Measurement Units and Dimensions

Overview

Units and dimensions allow physicists to express measurements consistently, compare results, and test equations logically.

This page deepens the unit-based part of Measurement and focuses on:

• SI base quantities and units
• derived quantities and units
• dimensional formulae
• homogeneity of equations
• SI prefixes
• worked examples

Understanding this topic is essential for all later chapters such as Kinematics, Forces, and Current Electricity Fundamentals.

Why It Matters

Units and dimensions let physicists express results consistently, compare quantities properly, and test whether equations are plausible.

Definition

A physical quantity is a measurable property described by a numerical value and a unit.

Key Representations

Examples:

$$[v]=L{T}^{-1}$$ $$[F]=ML{T}^{-2}$$

Physical Quantities and Units

A physical quantity is a measurable property described by:

• numerical value
• unit

Example:

$$5.20\text{m}$$

The number alone is incomplete without the unit.

SI Base Quantities

The SI system is built on base quantities.

Base Quantity	Unit	Symbol
length	metre	m
mass	kilogram	kg
time	second	s
electric current	ampere	A
thermodynamic temperature	kelvin	K
amount of substance	mole	mol
luminous intensity	candela	cd

For H2 Physics, the most frequently used are:

• m
• kg
• s
• A
• K

Derived Quantities

Derived quantities are formed by combining base quantities through multiplication or division.

Quantity	Formula	Unit
area	length × length	m${}^2$
volume	length${}^3$	m${}^3$
speed	distance / time	m s${}^{-1}$
acceleration	velocity / time	m s${}^{-2}$
density	mass / volume	kg m${}^{-3}$
force	mass × acceleration	N
pressure	force / area	Pa
energy	force × distance	J
power	energy / time	W
charge	current × time	C

Named Derived Units

Some derived units are given special names.

Quantity	Unit Name	Equivalent Base Units
force	newton (N)	kg m s${}^{-2}$
energy	joule (J)	kg m${}^2$ s${}^{-2}$
power	watt (W)	kg m${}^2$ s${}^{-3}$
pressure	pascal (Pa)	kg m${}^{-1}$ s${}^{-2}$
charge	coulomb (C)	A s
potential difference	volt (V)	kg m${}^2$ s${}^{-3}$ A${}^{-1}$
resistance	ohm ($\Omega$)	kg m${}^2$ s${}^{-3}$ A${}^{-2}$

Dimensions

Dimensions describe the physical type of a quantity, independent of chosen units.

Common symbols:

• Mass: $M$
• Length: $L$
• Time: $T$
• Current: $I$
• Temperature: $\Theta$

Examples:

Quantity	Dimensional Formula
length	$L$
area	$L^2$
volume	$L^3$
velocity	$L{T}^{-1}$
acceleration	$L{T}^{-2}$
force	$ML{T}^{-2}$
momentum	$ML{T}^{-1}$
energy	$M{L}^2{T}^{-2}$
power	$M{L}^2{T}^{-3}$
pressure	$M{L}^{-1}{T}^{-2}$

How to Derive Dimensions

Example 1: Force

Using:

$$F=ma$$

Mass has dimension $M$.

Acceleration has dimension:

$$L{T}^{-2}$$

So:

$$[F]=M(L{T}^{-2})=ML{T}^{-2}$$

Example 2: Energy

Using:

$$E=Fs$$

Force is $ML{T}^{-2}$ and displacement is $L$.

Thus:

$$[E]=(ML{T}^{-2})(L)=M{L}^2{T}^{-2}$$

Example 3: Pressure

Using:

$$p=\frac{F}{A}$$ $$[p]=\frac{ML{T}^{-2}}{L^2}=M{L}^{-1}{T}^{-2}$$

Principle of Homogeneity

Every physically valid equation must have the same dimensions on both sides.

Example 1

$$v=u+at$$

• $u$ has dimension $L{T}^{-1}$
• $at=L{T}^{-2}(T)=L{T}^{-1}$

Hence dimensionally consistent.

Example 2

$$s=ut+\frac{1}{2}a{t}^2$$

• $ut=L$
• $a{t}^2=L{T}^{-2}(T^2)=L$

So valid dimensionally.

Important Warning

A dimensionally correct equation may still be physically wrong.

Example:

$$s=ut+a{t}^2$$

Dimensions are correct, but coefficient should be $\frac{1}{2}$.

So dimensional analysis checks consistency, not exact correctness.

Using Dimensions to Find Unknown Units

Example

Given:

$$P=\frac{W}{t}$$

Power = energy / time

Energy unit = J

So:

$$\text{unit of }P={\text{J s}}^{-1}=\text{W}$$

Example

Given:

$$R=\frac{V}{I}$$

Unit of resistance:

$$\frac{\text{V}}{\text{A}}=\Omega$$

SI Prefixes

Prefix	Symbol	Value
pico	p	$10^{-12}$
nano	n	$10^{-9}$
micro	$\mu$	$10^{-6}$
milli	m	$10^{-3}$
centi	c	$10^{-2}$
kilo	k	$10^3$
mega	M	$10^6$
giga	G	$10^9$
tera	T	$10^{12}$

Prefix Conversion Examples

Example 1

$$5.0\text{mm}=5.0\times 10^{-3}\text{m}$$

Example 2

$$3.2\text{kV}=3.2\times 10^3\text{V}$$

Example 3

$$8.0\mu \text{s}=8.0\times 10^{-6}\text{s}$$

Orders of Magnitude

Order of magnitude = nearest power of ten estimate.

Examples:

Quantity	Approximate Order
atom diameter	$10^{-10}\text{ m}$
cell size	$10^{-5}\text{ m}$
human height	$10^0\text{ m}$
Earth radius	$10^6\text{ m}$

Useful for checking whether answers are sensible.

Worked Examples

Example 1: Unit of Gravitational Field Strength

Using:

$$g=\frac{F}{m}$$

Unit:

$$\frac{N}{kg}=m{s}^{-2}$$

(also equivalent)

Example 2: Unit of Electric Field Strength

Using:

$$E=\frac{F}{Q}$$

Unit:

$$\frac{N}{C}$$

Also:

$$V{m}^{-1}$$

Example 3: Dimension of Frequency

Frequency:

$$f=\frac{1}{T}$$

So:

$$[f]=T^{-1}$$

Common Exam Mistakes

• forgetting unit conversions
• mixing cm and m
• writing N as base unit instead of named unit when asked
• confusing unit with dimension
• assuming dimensional correctness proves formula true
• missing powers such as m${}^2$, m${}^3$

Fast Revision Summary

• Base units form the SI foundation.
• Derived units come from combinations of base units.
• Dimensions describe quantity type.
• Homogeneous equations have matching dimensions.
• Prefixes simplify large or small values.
• Orders of magnitude help estimate sensible answers.

Links

• Measurement
• Measurement Uncertainty and Errors
• Uncertainty Propagation Methods
• Measurement Data Presentation
• Measurement Estimation and Experimental Design
• Kinematics
• Forces
• Current Electricity Fundamentals