Measurement Units and Dimensions

Branch note: This page deepens one part of Measurement.

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.

Core Ideas

• A dimensional physical quantity is reported using a numerical value and unit; a dimensionless physical quantity has SI unit one, whose symbol is usually omitted.
• SI base units define the measurement system; derived units are built from them.
• Dimensional analysis checks whether equations are plausible, but it cannot prove that an equation is physically correct.

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:

$$5.20\text{m}$$

For a dimensional quantity, the number alone is incomplete without the unit. Dimensionless ratios such as refractive index and strain have SI unit one, usually written without a unit symbol.

SI Base Quantities

The SI system is built on base quantities. For 9749, recall the following six base quantities and units.

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

Wider SI enrichment

The complete SI system also includes luminous intensity, measured in candela (cd). It is not part of the six-item 9749 recall list.

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${}^{-2}$

Checking Homogeneity with SI Base Units

The syllabus method is to express every term in SI base units. An equation is homogeneous only if terms that are added or equated have the same base unit.

For example, in $v=u+at$, both $u$ and $at$ have unit $\mathrm{m}{\mathrm{s}}^{-1}$. In $p+\rho gh+\frac{1}{2}\rho v^2=k$, all three terms have unit $\mathrm{kg}{\mathrm{m}}^{-1}{\mathrm{s}}^{-2}$, so $k$ has the same unit.

Homogeneity is a necessary condition for a correct physical equation, not a sufficient one: it cannot detect a wrong dimensionless number such as replacing $\frac{1}{2}a{t}^2$ by $3a{t}^2$.

Enrichment: Dimensional Symbols

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	$10^{-2}$
deci	d	$10^{-1}$
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

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

Prefixes on squared and cubed units

Apply the exponent to the whole conversion factor:

$$1\mathrm{c}{\mathrm{m}}^2=(10^{-2}\mathrm{m}{)}^2=10^{-4}{\mathrm{m}}^2,$$

not $10^{-2}{\mathrm{m}}^2$. Similarly, $1\mathrm{c}{\mathrm{m}}^3=10^{-6}{\mathrm{m}}^3$.

Orders of Magnitude

In this note, an order-of-magnitude estimate means the power-of-ten scale of a quantity when written as $a\times 10^n$ with $1\le a<10$. It is used for plausibility checks; questions that explicitly request the nearest power of ten require comparing with the boundary $\sqrt{10}\times 10^n$.

Examples:

Quantity	Approximate Order
atom diameter	$10^{-10}\text{ m}$
cell size	$10^{-5}\text{ m}$
human height	$10^0\text{ m}$
Earth radius ($6.4\times 10^6\mathrm{m}$)	$10^6\mathrm{m}$ scale; nearest power is $10^7\mathrm{m}$

Useful for checking whether answers are sensible.

Worked Examples

Example 1: Unit of Gravitational Field Strength

Using:

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

Unit:

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

(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:

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

Exam Relevance

In exams, units and dimensions are commonly tested through base-unit conversion, homogeneity checks, prefix conversion, and order-of-magnitude estimates.

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