SI units, prefixes and homogeneity

A number on its own says nothing in physics: 5 could be five metres, five seconds or five kilograms. Every measured quantity is a number multiplied by a unit, and every unit used in physics can be built from a handful of SI base units. This lesson sets out the base units and prefixes, shows how to write any derived unit in base units, and uses base units to check whether an equation can be right.

SI base quantities and units

The SI system is built on a small set of base quantities, each with its own unit. You need these six:

SI also has luminous intensity as a base quantity, with unit candela; it is not needed here. A derived quantity is defined by an equation linking it to base quantities. Its unit follows from that equation.

Prefixes

Prefixes stand for powers of ten, so that very large and very small values can be written compactly.

To convert, replace the prefix by its power of ten: 7.2 GW = 7.2 × 10⁹ W. Capital and small letters mean different things: m is milli, M is mega.

When a prefixed unit is squared or cubed, the prefix is squared or cubed too, because it belongs to every length in the product:

The base unit of mass already carries a prefix: it is the kilogram, not the gram. So 1 g = 10⁻³ kg and 1 mg = 10⁻⁶ kg.

Guided practice 1

Milligrams and square millimetres

About 4 min

Problem

Convert 35 mg to the SI base unit of mass. A rectangle has sides 3.0 mm and 2.0 mm: find its area in m².

Convert both quantities

Mass

kgUnit: kg

Show the steps Optional working

Step 1: First side length in metres

mUnit: m

Area

m²Unit: m²

Check answer

Hints

Hint 1

mass

The base unit is the kilogram, so convert milligrams to grams and then grams to kilograms.

Hint 2

area

An area has two length factors, so the factor 10⁻³ appears twice.

Show solution step by step

1. Convert the mass

   Reason

   Each prefix is a power of ten, and the base unit is the kilogram.

   Working

   Answer

   35 mg = 35 × 10⁻³ g = 35 × 10⁻⁶ kg = 3.5 × 10⁻⁵ kg

2. Convert the area

   Working

   Answer

   A = (3.0 × 10⁻³ m)(2.0 × 10⁻³ m) = 6.0 × 10⁻⁶ m²

Derived units in base units

To write a derived unit in base units, take the equation that defines the quantity and replace each quantity by its unit.

Work down the table: each row uses units already found above it. For the volt, J C⁻¹ = (kg m² s⁻²)/(A s) = kg m² s⁻³ A⁻¹.

Check your understanding 1

Use R = V/I and the table to express the ohm in SI base units.

Your unit derivation

Show answer

Ω = V A⁻¹ = (kg m² s⁻³ A⁻¹)/A = kg m² s⁻³ A⁻²

Checking that an equation is homogeneous

You can only add, subtract or equate quantities of the same kind: a length plus a time means nothing. So in a correct physical equation, every term added or subtracted, and the two sides of the equation, must have the same base units. Such an equation is homogeneous.

A homogeneity check can prove an equation wrong: if the base units do not match, the equation cannot be right. It cannot prove an equation right, because pure numbers such as 1/2 or 2π have no units, and an equation can have the right units but the wrong physics.

Worked example 2

Test two equations for homogeneity

Problem

Show that p = ρ gh is homogeneous, where p is pressure, ρ density, g the acceleration of free fall and h depth. Then test s = ut + at² and s = ut + (1/2)at², where s is displacement, u initial velocity, t time and a acceleration.

Show full solution

1. Write the base units of each side

   Method

   Find the base units of p, then of ρ gh.

   Reason

   A homogeneous equation has the same base units on both sides.

   Working

   Answer

   Left: Pa = kg m⁻¹ s⁻². Right: (kg m⁻³)(m s⁻²)(m) = kg m⁻¹ s⁻². The units match, so p = ρ gh is homogeneous.

2. Check every term in a sum

   Method

   Find the base units of s, ut and at².

   Reason

   Terms that are added must have the same units as each other and as the left-hand side.

   Working

   Answer

   s: m. ut: (m s⁻¹)(s) = m. at²: (m s⁻²)(s²) = m.

3. Draw the conclusion

   Working

   Answer

   Both displacement equations are homogeneous, but only one is correct. Homogeneity cannot tell them apart, because the factor 1/2 has no units.

Try it yourself 3

The constant in a pendulum equation

Minimal support

Problem

The period of a pendulum is proposed to be T = C square root of (l/g), where l is the length of the pendulum and g the acceleration of free fall. Find the base units of C. What can a homogeneity check not tell you about C?

Find the units of C

Units of C, and what the check cannot tell you

Hints

Hint 1

inside the root

Find the base units of l/g first, then take the square root and compare with the unit of T.

Show solution step by step

1. Find the units of the square root

   Reason

   C must supply whatever is needed to turn square root of (l/g) into seconds.

   Working

   Answer

   l/g: m/(m s⁻²) = s², so square root of (l/g) has unit s.

2. Conclude

   Working

   Answer

   square root of (l/g) already has the unit of T, so C has no units. The check cannot find its value (2π), and it cannot show whether a real pendulum obeys the equation.

Try it yourself 4

Units in a homogeneous sum

Minimal support

Problem

The equation z² = p + qt is homogeneous. t is a time in seconds and q has unit m s⁻¹. Find the SI units of p and of z.

Find the units

Units of p and z, with your reasoning

Hints

Hint 1

start inside the sum

Find the unit of qt. Every other term, and z², must have the same unit.

Show solution step by step

1. Find the unit of the known term

   Reason

   q is a speed and t a time.

   Working

   Answer

   qt: (m s⁻¹)(s) = m

2. Match the other terms

   Working

   Answer

   p is added to qt, so p has unit m. z² also has unit m, so z has unit m^(1/2).

Quick check

An unusual unit such as m^(1/2) is allowed for a quantity defined this way.

Common mistakes

• Treating the gram as the base unit of mass. It is the kilogram.
• Converting cm² or cm³ with 10⁻² once, instead of squaring or cubing it.
• Mixing up m (milli) and M (mega).
• Saying an equation is correct because it is homogeneous. Homogeneity can only show that an equation is wrong.

Before you move on

Check your understanding 2

Convert 2.0 (μ m)² to m².

Area in square metres

m²Unit: m²

Check answer

Show answer

The area is 2.0 × 10⁻¹² m² because the micro prefix is squared.

Check your understanding 3

A proposed energy equation is E = kmv³, where m is mass and v speed. Find the SI base units required for k. Could k be a pure number such as 1/2?

Your derivation of the units of k and conclusion

Show answer

Dividing the energy unit by the unit of mv³ gives [k] = (kg m² s⁻²)/(kg m³ s⁻³) = m⁻¹ s. A pure number has no units, so it cannot make this equation homogeneous.