SI Units: Fundamental and Derived Quantities

Understanding the building blocks of measurement in mechanics

SPEC 6.1 • MECHANICS

Why Units Matter in Mechanics

In mechanics, every quantity must have consistent units. Using incorrect units can lead to catastrophic errors.

Real World Example

Mars Climate Orbiter (1999): NASA lost a $125 million spacecraft because one team used metric units (Newtons) while another used imperial units (pounds-force). The probe crashed into Mars.

Key Concept

The SI system (Système International) provides a universal standard for scientific measurements. All mechanics calculations must use consistent SI units.

Fundamental SI Base Units

In A-Level Mechanics, we use three fundamental base units from which all other units are derived:

Mass

kilogram

Length

metre

Time

second

Definition

Base units are the fundamental units that cannot be expressed in terms of other units. All other units in mechanics are derived from these three.

Derived Units: Velocity and Acceleration

Velocity

v = \frac{s}{t}

[v] = \frac{m}{s} = m s^{- 1}

Acceleration

a = \frac{v}{t}

[a] = \frac{m s ^{- 1}}{s} = m s^{- 2}

Notation

We can write units as fractions (m/s) or using negative indices (m s⁻¹). In A-Level, the index notation is preferred: m s⁻¹, m s⁻²

Common Value

Acceleration due to gravity:

g = 9.8 m s^{- 2}

(often approximated as 10 m s⁻²)

Derived Units: Force

Force is defined by Newton's Second Law:

Newton's Second Law

F = ma

The SI unit of force is the Newton (N). Derived from base units:

Derivation

[F] = [m] [a] = kg \times m s^{- 2} = kg m s^{- 2}

Definition

1 Newton is the force required to accelerate a mass of 1 kg at 1 m s⁻².

1 N = 1 kg m s^{- 2}

Derived Units: Weight and Moment

Weight

W = m g

Unit: N (Newton)

Weight is a force!

Moment

M = F \times d

Unit: N m

= kg m² s⁻²

Important

N m (Newton metres) for moments is NOT the same as J (Joules) for energy, even though they have the same base units. Context matters!

More Derived Units

Momentum

p = m v

[p] = kg \times m s^{- 1} = kg m s^{- 1}

Also written as N s

Kinetic Energy

K E = \frac{1}{2} m v^{2}

[K E] = kg \times m^{2} s^{- 2} = kg m^{2} s^{- 2}

= J (Joule)

Remember

Any quantity can be expressed in base units by substituting the formulas. This is essential for dimensional analysis.

Dimensional Analysis

Dimensional analysis is a technique to check if equations are dimensionally consistent — all terms must have the same units.

Method

1. Write out the units of each term in base units

2. Check that all terms on both sides have identical units

3. If units match → equation is dimensionally consistent

4. If units don't match → equation is definitely wrong

Example: Checking v² = u² + 2as

LHS:

[v^{2}] = (m s^{- 1})^{2} = m^{2} s^{- 2}

RHS:

[u^{2}] = m^{2} s^{- 2}

and

[2 a s] = m s^{- 2} \times m = m^{2} s^{- 2}

All terms have units m² s⁻² ✓

Limitation

Dimensional consistency doesn't prove an equation is correct — it only proves it's not obviously wrong. Dimensionless constants (like 2 or π) cannot be checked.

Unit Conversions

The most common conversion in mechanics is between km h⁻¹ and m s⁻¹.

Conversion Factors

1 km = 1000 m

1 hour = 3600 s

1 km h⁻¹ =

\frac{1000}{3600}

m s⁻¹ =

\frac{1}{3.6}

m s⁻¹

km h⁻¹ → m s⁻¹

Divide by 3.6

m s⁻¹ → km h⁻¹

Multiply by 3.6

Quick Check

72 km h⁻¹ = 72 ÷ 3.6 = 20 m s⁻¹

15 m s⁻¹ = 15 × 3.6 = 54 km h⁻¹

Summary Table of Units

QuantitySymbolSI UnitBase Units

Mass

m

kgkg (base)

Length/Displacement

s, x, d

mm (base)

Time

t

ss (base)

Velocity

v, u

m s⁻¹m s⁻¹

Acceleration

a

m s⁻²m s⁻²

Force/Weight

F, W

Nkg m s⁻²

Moment

M

N mkg m² s⁻²

Momentum

p

N skg m s⁻¹

Summary

Base SI Units in Mechanics

Mass: kg | Length: m | Time: s

Key Exam Skills

Derive units for any mechanical quantity using formulas

Use dimensional analysis to check if equations are valid

Convert between km h⁻¹ and m s⁻¹ (divide/multiply by 3.6)

Express derived units in base unit form (kg, m, s)

Key Formula

All force-related units derive from F = ma, so:

1 N = 1 kg m s⁻²