Units and Measurements · Lesson 1 of 7

Measurement and the SI System

“Express physical quantities using consistent standards, base units, derived units and meaningful conversions.”

Learning Objectives

• Explain a measurement as comparison with a standard. • Identify the seven SI base quantities and build derived units. • Convert compound units without losing powers. • Explain radians and steradians as ratios.

A number needs a reference

Saying that a table has length 2 does not tell another person how long it is. A measurement compares a physical quantity with an agreed standard called a unit. A length of 2 metres means that the length is twice the standard metre. The physical length stays the same when it is expressed as 200 centimetres; its numerical value changes because the unit changes.

Physical quantityLaTeX

Q = n u

Q is the quantity, n its numerical value and u its unit. The product represents the same quantity in every consistent unit system.

Words such as large, small and fast also need a comparison. An atom is small compared with a dust grain, and a jet is fast compared with a walking person. A proton being more massive than an electron already states a comparison. A count of molecules is a dimensionless number, so it differs from a mass or speed that requires a unit.

Base quantities and derived quantities

A system of units starts with a small set of independent base quantities. Other quantities are defined through relationships among them. Area is length multiplied by length, speed is distance divided by time, and density is mass divided by volume. Their units follow the same operations. CGS uses centimetre, gram and second for length, mass and time; FPS uses foot, pound and second; MKS uses metre, kilogram and second. SI extends the metre–kilogram–second framework to seven base quantities.

Base quantity	SI 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

Modern SI definitions use fixed values of physical constants: the caesium frequency for the second, the speed of light for the metre, and Planck’s constant for the kilogram. The ampere, kelvin, mole and candela are linked respectively to elementary charge, the Boltzmann constant, the Avogadro constant and a specified luminous efficacy. The purpose is a stable, reproducible standard. The long numerical definitions need not be memorised here. When using a mole, specify the entities being counted, such as atoms or molecules.

Derived quantity	Relationship	SI unit
Area	length × length	m²
Volume	length × length × length	m³
Speed	distance / time	m s⁻¹
Density	mass / volume	kg m⁻³
Force	mass × acceleration	kg m s⁻², called newton (N)
Energy	force × distance	kg m² s⁻², called joule (J)

Converting units with powers

Prefixes multiply a unit by powers of ten: kilo means one thousand, centi one hundredth, milli one thousandth and micro one millionth. Unit symbols are case sensitive and do not acquire a plural s. Some familiar units outside SI, such as minute, hour and litre, remain useful, but a calculation must use consistent units.

Converting volume and speed

Problem
Convert a cubic centimetre to cubic metres, and 18 kilometres per hour to metres per second.

1.Because $1 cm = 1 0^{- 2} m$ , cube the entire conversion: $1 c m^{3} = (1 0^{- 2})^{3} m^{3} = 1 0^{- 6} m^{3}$ .
2.Use $1 km = 1000 m$ and $1 h = 3600 s$ : $18 \times 1000/3600 = 5$ . Retaining two significant figures, the measured speed is $5.0 m s^{- 1}$ .
3.An area conversion requires the square of the length factor; an acceleration conversion requires the square of the time factor.

Angles measured as ratios

Plane and solid angleLaTeX

θ = \frac{s}{r}, Ω = \frac{A}{r ^{2}}

s is arc length, r is radius, and A is an intercepted area on a sphere centred at the vertex. Radian (rad) and steradian (sr) are named units for dimensionless quantities.

A plane angle compares a circular arc with its radius. A solid angle describes a spread of directions in space and compares the intercepted spherical area with the square of the radius. Length cancels in the first ratio and area cancels in the second. A full circle contains 2π radians, while a whole sphere subtends 4π steradians.

Plane angle and solid angle— The green patch lies on a sphere centred at O; it is curved, not a flat cross-section.

Reading the SI definitions

A definition fixes the value of a constant exactly; an actual laboratory realisation still has measurement uncertainty. For example, fixing the speed of light and defining the second lets a length be reproduced through a time measurement. Fixing Planck’s constant links the kilogram to units involving length and time. This replaces dependence on a single physical artefact with reproducible relationships.

Unit	Defining constant and fixed numerical value
second	Caesium-133 hyperfine transition frequency: 9 192 631 770 Hz
metre	Speed of light in vacuum: 299 792 458 m s⁻¹
kilogram	Planck constant: 6.62607015 × 10⁻³⁴ J s
ampere	Elementary charge: 1.602176634 × 10⁻¹⁹ C
kelvin	Boltzmann constant: 1.380649 × 10⁻²³ J K⁻¹
mole	Avogadro constant: 6.02214076 × 10²³ mol⁻¹
candela	Luminous efficacy: 683 lm W⁻¹ for radiation of frequency 540 × 10¹² Hz

These values explain the standards and serve as reference information. The central learning task is to connect each base quantity with its unit and symbol. A mole contains exactly 6.02214076 × 10²³ specified entities; the type of entity must always be stated.

Useful units outside SI

The following conversions reproduce the scope of the chapter’s reference table. Some are familiar everyday units; others occur in specialist measurements. Recognising a conversion is useful even when the corresponding physical topic is taught later. Older units such as curie and roentgen are included as reference, rather than recommended replacements for SI units.

Unit	Symbol	Equivalent
minute	min	60 s
hour	h	3600 s
day	d	86 400 s
year, using 365.25 days	y	31 557 600 s ≈ 3.15576 × 10⁷ s
degree	°	π/180 rad
litre	L	1 dm³ = 10⁻³ m³
tonne	t	10³ kg
carat	c in the source table	200 mg
bar	bar	10⁵ Pa = 0.1 MPa
curie	Ci	3.7 × 10¹⁰ s⁻¹
roentgen	R	2.58 × 10⁻⁴ C kg⁻¹
quintal	q	100 kg
barn	b	100 fm² = 10⁻²⁸ m²
are	a	100 m²
hectare	ha	10⁴ m²
standard atmosphere	atm	101 325 Pa ≈ 1.013 × 10⁵ Pa

Prefix	Symbol	Multiplier
giga	G	10⁹
mega	M	10⁶
kilo	k	10³
deci	d	10⁻¹
centi	c	10⁻²
milli	m	10⁻³
micro	µ	10⁻⁶
nano	n	10⁻⁹
pico	p	10⁻¹²
femto	f	10⁻¹⁵

Write a space between a numerical value and its unit symbol, such as 5 m. Use kg for kilogram and K for kelvin; kg and Kg are not interchangeable. Unit symbols do not take a full stop unless it ends a sentence. A prefix belongs to the unit before an exponent is applied: a square centimetre means the square of one centimetre. The watt is another named derived unit: one watt is one joule per second, so its base-unit expression is kg m² s⁻³.

One radian and one steradian

Interpreting an angular unit

Problem
What do one radian and one steradian represent?

1.If the arc length equals the radius, $s = r$ , then $θ = s / r = 1 rad$ . This angle is about $57. 3^{\circ}$ .
2.A complete circumference is $2 π r$ , giving $2 π$ radians. Since it is also $36 0^{\circ}$ , $1^{\circ} = π /180$ radians.
3.If the spherical patch area is $A = r^{2}$ , its solid angle is $1 sr$ . The whole sphere has area $4 π r^{2}$ , so it subtends $4 π$ steradians.

Using the angle ratios

Problem
An arc is 6 cm long on a circle of radius 12 cm. A patch of area 8 m² lies on a sphere of radius 2 m. Find the angles.

1.The plane angle is $6/12 = 0.5 rad$ . Convert to degrees by multiplying by $180/ π$ , giving approximately $28. 6^{\circ}$ .
2.The solid angle is $8/ 2^{2} = 2 sr$ . The denominator is radius squared, not radius.
3.These simple values illustrate the geometry; measurement uncertainty is not specified.

Relative density and density

Relative densityLaTeX

ρ_{relative} = \frac{ρ}{ρ _{reference}}

For a solid or liquid, the reference is usually water under specified conditions. The ratio has no unit.

Recovering the density of lead

Problem
Lead has relative density 11.3. Use water density 1.00 g/cm³.

1.Multiply the ratio by the reference density: $ρ = 11.3 \times 1.00 = 11.3 g c m^{- 3}$ .
2.Because $1 g = 1 0^{- 3} kg$ and $1 c m^{3} = 1 0^{- 6} m^{3}$ , $1 g c m^{- 3} = 1 0^{3} kg m^{- 3}$ .
3.Thus the lead density is $1.13 \times 1 0^{4} kg m^{- 3}$ .

Distances defined using light

A light-year is a unit of distance: the distance light travels in vacuum during one year. The word year describes the time used to define the unit; it does not make a light-year a time unit. Distance is speed multiplied by elapsed time.

Building and using a light-year

Problem
Use c = 3.00 × 10⁸ m/s and a year of 365.25 days to find one light-year and express one metre in light-years.

1.Convert the year: $365.25 \times 24 \times 3600 = 3.15576 \times 1 0^{7} s$ .
2.Use $d = c t$ : $1 ly \approx (3.00 \times 1 0^{8}) (3.15576 \times 1 0^{7}) = 9.47 \times 1 0^{15} m$ to three significant figures.
3.Invert the conversion: $1 m \approx 1.06 \times 1 0^{- 16} ly$ . The rounded value of c sets the precision in this calculation.

A light-second is the distance light travels in one second. If the time unit remains the second and the length unit becomes the light-second, the numerical speed of light is 1. A travel time of 8 minutes 20 seconds is 500 seconds, so the Sun–Earth distance is 500 light-seconds. Both the chosen length unit and retained time unit must be stated.

Quick check

Which conversion is correct?

Key Takeaways

• A measurement combines a numerical value and a unit. • Derived units follow the algebra of the defining relationship. • Apply conversion factors to every power in a compound unit. • Plane and solid angles are dimensionless ratios with named units.

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Next · Lesson 2

Precision and Significant Figures