A unit (the standard of reference)

Example: If a car travels with a speed of \(25\text{ m}\cdot\text{s}^{-1}\), the number \(25\) is the numerical magnitude, and \(\text{m}\cdot\text{s}^{-1}\) is the unit. Giving just a number (e.g. "the speed is 25") is meaningless in physics!


2. SI Base Quantities and Base Units

In the SI system, there are fundamental building blocks called base quantities. These are quantities that cannot be defined in terms of other physical quantities. Each base quantity has an official SI base unit.

For your CCEA AS Physics examination, you must know these six core base quantities and their exact standard symbols:

1. Mass: kilogram (\(\text{kg}\))
2. Length: metre (\(\text{m}\))
3. Time: second (\(\text{s}\))
4. Electric Current: ampere (\(\text{A}\))
5. Thermodynamic Temperature: kelvin (\(\text{K}\))
6. Amount of Substance: mole (\(\text{mol}\))

Note: The candela (\(\text{cd}\)) is also recognised as an SI base unit for luminous intensity, but the CCEA syllabus focuses primarily on the core six listed above.

Memory Trick: Think of base units as primary colours (red, blue, yellow). You cannot make them by mixing other colours, but you can combine them to make every other colour!

Key Takeaway: Base units are the foundational units from which all other mechanical and electrical units are constructed.


3. Standard SI Prefixes & Formatting Rules

Physical quantities can be enormous (like the distance between galaxies) or tiny (like the size of an atom). To avoid writing endless zeros, we use standard SI prefixes.

SI Multiples and Submultiples to Memorise:

pico (\(\text{p}\)): \(10^{-12}\)
nano (\(\text{n}\)): \(10^{-9}\)
micro (\(\mu\)): \(10^{-6}\)
milli (\(\text{m}\)): \(10^{-3}\)
centi (\(\text{c}\)): \(10^{-2}\)
kilo (\(\text{k}\)): \(10^{3}\)
mega (\(\text{M}\)): \(10^{6}\)
giga (\(\text{G}\)): \(10^{9}\)
tera (\(\text{T}\)): \(10^{12}\)

Important Exam Convention: Linear Index Notation

CCEA strictly requires you to write units using linear index (exponential) notation with negative powers rather than a slash or solidus. For example:

Always write: \(\text{m}\cdot\text{s}^{-1}\) (NOT \(\text{m/s}\))
Always write: \(\text{kg}\cdot\text{m}^{-3}\) (NOT \(\text{kg/m}^3\))
Always write: \(\text{m}\cdot\text{s}^{-2}\) (NOT \(\text{m/s}^2\))

Prefix Pitfall: Squaring and Cubing Units

Don't worry if converting areas or volumes feels confusing at first! Remember that when a unit has an exponent, the prefix is also raised to that power:

\(1\text{ cm} = 10^{-2}\text{ m} \implies (1\text{ cm})^2 = (10^{-2}\text{ m})^2 = 10^{-4}\text{ m}^2\)
\(1\text{ mm} = 10^{-3}\text{ m} \implies (1\text{ mm})^3 = (10^{-3}\text{ m})^3 = 10^{-9}\text{ m}^3\)

Key Takeaway: Always express final units in linear format (\(\text{m}\cdot\text{s}^{-1}\)) and convert prefixes before substituting them into formulas.


4. Derived Quantities and Expressing Units in Base Units

A derived quantity is any quantity defined by an algebraic combination of base quantities. Its unit is called a derived unit.

In the AS 1 exam, you are frequently asked to "express the unit in SI base units". To do this, replace each variable in a defining formula with its base units and simplify.

Step-by-Step Derivations:

1. Force (Newton, \(\text{N}\))
Formula: \(F = ma\)
Units: \(\text{mass } (\text{kg}) \times \text{acceleration } (\text{m}\cdot\text{s}^{-2})\)
Base Unit Equivalent: \(\text{kg}\cdot\text{m}\cdot\text{s}^{-2}\)

2. Energy and Work Done (Joule, \(\text{J}\))
Formula: \(W = Fs\) (Work = Force \(\times\) distance)
Units: \((\text{kg}\cdot\text{m}\cdot\text{s}^{-2}) \times (\text{m})\)
Base Unit Equivalent: \(\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}\)
Check: Using \(E_k = \frac{1}{2}mv^2\) gives \((\text{kg}) \times (\text{m}\cdot\text{s}^{-1})^2 = \text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}\), which matches perfectly!

3. Power (Watt, \(\text{W}\))
Formula: \(P = \frac{W}{t}\) (Power = Work \(\div\) time)
Units: \(\frac{\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}}{\text{s}}\)
Base Unit Equivalent: \(\text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}\)

4. Pressure and Stress (Pascal, \(\text{Pa}\))
Formula: \(p = \frac{F}{A}\) (Pressure = Force \(\div\) Area)
Units: \(\frac{\text{kg}\cdot\text{m}\cdot\text{s}^{-2}}{\text{m}^2}\)
Base Unit Equivalent: \(\text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-2}\)

5. Electric Charge (Coulomb, \(\text{C}\))
Formula: \(Q = It\) (Charge = Current \(\times\) time)
Units: \((\text{A}) \times (\text{s})\)
Base Unit Equivalent: \(\text{A}\cdot\text{s}\)

6. Potential Difference / EMF (Volt, \(\text{V}\))
Formula: \(V = \frac{W}{Q}\) (Voltage = Work Done \(\div\) Charge)
Units: \(\frac{\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}}{\text{A}\cdot\text{s}}\)
Base Unit Equivalent: \(\text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}\cdot\text{A}^{-1}\)

7. Electrical Resistance (Ohm, \(\Omega\))
Formula: \(R = \frac{V}{I}\) (Resistance = Voltage \(\div\) Current)
Units: \(\frac{\text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}\cdot\text{A}^{-1}}{\text{A}}\)
Base Unit Equivalent: \(\text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}\cdot\text{A}^{-2}\)

Key Takeaway: When asked for SI base units, break the formula down into basic definitions until only \(\text{kg}\), \(\text{m}\), \(\text{s}\), and \(\text{A}\) remain.


5. Homogeneity of Physical Equations

An equation is homogeneous with respect to units if every term separated by an addition sign (\(+\)), subtraction sign (\(-\)), or equals sign (\(=\)) has the exact same SI base units.

How to Test for Homogeneity (Step-by-Step):

Let us test the equation of motion: \(s = ut + \frac{1}{2}at^2\)

Step 1: Find the base units of the Left-Hand Side (LHS):
Displacement \(s\) has base unit: \(\text{m}\)

Step 2: Find the base units of each term on the Right-Hand Side (RHS):
First term \(ut\): \((\text{m}\cdot\text{s}^{-1}) \times (\text{s}) = \text{m}\)
Second term \(\frac{1}{2}at^2\): Pure numbers like \(\frac{1}{2}\) have no units. Acceleration \(\times\) time squared: \((\text{m}\cdot\text{s}^{-2}) \times (\text{s})^2 = \text{m}\)

Step 3: Compare all terms:
LHS has unit \(\text{m}\); RHS terms both have unit \(\text{m}\).
Since all terms share the same base units, the equation is homogeneous.

Crucial Examiner Note on Homogeneity:

Homogeneity is a necessary condition for an equation to be physically valid, but it is not sufficient on its own to prove the equation is correct.

Why? Unit analysis cannot detect incorrect pure dimensionless constants. For instance, the incorrect equation \(s = ut + 99at^2\) is still homogeneous because the number \(99\) has no units, even though the equation is physically wrong.

Key Takeaway: All physically valid equations must be homogeneous, but being homogeneous does not guarantee that dimensionless numerical values are correct.


6. Summary of Common Mistakes to Avoid

1. Using the Slash / Solidus: Writing \(\text{N/m}^2\) or \(\text{m/s}\) instead of \(\text{N}\cdot\text{m}^{-2}\) or \(\text{m}\cdot\text{s}^{-1}\) will lose you marks under CCEA formatting rules.
2. Naming Derived Units instead of Base Units: If the question asks for the base units of energy, writing "Joules (\(\text{J}\))" scores zero. You must write \(\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}\).
3. Base Mass Confusion: Remember that the base unit of mass is the kilogram (\(\text{kg}\)), not the gram (\(\text{g}\)). Do not remove the "kilo" from \(\text{kg}\) when converting to base units!
4. Area and Volume Conversions: Always square or cube the power of ten multiplier when converting \(\text{cm}^2\), \(\text{mm}^2\), \(\text{cm}^3\), or \(\text{mm}^3\) to \(\text{m}^2\) or \(\text{m}^3\).


Quick Review

Physical Quantity = Magnitude \(+\) Unit
6 Core Base Units: \(\text{kg}\), \(\text{m}\), \(\text{s}\), \(\text{A}\), \(\text{K}\), \(\text{mol}\)
Force in Base Units: \(\text{kg}\cdot\text{m}\cdot\text{s}^{-2}\)
Work / Energy in Base Units: \(\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}\)
Homogeneity Condition: Every additive term on both sides of an equation must share identical base units.