Physical quantities — fundamental and derived

WHAT are physical quantities?

WHY the number AND the unit? A bare "5" is meaningless — 5 what? Seconds? Kilograms? The unit tells you which standard you compared against, and the number tells you how many copies of that standard fit.

Fundamental vs Derived

Definition Fundamental (base) quantities

Quantities chosen as independent building blocks — they are not defined in terms of any other quantity. In SI there are seven:

Quantity	Symbol	SI Unit	Unit symbol
Length	$L$	metre	m
Mass	$M$	kilogram	kg
Time	$T$	second	s
Electric current	$I$ (or $A$ )	ampere	A
Temperature	$\Theta$	kelvin	K
Amount of substance	$N$	mole	mol
Luminous intensity	$J$	candela	cd

WHY only seven? Because seven is the minimum set that is (a) independent — none can be made from the others — and (b) sufficient — every known quantity can be reached from them. Fewer would leave gaps; more would be redundant.

Figure — Physical quantities — fundamental and derived

HOW to build a derived unit — derive it, don't memorise

Formula Deriving the unit of force (Newton)

Start from the physics, not the answer. Newton's second law defines force as the rate of change of momentum, and for constant mass: $F = ma$ Now substitute the definitions of $a$ and trace each unit:

acceleration $a = \dfrac{\text{velocity}}{\text{time}} = \dfrac{\text{length}/\text{time}}{\text{time}} = \dfrac{L}{T^2}$ ⇒ $[\text{m s}^{-2}]$
mass has unit $[\text{kg}]$

Therefore: $[F] = [\text{kg}]\cdot[\text{m s}^{-2}] = \text{kg m s}^{-1\cdot 2}= \boxed{\text{kg m s}^{-2}} \equiv 1\ \text{N}$ So 1 newton = the force that gives 1 kg an acceleration of 1 m s $^{-2}$ . The unit means something.

Steel-manning the classic mistakes

Active recall

Recall Try before peeking

Why must every physical quantity carry a unit?
List the seven SI base quantities with their units.
Derive the base units of the joule from first principles.
Is an angle fundamental or derived? Why?
State the relation $n_1u_1=n_2u_2$ and explain it.

(Answers are throughout the note — close it and reconstruct.)

Recall Feynman: explain to a 12-year-old

Imagine you only own three measuring tools: a ruler (length), a kitchen scale (mass), and a stopwatch (time). With just these you can figure out how fast a car goes (length ÷ time), how hard you push something (mass × how fast its speed grows), and how much energy a punch has. You never needed a special "energy ruler" — you built energy out of the three simple tools. The seven SI base units are just the world's official "simple tools", and everything else in physics is made by combining them, like LEGO.

Connections

Dimensional Analysis — uses these base quantities as the symbols $[M^aL^bT^c]$ .
SI Units and Prefixes — the standards and the $n_1u_1=n_2u_2$ conversions.
Newton's Second Law — source of the newton's derivation.
Errors and Significant Figures — every measured value has uncertainty.
Vectors — many derived quantities (force, velocity) are vectors.

A physical quantity is expressed as

a numerical value × a unit

How many SI base (fundamental) quantities are there?

Seven

Name the seven SI base quantities

length, mass, time, electric current, temperature, amount of substance, luminous intensity

SI unit of luminous intensity

candela (cd)

SI unit of amount of substance

mole (mol)

Define a derived quantity

a quantity built from fundamental quantities by multiplication/division (and powers)

Base units of force (newton)

kg m s⁻² (from F = ma)

Base units of energy (joule)

kg m² s⁻² (from W = F·d)

Base units of pressure (pascal)

kg m⁻¹ s⁻² (from P = F/area)

Why is an angle dimensionless?

it is arc length ÷ radius = length/length, so units cancel

Relation between number and unit for a fixed quantity

n₁u₁ = n₂u₂ (smaller unit ⇒ larger number)

Is mass or weight fundamental?

mass is fundamental (kg); weight is a derived force (W = mg, in N)

Convert 72 km/h to m/s

20 m/s

Why exactly seven base quantities?

minimal set that is both independent and sufficient to build all others

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, physics ka pura khel hai measurement ka. Lekin har cheez ko zero se measure karna impossible hai — toh humne 7 "starter" quantities chun li: length, mass, time, current, temperature, amount of substance, aur luminous intensity. Inko bolte hain fundamental (base) quantities. Inke alawa jo bhi quantity hai — speed, force, energy, pressure — woh in saaton ko multiply/divide karke banti hai, isliye unhe derived quantities kehte hain. Bilkul LEGO jaisa: thode se blocks, par anginat structures.

Sabse important baat: koi bhi quantity = number × unit. Sirf "5" bekaar hai, "5 metre" matlab rakhta hai. Aur ek mast rule yaad rakho: $n_1u_1 = n_2u_2$ . Matlab same length ko chhoti unit me likho toh number bada ho jayega (2 m = 200 cm). Quantity wahi rehti hai, sirf label badalta hai.

Derived unit ko ratta mat maro — derive karo. Jaise force: Newton ka law $F = ma$ , aur $a$ ki unit $\text{m s}^{-2}$ , mass ki $\text{kg}$ , toh force = $\text{kg m s}^{-2}$ = 1 Newton. Energy ke liye $W = F \times d$ lagao, mil jayegi $\text{kg m}^2\text{ s}^{-2}$ = Joule. Bas units ko bhi numbers jaisa multiply/divide karo, exponents add/subtract karo.

Do common galtiyan: (1) Weight aur mass same nahi hain — mass fundamental (kg) hai, weight ek force ( $W=mg$ , Newton me) hai jo derived hai. (2) Angle fundamental nahi hai — woh arc/radius = length/length hai, toh dimensionless. In points ko clear rakho toh exam me dimensional analysis aur conversions bilkul easy ho jayenge.

Physical quantities — fundamental and derived

WHAT are physical quantities?

Fundamental vs Derived

HOW to build a derived unit — derive it, don't memorise

Steel-manning the classic mistakes

Active recall

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Measurement, Vectors & Kinematics

Connections