CDS Physics · Units, Measurement and Dimensions

Units, Prefixes and Dimensions

The SI units CDS asks about, how to read a unit off a formula, the powers of ten behind each prefix, and how to write and compare dimensions.

All Units, Measurement and Dimensions notes

Why this matters

Five CDS questions from 2016 (II) to 2021 (I), two of them in the 2016 (II) paper. Each one is a single step: name a unit, convert a prefix, or match two quantities by their dimensions.

Concept 1 of 3: SI units and reading a unit off a formula

Every unit in physics is built from a few base units: the metre, the kilogram and the second. If you know the formula for a quantity, you know its unit: put the units of each term into the formula and simplify. A spring constant is force per extension, so its unit is newton per metre. Pressure is force per area, so its unit is newton per square metre, which is called the pascal.

Definition

Units CDS has asked about:

Pressure: SI unit the pascal, 1 Pa = 1 N/m².
Standard atmosphere: the pressure of a 760 mm column of mercury at 0 °C, about 101325 Pa.
Bar: exactly $10^{5}$ Pa, a little less than one atmosphere.
Poise: the CGS unit of viscosity, not of pressure.
Spring constant $k$ : from Hooke's law $F = kx$ , $k = F/x$ , so its unit is newton per metre (N/m).
Method: rearrange the formula so the quantity stands alone, then replace each symbol by its unit.

Quantity or unit	What to remember
Pressure	Force per area; SI unit pascal (N/m²)
1 atmosphere	760 mm of mercury at 0 °C, about 101325 PaQ CDS 2016 (II): a 760 mm mercury column at 0 °C is one atmosphere, not one pascal or one bar.
1 bar	10⁵ Pa, slightly less than 1 atmosphere
Poise	Unit of viscosity (CGS)
Spring constant k	Force per extension: N/mQ CDS 2020 (I): the unit of the force constant of a spring is N/m.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2016 · CDS (II) 2016 — General Knowledge · Q19Easy

Example 1 · Units, Measurement and Dimensions · Units, Prefixes and Dimensions

The pressure exerted by a 760 mm column of mercury at

0\ ^\circ\text{C}

is known as

Atmosphere, bar and pascal are three different sizes

One atmosphere is the pressure of 760 mm of mercury, about 101325 Pa. One bar is exactly 100000 Pa. One pascal is tiny, one newton on a square metre. The mercury column always names the atmosphere.

Spring constant is per metre, not per square metre

k = F/x

divides a force by a length, so its unit is N/m. N/m² divides a force by an area: that is the unit of pressure or stress.

Drill 1 more on si units and reading a unit off a formula

Concept 2 of 3: Prefixes, powers of ten and the nanoscale

A prefix is a shorthand for a power of ten stuck in front of a unit. Kilo means a thousand, milli means a thousandth, nano means a billionth. To change a length from one prefix to another, write both as powers of ten of the metre and divide. Nanotechnology gets its name from the nanometre, because it works with things between about 1 and 100 nanometres across.

Definition

Lengths as powers of ten of the metre:

kilometre $10^{3}$ m, centimetre $10^{-2}$ m, millimetre $10^{-3}$ m.
micrometre (micron, μm) $10^{-6}$ m, nanometre $10^{-9}$ m, picometre $10^{-12}$ m.
Ångström (Å) $10^{-10}$ m, so 1 nm = 10 Å.
To convert, divide the powers: from metres to centimetres, multiply by $10^{2}$ .
Nanotechnology studies and builds structures of about 1 to 100 nm, where materials behave differently from the bulk solid.

Prefix or unit	Power of ten (in metres)
kilo (km)	10³
centi (cm)	10⁻²
milli (mm)	10⁻³
micro (μm)	10⁻⁶
nano (nm)	10⁻⁹, which is 10⁻⁷ cmQ CDS 2018 (I): one nanometre is 10⁻⁷ cm.
ångström (Å)	10⁻¹⁰
Nanoscale	About 1 to 100 nmQ CDS 2021 (I): the particles studied in nanotechnology are about 1 to 100 nm in size.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2018 · CDS (I) 2018 — General Knowledge · Q9Moderate

Example 2 · Units, Measurement and Dimensions · Units, Prefixes and Dimensions

Which one of the following is the value of one nanometer?

A nanometre is 10⁻⁹ metre, not 10⁻⁹ centimetre

The power

10^{-9}

belongs to the metre. In centimetres one nanometre is

10^{-9} \times 10^{2} = 10^{-7}

cm. Converting the unit changes the power.

The nanoscale runs to 100 nm

Nanotechnology covers about 1 to 100 nm. A range like 1 Å to 10 nm is too small at both ends, and anything in micrometres or millimetres is far too big.

Drill 1 more on prefixes, powers of ten and the nanoscale

Concept 3 of 3: Dimensions of a physical quantity

The dimensions of a quantity say how it is built from mass (M), length (L) and time (T). Speed is length over time, so it is L T⁻¹. Two quantities with the same dimensions can be measured in the same unit, even when they mean different things. To find dimensions, start from a formula and replace every symbol by its dimensions.

Definition

How to write and compare dimensions:

Velocity $[\text{L}\text{T}^{-1}]$ , acceleration $[\text{L}\text{T}^{-2}]$ , force = mass × acceleration $[\text{M}\text{L}\text{T}^{-2}]$ .
Energy and work = force × distance $[\text{M}\text{L}^{2}\text{T}^{-2}]$ ; torque has the same dimensions.
Linear momentum = mass × velocity $[\text{M}\text{L}\text{T}^{-1}]$ .
Angular momentum = linear momentum × distance $[\text{M}\text{L}^{2}\text{T}^{-1}]$ .
Planck's constant $h = E/\nu$ is energy × time (unit J s), so it has the dimensions $[\text{M}\text{L}^{2}\text{T}^{-1}]$ , the same as angular momentum.

Common dimensional formulas

[F] = [\text{M}\text{L}\text{T}^{-2}] \qquad [E] = [\text{M}\text{L}^{2}\text{T}^{-2}] \qquad [p] = [\text{M}\text{L}\text{T}^{-1}] \qquad [L] = [h] = [\text{M}\text{L}^{2}\text{T}^{-1}]

Fforce
Eenergy or work
plinear momentum
L, hangular momentum and Planck's constant

Worked example

Pressure is force divided by area. Find the dimensions of pressure.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2016 · CDS (II) 2016 — General Knowledge · Q20Moderate

Example 3 · Units, Measurement and Dimensions · Units, Prefixes and Dimensions

Which one of the following physical quantities has the same dimensions as that of Planck's constant

h

Planck's constant matches angular momentum, not linear momentum

Planck's constant has the unit joule second,

[\text{M}\text{L}^{2}\text{T}^{-1}]

. Linear momentum is only

[\text{M}\text{L}\text{T}^{-1}]

: it lacks one power of length. Angular momentum, which is momentum × distance, has that extra L.

Numbers carry no dimensions

A pure number such as the half in

\tfrac{1}{2}mv^{2}

, or

2\pi

, has no dimensions. Drop it before comparing two quantities.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

Dimensions of a physical quantity
Common dimensional formulas
$[F] = [\text{M}\text{L}\text{T}^{-2}] \qquad [E] = [\text{M}\text{L}^{2}\text{T}^{-2}] \qquad [p] = [\text{M}\text{L}\text{T}^{-1}] \qquad [L] = [h] = [\text{M}\text{L}^{2}\text{T}^{-1}]$

Reference tables (2)

SI units and reading a unit off a formula5 rows

Quantity or unit	What to remember
Pressure	Force per area; SI unit pascal (N/m²)
1 atmosphere	760 mm of mercury at 0 °C, about 101325 PaQ CDS 2016 (II): a 760 mm mercury column at 0 °C is one atmosphere, not one pascal or one bar.
1 bar	10⁵ Pa, slightly less than 1 atmosphere
Poise	Unit of viscosity (CGS)
Spring constant k	Force per extension: N/mQ CDS 2020 (I): the unit of the force constant of a spring is N/m.

Prefixes, powers of ten and the nanoscale7 rows

Prefix or unit	Power of ten (in metres)
kilo (km)	10³
centi (cm)	10⁻²
milli (mm)	10⁻³
micro (μm)	10⁻⁶
nano (nm)	10⁻⁹, which is 10⁻⁷ cmQ CDS 2018 (I): one nanometre is 10⁻⁷ cm.
ångström (Å)	10⁻¹⁰
Nanoscale	About 1 to 100 nmQ CDS 2021 (I): the particles studied in nanotechnology are about 1 to 100 nm in size.

Watch out for (6)

Atmosphere, bar and pascal are three different sizes
→ SI units and reading a unit off a formula
Spring constant is per metre, not per square metre
→ SI units and reading a unit off a formula
A nanometre is 10⁻⁹ metre, not 10⁻⁹ centimetre
→ Prefixes, powers of ten and the nanoscale
The nanoscale runs to 100 nm
→ Prefixes, powers of ten and the nanoscale
Planck's constant matches angular momentum, not linear momentum
→ Dimensions of a physical quantity
Numbers carry no dimensions
→ Dimensions of a physical quantity

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