Physics (9702) | SI units

Welcome to SI Units: The Language of Physics!

Hey there, future physicist! Starting out in A-Level Physics means establishing a solid foundation, and nothing is more fundamental than understanding units. Think of units as the language scientists use to communicate measurements clearly and consistently across the world. If you use the wrong units, things can go disastrously wrong (like the famous NASA Mars orbiter that crashed because one team used imperial units instead of metric!).

This chapter is all about the international standard—the Système International d’Unités (SI Units). Don't worry, it's straightforward, logical, and vital for every calculation you'll do!

1. The Foundation: SI Base Quantities and Units (1.2.1)

In physics, everything we measure can be broken down into a handful of fundamental, independent components. These are called Base Quantities, and they have corresponding SI Base Units.

What is a Base Quantity?

A Base Quantity is a physical quantity that is not defined in terms of other quantities. They are the 'building blocks' of all other measurements.

The syllabus requires you to recall five core base quantities and their units (plus 'amount of substance' which appears later in the syllabus but is officially the sixth primary base unit you need to know):

Quick List of Essential SI Base Units

• Mass: kilogram (kg)
• Length: metre (m)
• Time: second (s)
• Electric Current: ampere (A)
• Thermodynamic Temperature: kelvin (K)
• Amount of Substance: mole (mol) (Covered fully in topic 15.1)

(Side Note: The seventh base unit, the candela (cd) for luminous intensity, is not covered in the AS/A Level Physics 9702 syllabus.)

2. Building Blocks: Derived Units (1.2.2)

Most quantities you deal with—like speed, force, or energy—are combinations of the base quantities. Their units, therefore, are combinations of the base units. These are called Derived Units.

How to Express Derived Units

You must be able to express any derived unit as a product or quotient (multiplication or division) of the SI base units.

The Trick: Always start with the defining equation for the quantity.

Step-by-Step Example: Finding the Base Units for Force

Quantity: Force (\(F\))

1. Start with the equation: Newton's Second Law: \(F = ma\)
2. Substitute units for quantities:
   • Unit of mass (\(m\)): \(\text{kg}\)
   • Unit of acceleration (\(a\)): \(\text{m} / \text{s}^2\) or \(\text{m} \cdot \text{s}^{-2}\)
3. Combine them:
   Unit of Force = \(\text{kg} \cdot \text{m} \cdot \text{s}^{-2}\)
4. Name the Derived Unit: We call this the Newton (N).

Therefore, \(1 \text{ N} = 1 \text{ kg} \cdot \text{m} \cdot \text{s}^{-2}\)

Example 2: Finding the Base Units for Energy (Work Done)

Quantity: Work or Energy (\(W\))

1. Start with the equation: Work done = Force \(\times\) distance: \(W = Fd\)
2. Substitute base units:
   • Unit of Force (\(F\)): \(\text{kg} \cdot \text{m} \cdot \text{s}^{-2}\) (from Example 1)
   • Unit of distance (\(d\)): \(\text{m}\)
3. Combine them:
   Unit of Work = \((\text{kg} \cdot \text{m} \cdot \text{s}^{-2}) \times \text{m}\)
   Unit of Work = \(\text{kg} \cdot \text{m}^2 \cdot \text{s}^{-2}\)
4. Name the Derived Unit: We call this the Joule (J).

Key Takeaway: If you forget the base units for a derived quantity (like the Joule), just remember the key formula (like \(W=Fd\)) and substitute the base units you know!

3. Using Prefixes: Handling Big and Small Numbers (1.2.4)

Physics often deals with incredibly huge numbers (like the speed of light) or incredibly tiny numbers (like the size of an atom). To make these numbers manageable, we use Prefixes.

Prefixes and Powers of Ten

You must recall and use the following prefixes, which represent decimal multiples or submultiples of base or derived units:

Multiples (Making Units Bigger)

• Tera (T): \(10^{12}\) (e.g., 1 TB hard drive)
• Giga (G): \(10^9\) (e.g., 1 GHz processor)
• Mega (M): \(10^6\) (e.g., 1 MW of power)
• kilo (k): \(10^3\) (e.g., 1 km = 1000 m)

Submultiples (Making Units Smaller)

• deci (d): \(10^{-1}\)
• centi (c): \(10^{-2}\) (e.g., 1 cm = 0.01 m)
• milli (m): \(10^{-3}\)
• micro (\(\mu\)): \(10^{-6}\)
• nano (n): \(10^{-9}\)
• pico (p): \(10^{-12}\)

Conversion Tip: Always Go Back to Base

When solving problems, especially complex ones, the safest way to avoid calculation errors is to convert all given values into their basic SI units (no prefixes) before starting the calculation.

Example: Convert 20 \(\mu\text{F}\) (microfarads) to Farads (\text{F}).
\(20 \mu\text{F} = 20 \times 10^{-6} \text{ F}\)

Example: Convert 5 GJ (gigajoules) to Joules (\text{J}).
\(5 \text{ GJ} = 5 \times 10^9 \text{ J}\)

4. Checking Equations: Homogeneity (1.2.3)

One of the most powerful tools in Physics is called dimensional analysis, or what the syllabus refers to as checking the homogeneity of physical equations.

What does 'Homogeneous' mean?

An equation is homogeneous if the units on the Left-Hand Side (LHS) are exactly the same as the units on the Right-Hand Side (RHS).

Why is this important?

If an equation is not homogeneous, it is mathematically incorrect and cannot describe a real physical process. If it is homogeneous, it might be correct (but unit checking can't prove numerical factors like \(\frac{1}{2}\) or \(2\pi\)).

Step-by-Step Example: Checking the Equation for Displacement

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

Goal: Check if the unit of \(s\) matches the units of \(ut\) and \(\frac{1}{2}at^2\).

1. Units of LHS (Displacement, \(s\)):
   The unit of displacement is \(\text{m}\).
2. Units of the first term on RHS (\(ut\)):
   • \(u\) (initial velocity): \(\text{m} \cdot \text{s}^{-1}\)
   • \(t\) (time): \(\text{s}\)
   • Unit of \(ut\): \((\text{m} \cdot \text{s}^{-1}) \cdot (\text{s}) = \text{m} \cdot \text{s}^0 = \B \text{m}\)
3. Units of the second term on RHS (\(\frac{1}{2}at^2\)):
   • The constant \(\frac{1}{2}\) has no units.
   • \(a\) (acceleration): \(\text{m} \cdot \text{s}^{-2}\)
   • \(t^2\) (time squared): \(\text{s}^2\)
   • Unit of \(\frac{1}{2}at^2\): \((\text{m} \cdot \text{s}^{-2}) \cdot (\text{s}^2) = \text{m} \cdot \text{s}^0 = \B \text{m}\)

Conclusion: Since the units of all terms in the equation are metres (\(\text{m}\)), the equation is homogeneous (or dimensionally consistent).

Example 2: Checking the Formula for Power (\(P\))

Suppose you derive a new formula for power: \(P = Fv^2\) (Force times velocity squared).

1. Units of LHS (Power, \(P\)):
Power is Work/Time. We found Work is \(\text{kg} \cdot \text{m}^2 \cdot \text{s}^{-2}\).
So, Power \(P\) unit: \((\text{kg} \cdot \text{m}^2 \cdot \text{s}^{-2}) / \text{s} = \text{kg} \cdot \text{m}^2 \cdot \text{s}^{-3}\) (The Watt, W).

2. Units of RHS (\(Fv^2\)):
• \(F\) (Force): \(\text{kg} \cdot \text{m} \cdot \text{s}^{-2}\)
• \(v^2\) (Velocity squared): \((\text{m} \cdot \text{s}^{-1})^2 = \text{m}^2 \cdot \text{s}^{-2}\)
• Unit of \(Fv^2\): \((\text{kg} \cdot \text{m} \cdot \text{s}^{-2}) \cdot (\text{m}^2 \cdot \text{s}^{-2}) = \text{kg} \cdot \text{m}^3 \cdot \text{s}^{-4}\)

Conclusion: \(\text{kg} \cdot \text{m}^2 \cdot \text{s}^{-3}\) is NOT equal to \(\text{kg} \cdot \text{m}^3 \cdot \text{s}^{-4}\).
This proposed equation is not homogeneous and must be incorrect! (The correct equation is \(P=Fv\)).

Chapter Key Takeaways

• Units ensure universal scientific communication.
• Recall the six primary SI Base Units (kg, m, s, A, K, mol).
• Derived units are combinations of base units; use the defining formula to find them.
• Always convert values using prefixes (\(10^x\)) into base units before calculation.
• Checking for homogeneity (unit consistency across an equation) confirms if a formula is possibly correct.