CDS Physics · Formula sheet

Units, Measurement and Dimensions formulas

1 formula, 2 reference tables and 6 common traps for CDS Physics Units, Measurement and Dimensions, grouped by subtopic.

Full notes with worked examples

Units, Prefixes and Dimensions

Learn this subtopic in the notes

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}]

$F$ force
$E$ energy or work
$p$ linear momentum
$L, h$ angular momentum and Planck's constant

SI units and reading a unit off a formula

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 nanoscale

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.

Common traps

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.

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.

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.