NotesShala Class 11 · Physics

Chapter01

Units and Measurements

System of Units · Significant Figures & Rounding Off · Dimensions of a Physical Quantity

Reading key Definition Note Remember Derivation Strategy Formulas sit on graph paper · coloured words are the key terms

Part 1

System of Units

Definition

System of Units

A few quantities are chosen first, and each one is given its own unit. These units are called base (or fundamental) units. All other units are made by multiplying or dividing powers of the base units. They are called derived units. The complete set of base units and derived units is called a system of units.

1.1Physical Quantities

A quantity that can be measured is called a physical quantity. It is measured directly with an instrument or found by calculation. The laws of physics are written using these quantities. Examples: length, mass, speed and force. Physical quantities are of two types:

iFundamental Quantities

These quantities are not defined using any other quantity. They are independent. They are also called base quantities.

e.g. Mass, length, time, temperature, electric current, luminous intensity and amount of substance.

iiDerived Quantities

These quantities are made from the fundamental quantities by multiplying or dividing them.

e.g. Velocity (length ÷ time), acceleration and linear momentum.

1.2Measurement of Physical Quantities

Measurement means comparing a quantity with a fixed standard of the same kind. This fixed standard is called the unit. So every measurement has two parts:

1Unit

(u)

The standard that the quantity is compared with. For example, the metre, the second or the kilogram.

2Numerical Value

(n)

The number written before the unit. It tells how many times the unit $(u)$ is contained in the quantity.

Note

Smaller Unit, Bigger Number

For the same quantity, the numerical value $(n)$ is inversely proportional to the size of the unit $(u)$ :

n \propto \dfrac{1}{u}\quad\Rightarrow\quad nu = \text{constant}

e.g. the same length written in three different units:

in metre (m)→n = 1largest unit

in centimetre (cm)→n = 100

in millimetre (mm)→n = 1000smallest unit

KeyA smaller unit (u ↓) gives a bigger number (n ↑). The product n · u never changes

Changing the unit does not change the quantity itself. Let $n_1$ and $n_2$ be the numerical values of the same quantity in units $u_1$ and $u_2$ . Then $n_1u_1 = n_2u_2$ .

Physical Quantity

Q = nu

$Q$ = physical quantity, $n$ = numerical value, $u$ = unit

1.3Physical Unit

Definition

A physical unit is a fixed standard amount of a quantity. It is used to measure other amounts of the same kind of quantity.

A good unit must meet a few basic requirements:

What Makes a Good Unit

It should be of a convenient size. It should not be too large or too small for everyday use.
It should be easily available to anyone who needs it.
It should not change with time.
It should be easy to reproduce anywhere in the world.
It should not change with physical conditions such as pressure, volume or temperature.

Units are of two types:

iFundamental Units

These units cannot be obtained from one another. They cannot be broken down into simpler units. They are the units of fundamental quantities, such as length, mass and time. They are also called base units.

iiDerived Units

These are the units of all other quantities. They are formed by combining fundamental units.

e.g. Speed is distance divided by time. So its unit is made from the units of distance and time. It is a derived unit:

\begin{aligned}\text{Speed} &= \dfrac{\text{Distance}}{\text{Time}} \\[3pt] \therefore\ \text{Unit of speed} &= \dfrac{\text{Unit of distance}}{\text{Unit of time}} \\[3pt] &= \dfrac{\text{m}}{\text{s}} = \text{ms}^{-1}\end{aligned}

1.4The International System of Units

A system of units gives units for all physical quantities, both fundamental and derived. Different systems have been used over the years. The most common ones in mechanics are:

iFPS System

It is the British engineering system. It measures length in foot, mass in pound and time in second.

iiCGS System

It is also called the Gaussian system. It measures length in centimetre, mass in gram and time in second.

iiiMKS System

It measures length in metre, mass in kilogram and time in second.

ivInternational System of Units (SI Units)

SI is short for Système International d'Unités. This French name means International System of Units. It is the system accepted all over the world. The General Conference on Weights and Measures adopted it in 1960. In 1971, the mole was added as its seventh base unit. It gives one standard set of symbols, units and abbreviations for science, technology, industry and trade.

Note

SI was developed from the MKS system. So it is also called the rationalised MKS system. It has units for every physical quantity used in physics.

1.5SI Base Quantities and Units

SI has seven base units. It also has two supplementary units for measuring angles.

The Seven Base Units at a Glance

Lengthmmetre

Masskgkilogram

Timessecond

Electric currentAampere

Thermodynamic temperatureKkelvin

Amount of substancemolmole

Luminous intensitycdcandela

Formal Definitions (since 2019)

Base quantity	Name	Symbol	Definition
Length	Metre	m	Fixed by the speed of light: $c$ is exactly $299\,792\,458\ \text{m s}^{-1}$ . So one metre is the distance light travels in vacuum in $\tfrac{1}{299\,792\,458}$ of a second. 2019Same meaning as the 1983 definition.
Mass	Kilogram	kg	Fixed by the Planck constant: $h$ is exactly $6.626\,070\,15\times10^{-34}\ \text{J s}$ , where $\text{J s}=\text{kg m}^2\text{s}^{-1}$ . With the metre and the second already fixed, this sets the kilogram. 2019Earlier (1889–2019): the mass of a platinum–iridium cylinder kept near Paris.
Time	Second	s	Fixed by the caesium-133 atom: the frequency of its hyperfine transition, $\Delta\nu_{\text{Cs}}$ , is exactly $9\,192\,631\,770$ Hz. So one second is the time taken by $9\,192\,631\,770$ periods of that radiation. 2019Same meaning as the 1967 definition.
Electric current	Ampere	A	Fixed by the elementary charge: $e$ is exactly $1.602\,176\,634\times10^{-19}$ C, where C = A s. So a current of 1 A carries about $6.24\times10^{18}$ elementary charges every second. 2019Earlier (1948–2019): defined by the force between two long, parallel current-carrying wires.
Thermodynamic temperature	Kelvin	K	Fixed by the Boltzmann constant: $k$ is exactly $1.380\,649\times10^{-23}\ \text{J K}^{-1}$ (here $1\ \text{J}=1\ \text{kg m}^2\text{s}^{-2}$ ). 2019Earlier (1967–2019): $\tfrac{1}{273.16}$ of the thermodynamic temperature of the triple point of water.
Amount of substance	Mole	mol	One mole contains exactly $6.022\,140\,76\times10^{23}$ elementary entities (atoms, molecules, ions, …). This number is the Avogadro constant $N_A$ , expressed in $\text{mol}^{-1}$ . 2019Earlier (1971–2019): as many entities as there are atoms in 0.012 kg of carbon-12.
Luminous intensity	Candela	cd	Fixed by the luminous efficacy $K_{\text{cd}}$ of monochromatic light of frequency $540\times10^{12}$ Hz, which is exactly $683\ \text{lm W}^{-1}$ . 2019Same meaning as the 1979 definition (radiant intensity of $\tfrac{1}{683}$ watt per steradian).

Good to Know

What Changed in 2019?

Since 20 May 2019, every SI base unit is defined using a constant of nature. Each constant is given an exact, fixed value. Now no unit depends on a physical object. Earlier, the kilogram was defined by a metal cylinder. This cylinder could get scratched or slowly change. Today the kilogram can be reproduced anywhere in the world from the Planck constant.

$\Delta\nu_{\text{Cs}}$ → second $c$ → metre $h$ → kilogram $e$ → ampere $k$ → kelvin $N_A$ → mole $K_{\text{cd}}$ → candela

Definition

Two Supplementary Quantities

Apart from the seven base quantities, SI has two supplementary quantities for measuring angles. These are the plane angle and the solid angle.

Plane AngleRadian · rad

One radian is the angle made at the centre of a circle by an arc whose length is equal to the radius.

d\theta=\dfrac{ds}{r}

Arc

s

makes a plane angle

\theta

at the centre

O

Solid AngleSteradian · sr

One steradian is the solid angle made at the centre of a sphere by a part of its surface whose area is equal to the square of the radius.

d\Omega=\dfrac{dA}{r^{2}}

Surface area

A

makes a solid angle

\Omega

at the centre

O

Note

The FPS system is not a metric system. It is rarely used today.
The CGS system has a weakness. Many of its derived units are too small to be convenient.
The MKS system has an advantage. Many of its derived units are of a practical size.

1.6SI Derived Units with Special Names

Many derived units have special names. Most of them are named in honour of famous scientists. The common ones are:

Physical Quantity	Name	Symbol	In terms of other units	In terms of SI base units
Frequency	Hertz	Hz	–	$\text{s}^{-1}$
Force	Newton	N	–	$\text{kg m s}^{-2}$
Pressure, stress	Pascal	Pa	$\text{N/m}^2$ or $\text{N m}^{-2}$	$\text{kg m}^{-1}\text{s}^{-2}$
Energy, work, quantity of heat	Joule	J	N m	$\text{kg m}^{2}\text{s}^{-2}$
Power, radiant flux	Watt	W	J/s or J s $^{-1}$	$\text{kg m}^{2}\text{s}^{-3}$
Quantity of electricity, electric charge	Coulomb	C	–	A s
Electric potential, potential difference, EMF	Volt	V	W/A or W A $^{-1}$	$\text{kg m}^{2}\text{s}^{-3}\text{A}^{-1}$
Capacitance	Farad	F	C/V	$\text{A}^{2}\text{s}^{4}\text{kg}^{-1}\text{m}^{-2}$
Electric resistance	Ohm	$\Omega$	V/A	$\text{kg m}^{2}\text{s}^{-3}\text{A}^{-2}$
Conductance	Siemens	S	A/V	$\text{m}^{-2}\text{kg}^{-1}\text{s}^{3}\text{A}^{2}$
Magnetic flux	Weber	Wb	V s or J/A	$\text{kg m}^{2}\text{s}^{-2}\text{A}^{-1}$
Magnetic field, flux density, induction	Tesla	T	Wb/m $^2$	$\text{kg s}^{-2}\text{A}^{-1}$
Inductance	Henry	H	Wb/A	$\text{kg m}^{2}\text{s}^{-2}\text{A}^{-2}$
Luminous flux, luminous power	Lumen	lm	–	cd sr
Illuminance	Lux	lx	lm/m $^2$	$\text{m}^{-2}\text{cd sr}$
Activity (of a radio nuclide / radioactive source)	Becquerel	Bq	–	$\text{s}^{-1}$
Absorbed dose, absorbed dose index	Gray	Gy	J/kg	$\text{m}^{2}\text{s}^{-2}$

1.7Some SI Derived Units Expressed by Means of SI Units

Other derived units have no special name. They are simply written as a combination of SI units. Useful examples:

Physical quantity	Name	Symbol	In terms of SI base units
Magnetic moment	Joule per tesla	J T $^{-1}$	$\text{m}^{2}\text{A}$
Dipole moment	Coulomb metre	C m	s A m
Dynamic viscosity	Poiseuille or pascal second or newton second per square metre	Pl or Pa s or N s/m $^2$	$\text{m}^{-1}\text{kg s}^{-1}$
Torque, couple, moment of force	Newton metre	N m	$\text{m}^{2}\text{kg s}^{-2}$
Surface tension	Newton per metre	N/m	$\text{kg s}^{-2}$
Power density, irradiance, heat flux density	Watt per square metre	W/m $^2$	$\text{kg s}^{-3}$
Heat capacity, entropy	Joule per kelvin	J/K	$\text{m}^{2}\text{kg s}^{-2}\text{K}^{-1}$
Specific heat capacity, specific entropy	Joule per kilogram kelvin	J/kg K	$\text{m}^{2}\text{s}^{-2}\text{K}^{-1}$
Specific energy, latent heat	Joule per kilogram	J/kg	$\text{m}^{2}\text{s}^{-2}$
Radiant intensity	Watt per steradian	W sr $^{-1}$	$\text{m}^{2}\text{kg s}^{-3}\text{sr}^{-1}$
Thermal conductivity	Watt per metre kelvin	W m $^{-1}$ K $^{-1}$	$\text{m kg s}^{-3}\text{K}^{-1}$
Energy density	Joule per cubic metre	J/m $^3$	$\text{kg m}^{-1}\text{s}^{-2}$
Electric field strength	Volt per metre	V/m	$\text{m kg s}^{-3}\text{A}^{-1}$
Electric charge density	Coulomb per cubic metre	C/m $^3$	$\text{m}^{-3}\text{A s}$
Electric flux density	Coulomb per square metre	C/m $^2$	$\text{m}^{-2}\text{A s}$
Permittivity	Farad per metre	F/m	$\text{m}^{-3}\text{kg}^{-1}\text{s}^{4}\text{A}^{2}$
Permeability	Henry per metre	H/m	$\text{m kg s}^{-2}\text{A}^{-2}$
Molar energy	Joule per mole	J/mol	$\text{m}^{2}\text{kg s}^{-2}\text{mol}^{-1}$
Angular momentum, Planck constant	Joule second	J s	$\text{kg m}^{2}\text{s}^{-1}$
Molar entropy, molar heat capacity	Joule per mole kelvin	J/mol K	$\text{m}^{2}\text{kg s}^{-2}\text{K}^{-1}\text{mol}^{-1}$
Exposure (X-rays and $\gamma$ -rays)	Coulomb per kilogram	C/kg	$\text{kg}^{-1}\text{s A}$
Absorbed dose rate	Gray per second	Gy/s	$\text{m}^{2}\text{s}^{-3}$
Compressibility	Per pascal	Pa $^{-1}$	$\text{m kg}^{-1}\text{s}^{2}$
Elastic moduli	Newton per square metre	N/m $^2$ or N m $^{-2}$	$\text{kg m}^{-1}\text{s}^{-2}$
Pressure gradient	Pascal per metre	Pa/m or N m $^{-3}$	$\text{kg m}^{-2}\text{s}^{-2}$
Surface potential	Joule per kilogram	J/kg or N m/kg	$\text{m}^{2}\text{s}^{-2}$
Pressure energy	Pascal cubic metre	Pa m $^3$ or N m	$\text{kg m}^{2}\text{s}^{-2}$
Impulse	Newton second	N s	$\text{kg m s}^{-1}$
Angular impulse	Newton metre second	N m s	$\text{kg m}^{2}\text{s}^{-1}$
Specific resistance	Ohm metre	$\Omega$ m	$\text{kg m}^{3}\text{s}^{-3}\text{A}^{-2}$
Surface energy	Joule per square metre	J/m $^2$ or N/m	$\text{kg s}^{-2}$

1.8Advantages of SI over Other Systems of Units

iCoherent System

Every derived unit is obtained from the base units by simple multiplication or division. No extra numerical factors are needed.

iiRational System

It has only one unit for each quantity. For example, all forms of energy are measured in joule.

iiiMetric System

Larger and smaller units are formed using powers of 10, i.e. $a\times10^{\pm b}$ . This makes conversion between them easy.

ivAbsolute System

It does not use gravitational units. So its definitions do not depend on the value of $g$ .

1.9Some General Units & Unit Conversion

Some General Units (Outside SI)

Name	Symbol	Value in SI Unit
Minute	min	60 s
Hour	h	60 min = 3600 s
Day	d	24 h = 86400 s
Year	y	$365.25$ d $= 3.156\times10^{7}$ s
Degree	$^\circ$	$1^\circ=(\pi/180)$ rad
Litre	L	$1\,\text{dm}^3=10^{-3}\,\text{m}^3$
Tonne	t	$10^{3}$ kg
Carat	c	200 mg
Bar	bar	$0.1$ MPa $=10^{5}$ Pa
Curie	Ci	$3.7\times10^{10}\,\text{s}^{-1}$
Roentgen	R	$2.58\times10^{-4}$ C/kg
Quintal	q	100 kg
Barn	b	$100\,\text{fm}^2=10^{-28}\,\text{m}^2$
Are	a	$1\,\text{dam}^2=10^{2}\,\text{m}^2$
Hectare	ha	$1\,\text{hm}^2=10^{4}\,\text{m}^2$
Standard atmospheric pressure	atm	$101325$ Pa $=1.013\times10^{5}$ Pa

Problem Solving Strategy

Unit Conversion

Find what is needed. Note the given units and the units asked for. First convert everything to SI units (metre, kilogram, second). This usually makes the problem easiest.
Write the conversion factors. Units can be multiplied and divided like algebraic symbols. A factor such as $\dfrac{60\ \text{s}}{1\ \text{min}}$ is equal to 1. So multiplying by it changes the unit but not the quantity.
Multiply and cancel. Multiply the quantity by the factors. Cancel the same units until only the required unit is left.
Check the result. The answer should be in the unit asked for. The size of the number should be reasonable.

1.10Rules for Writing SI Units

1Rule 1

Unit symbols are normally written in small (lower-case) letters. Examples: m, kg, s.

2Rule 2

A symbol is never followed by a full stop. Write “5 kg”, not “5 kg.” with a dot.

3Rule 3

A symbol begins with a capital letter only when the unit is named after a scientist. Examples: N, Pa, J.

4Rule 4

A unit written in full always begins with a small letter. This is true even if it is named after a scientist. Examples: newton, joule.

5Rule 5

Symbols have no plural form. Write 10 kg, not 10 kgs.

1.11Abbreviations in Powers of Ten

Very large and very small values are easier to write using prefixes. Each prefix stands for a power of ten. It gives a convenient multiple or sub-multiple of the SI unit:

Multiple	Prefix	Symbol	Sub-multiple	Prefix	Symbol
$10^{1}$	deca	da	$10^{-1}$	deci	d
$10^{2}$	hecto	h	$10^{-2}$	centi	c
$10^{3}$	kilo	k	$10^{-3}$	milli	m
$10^{6}$	mega	M	$10^{-6}$	micro	$\mu$
$10^{9}$	giga	G	$10^{-9}$	nano	n
$10^{12}$	tera	T	$10^{-12}$	pico	p
$10^{15}$	peta	P	$10^{-15}$	femto	f
$10^{18}$	exa	E	$10^{-18}$	atto	a

Examples

1 megaohm

\text{M}\Omega=10^{6}\,\Omega

1 milliampere

1\,\text{mA}=10^{-3}\,\text{A}

1 kilometre

1\,\text{km}=10^{3}\,\text{m}

1 microvolt

1\,\mu\text{V}=10^{-6}\,\text{V}

1 decagram

1\,\text{dag}=10\,\text{g}

1 nanosecond

1\,\text{ns}=10^{-9}\,\text{s}

1 centimetre

1\,\text{cm}=10^{-2}\,\text{m}

1 picofarad

1\,\text{pF}=10^{-12}\,\text{F}

Note

Why CGS, MKS and SI Are "Decimal" Systems

In CGS, MKS and SI, the larger and smaller units are related to the basic units by powers of 10. That is why they are called metric or decimal systems of units.

1.12Some Important Practical Units

Some quantities are very large or very small. For them, special practical units are more convenient than SI units. They are grouped below by the quantity they measure:

For Length / Distance

Astronomical / Large Distances

Astronomical Unit (AU)The average distance between the Earth and the Sun.

1\,\text{AU}=1.496\times10^{11}\,\text{m}

Light Year (ly)The distance light travels in vacuum in one year.

1\,\text{ly}=9.46\times10^{15}\,\text{m}

Parallactic second (parsec)The distance at which an arc of 1 AU makes an angle of one second of arc.

1\,\text{parsec}=3.084\times10^{16}\,\text{m}=3.26\,\text{ly}

Sub-microscopic / Small Distances

Micron / micrometre

1\,\mu\text{m}=10^{-6}\,\text{m}

Nanometre

1\,\text{nm}=10^{-9}\,\text{m}

Angstrom unit (Å)

1\,\text{\AA}=10^{-10}\,\text{m}

FermiUsed for the sizes of nuclei.

1\,\text{fm}=10^{-15}\,\text{m}

For Mass

Conventional / Large Masses

Pound

1\,\text{lb}=0.4536\,\text{kg}

Slug

1\,\text{slug}=14.59\,\text{kg}

Quintal

1\,\text{q}=100\,\text{kg}

Tonne / Metric ton

1\,\text{t}=1000\,\text{kg}

Atomic Scale

Atomic mass unit (amu)One-twelfth (

\tfrac{1}{12}

) of the mass of one

_{6}\text{C}^{12}

atom.

1\,\text{amu}=1.66\times10^{-27}\,\text{kg}

For Time

Astronomical & Calendar Units

Solar dayThe time the Earth takes to rotate once on its own axis, measured with respect to the Sun.

Sidereal dayThe time the Earth takes to rotate once on its own axis, measured with respect to a distant star.

Solar yearThe time the Earth takes to go once around the Sun in its orbit.

1

solar year

=365.25

average solar days

=366.25

sidereal days.

Tropical yearThe time from one spring (vernal) equinox to the next. It is about

365.24

days. Our calendar year is based on it.

Leap yearA year divisible by 4, in which February has 29 days. A century year must be divisible by 400.

Lunar monthThe time the Moon takes to go once around the Earth.

1\text{ lunar month}=27.3\text{ days}

Smallest Practical Unit

ShakeThe smallest practical unit of time.

1\,\text{shake}=10^{-8}\,\text{s}

For Areas

Barn

1\,\text{barn}=10^{-28}\,\text{m}^2

Acre

1\,\text{acre}=4047\,\text{m}^2

Hectare

1\,\text{hectare}=10^{4}\,\text{m}^2

For Other Quantities

Volume

LitreHere cc means cubic centimetre (cm

^3

1\,\text{L}=10^{3}\,\text{cc}=10^{-3}\,\text{m}^3

GallonUSA:

1

gallon

=3.7854\,\text{L}

; UK:

1

gallon

=4.546\,\text{L}

Pressure

PascalAtmospheric pressure is about

1\,\text{atm}=1.01\times10^{5}\,\text{Pa}

1\,\text{Pa}=1\,\text{N m}^{-2}

BarVery close to atmospheric pressure (

1\,\text{atm}\approx1.013

bar).

1\,\text{bar}=10^{5}\,\text{Pa}

TorrThe pressure of a 1 mm column of mercury.

1\,\text{torr}=133.3\,\text{Pa}

Energy / Work

Electron volt

1\,\text{eV}=1.6\times10^{-19}\,\text{J}

Erg

1\,\text{erg}=10^{-7}\,\text{J}

Kilowatt hour

1\,\text{kWh}=3.6\times10^{6}\,\text{J}

Power, Optics & Angle

Horse power

1\,\text{HP}=746\,\text{W}

DioptreUnit of the power of a lens.

1\,\text{D}=1\,\text{m}^{-1}

DegreeUnit of plane angle.

1^\circ=(\pi/180)\,\text{rad}

Part 2

Significant Figures & Rounding Off

Definition

Significant Figures

Every measured value has some digits that are certain. They are followed by one last digit, which is only an estimate. The reliable digits and this first uncertain digit together are called the significant figures (or significant digits) of the measurement.

Reading a Measurement · 374.5 m

374

Certain & reliable

Uncertain (estimated)

= 4 significant figures

The digits 3, 7 and 4 are certain. The last digit, 5, is estimated. Any more digits would add no real information.

2.1Rules for Determining Significant Figures

significant not significant

1Rule 1

Every non-zero digit is significant.

1234→4 S.F.

2Rule 2

Zeros between non-zero digits are always significant. It does not matter where the decimal point is, or if there is none.

1007→4 S.F.

100.7→4 S.F.

3Rule 3

In a number smaller than one, the zeros after the decimal point but before the first non-zero digit are not significant.

0.005704→4 S.F.

4Rule 4

In a number without a decimal point, the trailing zeros (zeros at the end) are not significant.

3210→3 S.F.

5Rule 5

In a number with a decimal point, the trailing zeros are significant.

3.500→4 S.F.

6Rule 6

Exact numbers have unlimited significant figures. They are pure numbers used to multiply or divide. They are not measured or rounded.
e.g. In $r=\dfrac{d}{2}$ and $s=2\pi r$ , the factor 2 is exact. It can be treated as 2.0, 2.00, 2.000 and so on.

2→∞ S.F. (exact)

7Rule 7

Changing the unit never changes the number of significant figures. For example, $16.4$ cm, $0.164$ m and $0.000164$ km all have three.

16.4cm→3 S.F.

0.164m→3 S.F.

0.000164km→3 S.F.

2.2Ambiguity & Scientific Notation

Remember

Ambiguity in Significant Figures

Trailing zeros can cause confusion. Suppose a length is measured as $x=4.700$ m. Here the zeros show the precision of the measurement. So they are significant. The same length can also be written as $x=0.004700$ km, $x=470.0$ cm or $x=4700$ mm.

By Rule 4, $4700$ mm would wrongly seem to have only two significant figures. But $0.004700$ km clearly shows four. Changing units cannot change the number of significant figures (Rule 7). So we need a clearer way to write such values.

The Same Length, Four Ways · x = 4.700 m

4.700m→4 S.F.

0.004700km→4 S.F.

470.0cm→4 S.F.

4700mm→looks like 2 S.F.? — ambiguous!

The length is the same in every line. But 4700 mm seems to have only two significant figures. The solution:

Write every measurement in scientific notation, i.e. in powers of 10. Any number can be written in the form:

Scientific Notation

a \times 10^{b}

$a$ = a number between 1 and 10, $b$ = a positive or negative power of 10 (the order of magnitude)

e.g. $x=4.700$ m $=4.700\times10^{-3}$ km has 4 significant figures. The power of 10 is not counted.

2.3Rounding Off

A result calculated from approximate values often has too many uncertain digits. So it is rounded off. The usual rules are:

1Rule 1

If the digit to be dropped is less than 5, keep the digit before it unchanged.

7.82→7.8

2Rule 2

If the digit to be dropped is more than 5, increase the digit before it by one.

6.87→6.9

3Rule 3

If the digit to be dropped is 5 followed by non-zero digits, increase the digit before it by one.

16.351→16.4

4Rule 4

If the digit to be dropped is 5, or 5 followed only by zeros, keep the digit before it unchanged if it is even.

3.250→3.2(2 is even)

5Rule 5

If the digit to be dropped is 5, or 5 followed only by zeros, increase the digit before it by one if it is odd.

3.750→3.8(7 is odd)

2.4Rules for Arithmetical Operations with Significant Figures

Measured values that we add, subtract, multiply or divide may have different precisions. The answer can never be more precise than the least precise value used. So we round it using these rules:

Addition and Subtraction

Keep as many decimal places as the value that has the fewest decimal places.

RuleRound the answer to the fewest decimal places

\underset{\text{1 d.p.}}{2.1} + 1.78 + 2.046 = 5.926\text{ m}

round →5.9 m1 d.p. (set by 2.1)

Similarly, for $x=12.587$ m and $y=\underset{\text{1 d.p.}}{12.5}$ m:

x-y = 12.587 - 12.5 = 0.087\text{ m}

round →0.1 m1 d.p. (set by 12.5)

Multiplication and Division

Keep as many significant figures as the value that has the fewest significant figures.

RuleRound the answer to the fewest significant figures

e.g. The speed of light is $\underset{\text{3 S.F.}}{3.00}\times10^{8}$ m/s. One year is $3.1557\times10^{7}$ s. So one light year is:

3.00\times10^{8}\times3.1557\times10^{7} = 9.4671\times10^{15}\text{ m}

round →

9.47\times10^{15}

m3 S.F. (set by 3.00)

2.5Rules for Determining Uncertainty in the Results of Arithmetic Calculations

1Rule 1 · Product (Combining Percentage Errors)

A thin rectangular sheet is $15.4$ cm long and $10.2$ cm wide. Both values have three significant figures. They are precise to the first decimal place (±0.1 cm).

Convert each absolute error into a percentage error:

$l=(15.4\pm0.1)\text{ cm}=15.4\text{ cm}\pm\left(\dfrac{0.1}{15.4}\times100\right)\% = 15.4\text{ cm}\pm0.6\%$

$b=(10.2\pm0.1)\text{ cm}=10.2\text{ cm}\pm\left(\dfrac{0.1}{10.2}\times100\right)\% = 10.2\text{ cm}\pm1\%$
For a product, the percentage errors are added:

$A = 15.4\times10.2 \pm(0.6+1.0)\% = 157.08\text{ cm}^2 \pm1.6\% = (157.08\pm2.51)\text{ cm}^2$
Keep 3 S.F. in the area and 1 S.F. in its error:

$A=(157\pm3)\text{ cm}^2$

2Rule 2 · Result Retains the Significant Figures of the Data

If the data has $n$ significant figures, the result can be trusted only up to $n$ significant figures.

e.g. $x=13.7$ m and $y=8.08$ m each have three significant figures:

x-y = 13.7-8.08 = 5.62\text{ m}

round →5.6 mkeep the fewest decimal places (1 d.p., set by 13.7)

3Rule 3 · Relative Error Depends on the Number Itself

The relative error of a value depends on its number of significant figures. It also depends on the size of the number. For example, $m_1=(1.04\pm0.01)$ kg and $m_2=(9.24\pm0.01)$ kg have the same absolute error of $0.01$ kg:

1.04

kg→

\pm\dfrac{0.01}{1.04}\times100

\pm1\%

9.24

kg→

\pm\dfrac{0.01}{9.24}\times100

\pm0.1\%

InsightLarger value ⇒ smaller relative error

4Rule 4 · Keep One Extra Digit in Multi-Step Calculations

In a calculation with several steps, round off only at the end. In the middle steps, keep one more significant figure than the least precise measurement has.

e.g. Start with $x=9.58$ . Find $\dfrac{1}{x}$ . Then take the reciprocal again. Do we get $9.58$ back?

✗ Keep 3 digits

\dfrac{1}{9.58}=0.104

\dfrac{1}{0.104}=9.62

$\neq 9.58$ — an error appears

✓ Keep 4 digits

\dfrac{1}{9.58}=0.1044

\dfrac{1}{0.1044}=9.58

$=9.58$ — correct

Keeping one extra digit in the middle steps stops rounding errors from growing.

Part 3

Dimensions of a Physical Quantity

Definition

Dimensions of a Physical Quantity

The dimensions of a physical quantity are the powers to which the base quantities must be raised to express that quantity. Each of the seven base quantities has its own dimension symbol. It is written inside square brackets [ ]:

[L]Length [M]Mass [T]Time [A]Electric current [K]Temperature [cd]Luminous intensity [mol]Amount of substance

Example · Volume

Volume is length × length × length:

V = [\text{L}]\times[\text{L}]\times[\text{L}] = [\text{L}^3]

It has no mass and no time. So $\;V = \mathbf{[\text{M}^0\text{L}^3\text{T}^0]}$ .

Example · Force

Force is mass × acceleration:

F = \text{mass}\times\text{acceleration} = \text{mass}\times\dfrac{\text{length}}{(\text{time})^2}

$\therefore\; F = \dfrac{[\text{M}][\text{L}]}{[\text{T}^2]} = \mathbf{[\text{MLT}^{-2}]}$

Note

Square brackets $[\;]$ around a quantity show its dimensions, not its value.

3.1Dimensional Formulae & Dimensional Equations

Definition

Dimensional Formula

An expression that shows the base quantities in a physical quantity and their powers is called its dimensional formula.

e.g. powers colour-coded by base quantity — M mass, L length, T time:

Acceleration

[\textcolor{#0a7d73}{\text{M}^0}\,\textcolor{#1f5fcf}{\text{L}^1}\,\textcolor{#6a3fcf}{\text{T}^{-2}}]

Mass density

[\textcolor{#0a7d73}{\text{M}^1}\,\textcolor{#1f5fcf}{\text{L}^{-3}}\,\textcolor{#6a3fcf}{\text{T}^0}]

Volume

[\textcolor{#0a7d73}{\text{M}^0}\,\textcolor{#1f5fcf}{\text{L}^3}\,\textcolor{#6a3fcf}{\text{T}^0}]

Definition

Dimensional Equation

When a physical quantity is written equal to its dimensional formula, we get its dimensional equation.

e.g. each line shows the quantity, how it is defined, and its dimensions:

Linear momentum

=

mass

\times

velocity

=[\textcolor{#0a7d73}{\text{M}^1}\textcolor{#1f5fcf}{\text{L}^1}\textcolor{#6a3fcf}{\text{T}^{-1}}]

Impulse

=

force

\times

time

=[\textcolor{#0a7d73}{\text{M}^1}\textcolor{#1f5fcf}{\text{L}^1}\textcolor{#6a3fcf}{\text{T}^{-1}}]

Moment of inertia

=

mass

\times

(radius of gyration)

^2=[\textcolor{#0a7d73}{\text{M}^1}\textcolor{#1f5fcf}{\text{L}^2}\textcolor{#6a3fcf}{\text{T}^0}]

Problem Solving Strategy

Finding Dimensional Formulae

Read the question and note whether a formula or a law for the quantity is given.
Write down the defining formula of the quantity whose dimensions you need.
Write that formula in terms of fundamental quantities, e.g. acceleration $=\dfrac{\text{Length}}{\text{Time}^2}$ .
Replace each fundamental quantity by its symbol: mass $=[\text{M}]$ , length $=[\text{L}]$ , time $=[\text{T}]$ , and so on.
Simplify the powers algebraically to get the result.
Write the answer in the standard order $[\text{M}],\,[\text{L}],\,[\text{T}]\,\ldots$

3.2Dimensional Formulae of Some Important Quantities

Dimensional formulae are colour-coded by base quantity — M mass, L length, T time, A current.

Physical quantity	Relation with other quantities	Dimensional formula	SI unit
Area	length $\times$ breadth	$[\textcolor{#0a7d73}{\text{M}^0}\textcolor{#1f5fcf}{\text{L}^2}\textcolor{#6a3fcf}{\text{T}^0}]$	$\text{m}^2$
Volume	length $\times$ breadth $\times$ height	$[\textcolor{#0a7d73}{\text{M}^0}\textcolor{#1f5fcf}{\text{L}^3}\textcolor{#6a3fcf}{\text{T}^0}]$	$\text{m}^3$
Density	mass / volume	$[\textcolor{#0a7d73}{\text{M}^1}\textcolor{#1f5fcf}{\text{L}^{-3}}\textcolor{#6a3fcf}{\text{T}^0}]$	$\text{kg m}^{-3}$
Specific gravity	density of body / density of water at 4°C	$[\textcolor{#0a7d73}{\text{M}^0}\textcolor{#1f5fcf}{\text{L}^0}\textcolor{#6a3fcf}{\text{T}^0}]$ (dimensionless)	no units
Speed or velocity	distance / time	$[\textcolor{#0a7d73}{\text{M}^0}\textcolor{#1f5fcf}{\text{L}^1}\textcolor{#6a3fcf}{\text{T}^{-1}}]$	$\text{m s}^{-1}$
Linear momentum	mass $\times$ velocity	$[\textcolor{#0a7d73}{\text{M}^1}\textcolor{#1f5fcf}{\text{L}^1}\textcolor{#6a3fcf}{\text{T}^{-1}}]$	$\text{kg m s}^{-1}$
Acceleration	change in velocity / time	$[\textcolor{#0a7d73}{\text{M}^0}\textcolor{#1f5fcf}{\text{L}^1}\textcolor{#6a3fcf}{\text{T}^{-2}}]$	$\text{m s}^{-2}$
Acceleration due to gravity $(g)$	change in velocity / time	$[\textcolor{#0a7d73}{\text{M}^0}\textcolor{#1f5fcf}{\text{L}^1}\textcolor{#6a3fcf}{\text{T}^{-2}}]$	$\text{m s}^{-2}$
Force	mass $\times$ acceleration	$[\textcolor{#0a7d73}{\text{M}^1}\textcolor{#1f5fcf}{\text{L}^1}\textcolor{#6a3fcf}{\text{T}^{-2}}]$	N (newton)
Impulse	force $\times$ time	$[\textcolor{#0a7d73}{\text{M}^1}\textcolor{#1f5fcf}{\text{L}^1}\textcolor{#6a3fcf}{\text{T}^{-1}}]$	N s
Pressure	force / area	$[\textcolor{#0a7d73}{\text{M}^1}\textcolor{#1f5fcf}{\text{L}^{-1}}\textcolor{#6a3fcf}{\text{T}^{-2}}]$	$\text{N m}^{-2}$
Universal constant of gravitation $(G)$	$G=\dfrac{Fr^2}{m_1 m_2}$	$[\textcolor{#0a7d73}{\text{M}^{-1}}\textcolor{#1f5fcf}{\text{L}^3}\textcolor{#6a3fcf}{\text{T}^{-2}}]$	$\text{N m}^2\text{kg}^{-2}$
Work	force $\times$ distance	$[\textcolor{#0a7d73}{\text{M}^1}\textcolor{#1f5fcf}{\text{L}^2}\textcolor{#6a3fcf}{\text{T}^{-2}}]$	J (joule)
Energy (all types)	work	$[\textcolor{#0a7d73}{\text{M}^1}\textcolor{#1f5fcf}{\text{L}^2}\textcolor{#6a3fcf}{\text{T}^{-2}}]$	J (joule)
Moment of force	force $\times$ distance	$[\textcolor{#0a7d73}{\text{M}^1}\textcolor{#1f5fcf}{\text{L}^2}\textcolor{#6a3fcf}{\text{T}^{-2}}]$	N m
Power	work / time	$[\textcolor{#0a7d73}{\text{M}^1}\textcolor{#1f5fcf}{\text{L}^2}\textcolor{#6a3fcf}{\text{T}^{-3}}]$	W (watt)
Surface tension	force / length	$[\textcolor{#0a7d73}{\text{M}^1}\textcolor{#1f5fcf}{\text{L}^0}\textcolor{#6a3fcf}{\text{T}^{-2}}]$	$\text{N m}^{-1}$
Surface energy	energy / area	$[\textcolor{#0a7d73}{\text{M}^1}\textcolor{#1f5fcf}{\text{L}^0}\textcolor{#6a3fcf}{\text{T}^{-2}}]$	$\text{J m}^{-2}$
Force constant	force / displacement	$[\textcolor{#0a7d73}{\text{M}^1}\textcolor{#1f5fcf}{\text{L}^0}\textcolor{#6a3fcf}{\text{T}^{-2}}]$	$\text{N m}^{-1}$
Thrust	force	$[\textcolor{#0a7d73}{\text{M}^1}\textcolor{#1f5fcf}{\text{L}^1}\textcolor{#6a3fcf}{\text{T}^{-2}}]$	N (newton)
Tension	force	$[\textcolor{#0a7d73}{\text{M}^1}\textcolor{#1f5fcf}{\text{L}^1}\textcolor{#6a3fcf}{\text{T}^{-2}}]$	N (newton)
Stress, pressure	force / area	$[\textcolor{#0a7d73}{\text{M}^1}\textcolor{#1f5fcf}{\text{L}^{-1}}\textcolor{#6a3fcf}{\text{T}^{-2}}]$	$\text{N m}^{-2}$
Strain	change in dimension / original dimension	$[\textcolor{#0a7d73}{\text{M}^0}\textcolor{#1f5fcf}{\text{L}^0}\textcolor{#6a3fcf}{\text{T}^0}]$	no units
Coefficient of elasticity	stress / strain	$[\textcolor{#0a7d73}{\text{M}^1}\textcolor{#1f5fcf}{\text{L}^{-1}}\textcolor{#6a3fcf}{\text{T}^{-2}}]$	$\text{N m}^{-2}$
Radius of gyration $(K)$	distance	$[\textcolor{#0a7d73}{\text{M}^0}\textcolor{#1f5fcf}{\text{L}^1}\textcolor{#6a3fcf}{\text{T}^0}]$	m
Moment of inertia $(I)$	mass $\times$ (radius of gyration) $^2$	$[\textcolor{#0a7d73}{\text{M}^1}\textcolor{#1f5fcf}{\text{L}^2}\textcolor{#6a3fcf}{\text{T}^0}]$	$\text{kg m}^2$
Angle / angular displacement $(\theta)$	arc / radius	$[\textcolor{#0a7d73}{\text{M}^0}\textcolor{#1f5fcf}{\text{L}^0}\textcolor{#6a3fcf}{\text{T}^0}]$	radian
Angular velocity $(\omega)$	angle / time	$[\textcolor{#0a7d73}{\text{M}^0}\textcolor{#1f5fcf}{\text{L}^0}\textcolor{#6a3fcf}{\text{T}^{-1}}]$	$\text{rad s}^{-1}$
Angular acceleration $(\alpha)$	change in angular velocity / time	$[\textcolor{#0a7d73}{\text{M}^0}\textcolor{#1f5fcf}{\text{L}^0}\textcolor{#6a3fcf}{\text{T}^{-2}}]$	$\text{rad s}^{-2}$
Angular momentum	$I\omega$	$[\textcolor{#0a7d73}{\text{M}^1}\textcolor{#1f5fcf}{\text{L}^2}\textcolor{#6a3fcf}{\text{T}^{-1}}]$	$\text{kg m}^2\text{s}^{-1}$
Torque	force $\times$ distance	$[\textcolor{#0a7d73}{\text{M}^1}\textcolor{#1f5fcf}{\text{L}^2}\textcolor{#6a3fcf}{\text{T}^{-2}}]$	N m
Wavelength $(\lambda)$	length of one wave (distance)	$[\textcolor{#0a7d73}{\text{M}^0}\textcolor{#1f5fcf}{\text{L}^1}\textcolor{#6a3fcf}{\text{T}^0}]$	m
Frequency $(\nu)$	number of vibrations / time	$[\textcolor{#0a7d73}{\text{M}^0}\textcolor{#1f5fcf}{\text{L}^0}\textcolor{#6a3fcf}{\text{T}^{-1}}]$	$\text{s}^{-1}$ or Hz
Angular frequency $(\omega)$	$2\pi\times$ frequency	$[\textcolor{#0a7d73}{\text{M}^0}\textcolor{#1f5fcf}{\text{L}^0}\textcolor{#6a3fcf}{\text{T}^{-1}}]$	$\text{rad s}^{-1}$
Velocity of light in vacuum $(c)$	distance / time	$[\textcolor{#0a7d73}{\text{M}^0}\textcolor{#1f5fcf}{\text{L}^1}\textcolor{#6a3fcf}{\text{T}^{-1}}]$	$\text{m s}^{-1}$
Velocity gradient	velocity / distance	$[\textcolor{#0a7d73}{\text{M}^0}\textcolor{#1f5fcf}{\text{L}^0}\textcolor{#6a3fcf}{\text{T}^{-1}}]$	$\text{s}^{-1}$
Rate of flow	volume / time	$[\textcolor{#0a7d73}{\text{M}^0}\textcolor{#1f5fcf}{\text{L}^3}\textcolor{#6a3fcf}{\text{T}^{-1}}]$	$\text{m}^3\text{s}^{-1}$
Planck's constant $(h)$	energy / frequency	$[\textcolor{#0a7d73}{\text{M}^1}\textcolor{#1f5fcf}{\text{L}^2}\textcolor{#6a3fcf}{\text{T}^{-1}}]$	J s
Linear mass density $(m)$	mass / length	$[\textcolor{#0a7d73}{\text{M}^1}\textcolor{#1f5fcf}{\text{L}^{-1}}\textcolor{#6a3fcf}{\text{T}^0}]$	$\text{kg m}^{-1}$
Distance travelled in $n$ th second	distance / time	$[\textcolor{#0a7d73}{\text{M}^0}\textcolor{#1f5fcf}{\text{L}^1}\textcolor{#6a3fcf}{\text{T}^{-1}}]$	$\text{m s}^{-1}$
Avogadro's number $(N)$	number of molecules per mole	$[\textcolor{#0a7d73}{\text{M}^0}\textcolor{#1f5fcf}{\text{L}^0}\textcolor{#6a3fcf}{\text{T}^0}]$	$\text{mol}^{-1}$
Magnetic dipole moment $(M)$	$I\times A$ (current $\times$ area)	$[\textcolor{#0a7d73}{\text{M}^0}\textcolor{#1f5fcf}{\text{L}^2}\textcolor{#6a3fcf}{\text{T}^0}\textcolor{#c2570a}{\text{A}^1}]$	$\text{A m}^2$
Pole strength $(m)$	$m=\dfrac{M}{2l}$	$[\textcolor{#0a7d73}{\text{M}^0}\textcolor{#1f5fcf}{\text{L}^1}\textcolor{#6a3fcf}{\text{T}^0}\textcolor{#c2570a}{\text{A}^1}]$	A m
Magnetic permeability of free space $(\mu_0)$	$\mu_0=\dfrac{4\pi F r^2}{m_1 m_2}$	$[\textcolor{#0a7d73}{\text{M}^1}\textcolor{#1f5fcf}{\text{L}^1}\textcolor{#6a3fcf}{\text{T}^{-2}}\textcolor{#c2570a}{\text{A}^{-2}}]$	$\text{H m}^{-1}$
Resistance $(R)$	potential difference / current	$[\textcolor{#0a7d73}{\text{M}^1}\textcolor{#1f5fcf}{\text{L}^2}\textcolor{#6a3fcf}{\text{T}^{-3}}\textcolor{#c2570a}{\text{A}^{-2}}]$	$\Omega$ (ohm)
Capacitance $(C)$	charge / potential difference	$[\textcolor{#0a7d73}{\text{M}^{-1}}\textcolor{#1f5fcf}{\text{L}^{-2}}\textcolor{#6a3fcf}{\text{T}^4}\textcolor{#c2570a}{\text{A}^2}]$	F (farad)
Surface density of charge	charge / area	$[\textcolor{#0a7d73}{\text{M}^0}\textcolor{#1f5fcf}{\text{L}^{-2}}\textcolor{#6a3fcf}{\text{T}^1}\textcolor{#c2570a}{\text{A}^1}]$	$\text{C m}^{-2}$
Electric dipole moment $(p)$	$q\,(2d)$	$[\textcolor{#0a7d73}{\text{M}^0}\textcolor{#1f5fcf}{\text{L}^1}\textcolor{#6a3fcf}{\text{T}^1}\textcolor{#c2570a}{\text{A}^1}]$	C m
Specific resistance / resistivity $(\rho)$	$\dfrac{RA}{l}$	$[\textcolor{#0a7d73}{\text{M}^1}\textcolor{#1f5fcf}{\text{L}^3}\textcolor{#6a3fcf}{\text{T}^{-3}}\textcolor{#c2570a}{\text{A}^{-2}}]$	$\Omega$ m
Conductance $(G)$	$1/R$	$[\textcolor{#0a7d73}{\text{M}^{-1}}\textcolor{#1f5fcf}{\text{L}^{-2}}\textcolor{#6a3fcf}{\text{T}^3}\textcolor{#c2570a}{\text{A}^2}]$	S (siemens)
Conductivity $(\sigma)$	$1/\rho$	$[\textcolor{#0a7d73}{\text{M}^{-1}}\textcolor{#1f5fcf}{\text{L}^{-3}}\textcolor{#6a3fcf}{\text{T}^3}\textcolor{#c2570a}{\text{A}^2}]$	$\text{S m}^{-1}$
Electric flux	electric field $\times$ area	$[\textcolor{#0a7d73}{\text{M}^1}\textcolor{#1f5fcf}{\text{L}^3}\textcolor{#6a3fcf}{\text{T}^{-3}}\textcolor{#c2570a}{\text{A}^{-1}}]$	$\text{N m}^2\text{C}^{-1}$
Faraday constant	Avogadro number $\times$ elementary charge	$[\textcolor{#0a7d73}{\text{M}^0}\textcolor{#1f5fcf}{\text{L}^0}\textcolor{#6a3fcf}{\text{T}^1}\textcolor{#c2570a}{\text{A}^1}]$	$\text{C mol}^{-1}$
Mass defect $(\Delta m)$	sum of masses of nucleons $-$ mass of nucleus	$[\textcolor{#0a7d73}{\text{M}^1}\textcolor{#1f5fcf}{\text{L}^0}\textcolor{#6a3fcf}{\text{T}^0}]$	kg
Binding energy of nucleus	(mass defect) $\times$ (speed of light) $^2$	$[\textcolor{#0a7d73}{\text{M}^1}\textcolor{#1f5fcf}{\text{L}^2}\textcolor{#6a3fcf}{\text{T}^{-2}}]$	J (joule)

3.3Dimensional Analysis and Its Applications

Dimensional analysis is a quick and powerful tool. It is used to check equations for consistency (homogeneity). It can also find relations between quantities. It has three main uses:

Checking whether an equation is dimensionally consistent

see 3.4

Converting a value from one system of units to another

see 3.5

Finding the relation between physical quantities

see 3.6

3.4Checking the Dimensional Consistency of Equations

Only quantities with the same dimensions can be added or subtracted. For example, a mass cannot be added to a velocity. A current cannot be subtracted from a time. This simple idea gives the principle of homogeneity of dimensions. We use it to test equations.

Principle

Principle of Homogeneity of Dimensions

An equation is dimensionally correct only if every term on both sides has the same dimensions.

e.g. Test the equation of motion $\;s = ut + \tfrac{1}{2}at^2$ . Each term must have the dimensions $[\textcolor{#1f5fcf}{\text{L}}]$ :

[s]

[\textcolor{#1f5fcf}{\text{L}}]

✓

[ut]

[\textcolor{#1f5fcf}{\text{L}}\textcolor{#6a3fcf}{\text{T}^{-1}}][\textcolor{#6a3fcf}{\text{T}}] = [\textcolor{#1f5fcf}{\text{L}}]

✓

\left[\tfrac{1}{2}at^2\right]

[\textcolor{#1f5fcf}{\text{L}}\textcolor{#6a3fcf}{\text{T}^{-2}}][\textcolor{#6a3fcf}{\text{T}^2}] = [\textcolor{#1f5fcf}{\text{L}}]

✓

ResultEvery term reduces to $[\textcolor{#1f5fcf}{\text{L}}]$ . So the equation is dimensionally correct

3.5Conversion of One System of Units into Another

A quantity has the same size in every system of units. Its numerical value is inversely proportional to the size of the unit. So $n_1u_1 = n_2u_2$ , which gives $n_2 = \dfrac{n_1u_1}{u_2}$ . Let the two systems have base units $M_1,L_1,T_1$ and $M_2,L_2,T_2$ . Let the quantity have dimensions $[\textcolor{#0a7d73}{\text{M}^a}\textcolor{#1f5fcf}{\text{L}^b}\textcolor{#6a3fcf}{\text{T}^c}]$ . Then:

Conversion Between Systems

n_2 = n_1\left[\frac{M_1}{M_2}\right]^a\left[\frac{L_1}{L_2}\right]^b\left[\frac{T_1}{T_2}\right]^c

$n_1,n_2$ = numerical values in the two systems; $a,b,c$ = dimensions of the quantity in mass, length and time

3.6Deducing the Relation among the Physical Quantities

Suppose we know which quantities something depends on. Then dimensions can tell us how it depends on them. Take a simple pendulum. Its time period $t$ may depend on its length $(l)$ , the mass of the bob $(m)$ and the acceleration due to gravity $(g)$ .

Derivation

Setup

Assume $\;t = k\,m^a\,l^b\,g^c\quad\ldots(i)$ . Here $k$ is a dimensionless constant, and $a,b,c$ are the powers to be found.

Write both sides in terms of M, L and T:

$[\text{M}^0\text{L}^0\text{T}^1] = \text{M}^a\,\text{L}^b\,[\text{LT}^{-2}]^c = \text{M}^a\,\text{L}^{b+c}\,\text{T}^{-2c}$
By the principle of homogeneity, equate the powers of M, L and T on both sides:

$a=0,\quad -2c = 1 \Rightarrow c = -\tfrac{1}{2},\quad b+c = 0 \Rightarrow b = \tfrac{1}{2}$
Put these values of $a,b,c$ back into Eq. (i):

$t = k\,m^0\,l^{1/2}\,g^{-1/2} = k\sqrt{\dfrac{l}{g}}$

Time Period of a Simple Pendulum

t = k\sqrt{\dfrac{l}{g}}

$k$ = dimensionless constant (experiments give $k = 2\pi$ ); dimensional analysis alone cannot find $k$

Problem Solving Strategy

Deriving an Expression

Remember: dimensions can show that an equation is dimensionally valid. But they cannot prove the exact relation between the quantities.

Read the problem carefully. Understand the physics involved.
List the known and the unknown quantities.
Decide which physical principle connects these quantities.
Write the equation that links them. It must agree with the principle you chose.
Solve for the unknowns algebraically first. Substitute numbers only at the next step (except where some terms are zero).
Substitute the known values along with their units. This gives a numerical answer with units.
Check your answer. Are the units right? Is the size of the answer sensible? Compare it with a rough estimate.

3.7Limitations of Dimensional Analysis

✗Limitation 1

It cannot tell us whether a quantity is a scalar or a vector.

✗Limitation 2

It gives no value for dimensionless constants in a formula, such as 1, 2 or $\pi$ .

✗Limitation 3

It cannot derive formulas that contain trigonometric, logarithmic or exponential functions. These functions have no dimensions.

✗Limitation 4

The method fails if a quantity depends on more than three quantities that have dimensions. This is because comparing the powers of M, L and T gives only three equations.

End of Chapter 1 · notesshala.in