Physical Quantities and Measurement – Detailed Study Notes

Page 1 – Introduction & Chapter Overview

Central Theme: Any measurement requires:
1. A number answering “how much?”.
2. A unit answering “of what?”.
  The unit must be universally accepted so that science can be communicated without ambiguity.
Physical quantities highlighted: length, mass, time, temperature, area (derived from two lengths).
Skills to be learnt
▸ Using appropriate measuring devices.
▸ Converting between related units.
Sub-topics promised in the chapter
1. Measurement of Length – concept, ruler & tape, units.
2. Measurement of Time – concept, clock / watch / stopwatch, units.
3. Measurement of Mass – concept, balances, units.
4. Measurement of Temperature – concept, thermometers, normal body temperature, units.
5. Measurement of Area – concept, use of graph paper.
Units introduced (name ↔ symbol)
▸ Length: centimetre (cm), metre (m), kilometre (km), inch, foot (ft).
▸ Time: second (s), minute (min), hour (h).
▸ Mass: milligram (mg), gram (g), kilogram (kg).
▸ Temperature: degree Celsius (°C).

Page 2 – Learning Objectives & Meaning of Measurement

Learning objectives

Students should be able to …

Define length, mass, time.
Use correct units + symbols for length, mass, time, temperature, area.
Operate rulers, tapes, beam/electronic balances, clocks, stopwatches, thermometers, graph paper.
Convert a quantity from one unit to related units.

Why Measure?

Everyday illustrations: tailor measuring cloth, school peon timing periods, vendor weighing produce, doctor checking temperature.

Reliability of senses

Senses of touch/sight are subjective → need instruments for exact measurement.

Conceptual definition

Measurement = comparison with a standard unit.
Expressed as:
$\text{Measurement}=n\times u=nu$
where $u$ = chosen unit, $n$ = number of times unit fits into quantity.

Choice of a “good” unit

Convenient size.
Universally accepted (unchanged by place/time).

Page 3 – Historical Systems & S.I. Fundamentals

Historical unit systems

C.G.S. – centimetre, gram, second.
F.P.S. – foot, pound, second.
M.K.S. (metric) – metre, kilogram, second.

Adoption of S.I. (1960)

Basic physical quantities + symbols:

Quantity	S.I. unit	Symbol
Length	metre	m
Mass	kilogram	kg
Time	second	s
Temperature	kelvin	K

Prefixes

$\text{milli (m)} =10^{-3}$
$\text{centi (c)} =10^{-2}$
$\text{kilo (k)} =10^{3}$
Example: $0.0001\,\text{m}=0.1\,\text{mm}$ and $2000\,\text{m}=2\,\text{km}$ .

Page 4 – Conventions for Writing S.I. Units & Intro to Length

Conventions

Symbols (unless named after a scientist) in lower case: $\text{kg},\;\text{m},\;\text{s}$ .
Name of unit in words = lower case (ampere, newton, kelvin …).
Symbol for unit named after scientist = capital letter (A, N, K, F).
Symbols never pluralised.
Leave a space between compound-unit symbols, e.g. $\text{N\;s}$ .
Negative powers for quotients, e.g. speed: $\text{m\;s}^{-1}$ .

Length

Definition: distance between two points; breadth, depth, thickness, height, diameter are all lengths.

Earliest definition of metre: distance between two marks on a platinum–iridium bar at $0^{\circ}\text{C}$ .
Modern definition: distance light travels in $\tfrac{1}{299\,792\,458}\,\text{s}$ .

Page 5 – Units of Length & “Do-You-Know” Facts

Multiples / sub-multiples of metre

$1\,\text{km}=1000\,\text{m}=10^{3}\,\text{m}$ .
$1\,\text{cm}=\tfrac{1}{100}\,\text{m}=10^{-2}\,\text{m}$ .

$1\,\text{mm}=\tfrac{1}{1000}\,\text{m}=10^{-3}\,\text{m}$ .

FPS relations: $1\,\text{ft}=12\,\text{inch},\;1\,\text{inch}=2.54\,\text{cm},\;1\,\text{yd}=3\,\text{ft}=0.91\,\text{m}.$

Additional metric units: $1\,\text{dm}=\tfrac{1}{10}\,\text{m},\;1\,\text{dam}=10\,\text{m},\;1\,\text{hm}=100\,\text{m}.$

Devices

Metre ruler (wood/plastic, 1 m long, scaled in cm & mm; smallest division $1\,\text{mm}$ ).
Measuring tape (flexible; common lengths 1 m – 100 m).

Page 6 – Using a Metre Ruler Accurately

Steps

Place ruler along object, zero mark aligned with one end.
Read opposite end; keep eye perpendicular to scale to avoid parallax error.

Illustration: mis-reading 4.2 cm (eye at A) or 4.4 cm (eye at C) vs correct 4.3 cm (eye at B).

If zero end damaged: read positions of both ends $x1, x2$ then length $=x2-x1$ .

Do-You-Know

Thickness of a coin < 1 mm ⇒ measure stack of $n$ coins → thickness/coin $=\frac{\text{stack height}}{n}$ .
For $0.1\,\text{mm}$ accuracy use vernier calipers; for $0.01\,\text{mm}$ use screw gauge.

Metre ruler limits: straight objects only; accuracy $\pm1\,\text{mm}$ .

Page 7 – Measuring Tape & Introduction to Mass

Tape measurement

For curved line AB: lay tape along curve; read positions $xA,xB$ → length $=xB-xA$ . Example: $8.2\,\text{cm}-5.0\,\text{cm}=3.2\,\text{cm}$ .

Mass

Definition: quantity of matter in a body.

S.I. unit: kilogram (kg).

Original standard: Pt–Ir cylinder at Sèvres (1889).

Practical definition: mass of $1\,\text{L}$ ( $=1000\,\text{mL}$ ) of water at $4^{\circ}\text{C}$ .

Multiples/sub-multiples:

$1\,\text{quintal}=100\,\text{kg}$ .
$1\,\text{metric tonne}=10\,\text{quintal}=1000\,\text{kg}$ .
$1\,\text{g}=10^{-3}\,\text{kg},\;1\,\text{mg}=10^{-6}\,\text{kg}$ .

FPS relation: $1\,\text{lb}=453.59\,\text{g}$ .

Page 8 – Beam Balance & Standard Weights

Beam balance construction

Central support + pointer.
Two identical pans equidistant from fulcrum.

Procedure

Suspend empty → ensure beam horizontal.
Place object in left pan, standard weights in right until beam again horizontal.
Sum of weights = mass of object.

Standard weight series: $20,10,5,2,1\,\text{kg}$ plus smaller $500,200,100,50,20,10,5\,\text{g}$ .

Variants

Grocer’s balance – retail trade.
Physical (scientific) balance – high sensitivity used by goldsmiths & labs.

Page 9 – Electronic Balance

Components

Structure – mechanical support.
Load cell – converts force → electrical signal.
Signal conditioner – processes signal, displays mass (digital read-out).

Features

Wide capacity range (mg to quintal).
No weight box required; direct digital reading.

Page 10 – Time & Pendulum Clock

Definition

Time = interval between two events; based on mean solar day.

Units

S.I.: second (s). Defined as $\tfrac{1}{86400}$ of a mean solar day.
Conversions
$1\,\text{min}=60\,\text{s}$ ,
$1\,\text{h}=60\,\text{min}=3600\,\text{s}$ ,
$1\,\text{day}=24\,\text{h}=86400\,\text{s}$ ,
$1\,\text{year}=365\,\text{days}\approx3.15\times10^{7}\,\text{s}$ .

Pendulum clock

Pendulum period $T=2\,\text{s}$ (one–way $1\,\text{s}$ ).
Dial: 12 large marks, 60 small divisions; three hands: seconds, minutes, hours.
▸ Seconds hand: 1 division per $1\,\text{s}$ .
▸ Minutes hand: 1 division per $60\,\text{s}$ .
▸ Hours hand: 5 divisions per $60\,\text{min}$ (1 h).

Page 11 – Watch & Short-Interval Timing

Wrist / pocket watch

Works by gear wheels & wound spring; dial same graduation as clock.

Short intervals

Stop clock / mechanical stop watch – push-buttons for start/stop/reset.

Electronic stop watch – digital display, precision $0.01\,\text{s}$ ; used in athletics.

Page 12 – Temperature & Laboratory Thermometer

Heat flow concept

Body feels hot if heat flows to hand; cold if heat flows from hand. Heat flows from higher to lower temperature.

Definition: Temperature = measure of degree of hotness / coldness.

Units

S.I.: kelvin (K) – never written “degree K”.
Common: degree Celsius (°C) & degree Fahrenheit (°F).

Key fixed points & scale sizes

Scale	Ice point	Steam point	Divisions
Kelvin	$273\,\text{K}$	$373\,\text{K}$	100
Celsius	$0^{\circ}\text{C}$	$100^{\circ}\text{C}$	100
Fahrenheit	$32^{\circ}\text{F}$	$212^{\circ}\text{F}$	180
Relations
$1^{\circ}\text{C}=1^{\circ}\text{K}$ ;
$1^{\circ}\text{C}=\tfrac{9}{5}^{\circ}\text{F}$ .

Laboratory thermometer

Glass capillary + mercury bulb.
Stem markings $-10^{\circ}\text{C}\rightarrow110^{\circ}\text{C}$ .
Calibration: $0^{\circ}\text{C}$ in melting ice, $100^{\circ}\text{C}$ in boiling water.

Page 13 – Clinical Thermometer & Derived vs Fundamental

Clinical thermometer

Range $35^{\circ}\text{C}\rightarrow42^{\circ}\text{C}$ (or $95^{\circ}\text{F}\rightarrow110^{\circ}\text{F}$ ).
Constriction (kink) above bulb keeps mercury column fixed after removal.
Normal human body temp: $37^{\circ}\text{C}=98.6^{\circ}\text{F}$ (marked by red arrow).

Usage steps

Disinfect bulb, shake mercury below $37^{\circ}\text{C}$ .
Place under tongue or armpit ~1 min.
Read at eye-level; if > $37^{\circ}\text{C}$ → fever.

Modern replacement: digital thermometer (mercury-free).

Fundamental vs Derived quantities

Fundamental (independent): length, mass, time, temperature.
Derived (expressed via fundamentals):
▸ Area $=\text{length}\times\text{breadth}$ .
▸ Volume $=\text{l}\times\text{b}\times\text{h}$ .
▸ Speed $=\dfrac{\text{distance}}{\text{time}}$ .

Page 14 – Concept of Area & Formulae for Regular Shapes

Definition: Area = total surface occupied by an object.

Regular shape formulae

Square: $A=\ell^{2}$ .
Rectangle: $A=\ell\times b$ .
Triangle: $A=\tfrac{1}{2}\,\ell\,h$ .
Circle: $A=\pi r^{2}\;\bigl(\pi=\tfrac{22}{7}\approx3.14\bigr)$ .

Graph-paper method (regular or irregular)

Small square: $1\,\text{mm}\times1\,\text{mm}=1\,\text{mm}^{2}$ .
Count full squares + squares ≥ half inside outline; multiply by square area to get approximate area.

Page 15 – Examples of Graph-Paper Area Estimation

Example counts

Triangle outline: $7$ full + $3$ half ⇒ $A=(7+3)\times1\,\text{cm}^{2}=10\,\text{cm}^{2}$ .
Irregular surface: $13$ full + $7$ half ⇒ $A=20\,\text{cm}^{2}$ .
Another shape: $22$ full + $9$ half ⇒ $A=31\,\text{cm}^{2}$ .

Page 16 – Units of Area & Conversions

S.I. base unit

$1\,\text{m}^{2}=1\,\text{m}\times1\,\text{m}$ (see sketch of 1 m × 1 m square).

Multiples & sub-multiples

Are (square decametre): $1\,\text{are}=10\,\text{m}\times10\,\text{m}=100\,\text{m}^{2}$ .
Hectare: $1\,\text{ha}=100\,\text{m}\times100\,\text{m}=10^{4}\,\text{m}^{2}=100\,\text{are}$ .
Square kilometre: $1\,\text{km}^{2}=10^{6}\,\text{m}^{2}=100\,\text{ha}$ .
Square decimetre: $1\,\text{dm}^{2}=100\,\text{cm}^{2}$ .
Square centimetre: $1\,\text{cm}^{2}=10^{-4}\,\text{m}^{2}$ .
Square millimetre: $1\,\text{mm}^{2}=10^{-6}\,\text{m}^{2}$ .

Imperial conversions (approx.)

$1\,\text{yd}^{2}=0.836\,\text{m}^{2}$ .
$1\,\text{ft}^{2}=0.09290\,\text{m}^{2}$ .
$1\,\text{acre}=4046.856\,\text{m}^{2}$ .

Page 17 – Recapitulation (Key Take-aways)

Human senses subjective; instruments give objective measurements.
Four fundamental measurements in daily life: length, mass, time, temperature.
Measurement = comparison with standard unit; requires a number × unit.
Qualities of a good unit: convenient, universal, invariant.
Summary tables
• Length: $\text{m}$ (km, cm, mm).
• Mass: $\text{kg}$ (quintal, tonne, g, mg).
• Time: $\text{s}$ (min, h, day, year).
• Temperature: $\text{K}$ (°C, °F).
• Area: $\text{m}^{2}$ (are, hectare, km², cm² …).
Standard devices
• Length: metre ruler (avoid parallax), measuring tape, vernier calipers, screw gauge.
• Mass: beam/physical balance, electronic balance.
• Time: pendulum clock, watch, stop watch, electronic timer.
• Temperature: laboratory & clinical thermometers, digital thermometer.
• Area: formulae for regular shapes or counting squares on graph paper.
Normal body temp $=37^{\circ}\text{C}=98.6^{\circ}\text{F}$ .
Key derived relations
$\text{Speed}=\dfrac{\text{distance}}{\text{time}},\;\text{Area}=\ell\times b,\;\text{Volume}=\ell\times b\times h.$