Dimensional analysis — checking equations, deriving relations: Foundations

1.1.3 · D1 · Physics › Measurement, Vectors & Kinematics › Dimensional analysis — checking equations, deriving relation

Yeh page kuch bhi assume nahi karta. Pehle tum parent topic par equations check karo ya relations derive karo, tumhe symbols aur pictures ki ek chhoti si toolkit chahiye. Hum har ek cheez neeche se banate hain, aur har naya idea sirf unhi cheezOn par tikta hai jo pehle aa chuki hain.

0. "Physical quantity" kya hoti hai?

Humein yeh isliye chahiye kyunki poora topic usi labels ko straight rakhne ke baare mein hai.

1. Teen base kinds: $M$ , $L$ , $T$

Square brackets shorthand hain. Jab tum $[M]$ dekho, ise zor se padho "the kind mass." Yeh woh raw colours hain jisse baaki har colour mix hoti hai.

Figure — Dimensional analysis — checking equations, deriving relations

Figure mein teen coloured bars dekho: yeh primary paints hain. Length ke liye blue bar, time ke liye yellow bar, mass ke liye red bar. Mechanics mein har doosri physical quantity inhi teeno ki koi mixture hoti hai, kisi na kisi amount mein — aur woh amounts exactly wahi hain jo ek dimension record karta hai.

2. Unit vs dimension — zaroori fark

Aage badhne se pehle hum do aisi ideas alag karein jo log baar baar confuse karte hain.

Humein yeh distinction isliye chahiye kyunki dimensional analysis kinds ke level par kaam karta hai, isliye yeh is baat par depend nahi karta ki tumne kaunsi unit use ki. Yehi wajah hai ki yeh itna robust hai.

3. Notation $[Q]$ — " $Q$ ki dimension"

Ise ek sawaal ki tarah padho jo brackets poochhte hain: "yeh kis colour ka hai?" Jaise:

$[a distance] = L$
$[a duration] = T$
$[a heartbeat count] = M^{0} L^{0} T^{0}$ (ek pure number — bilkul bhi koi colour nahi)

Uss aakhri line mein poore topic ki sabse important habit introduce hoti hai: ek pure number ki koi dimension nahi hoti, jise hum saare exponents zero likh ke dikhate hain.

4. Dimensions par exponents — kinds ko multiply aur divide karna

Yahaan se $L^{2}$ aur $T^{- 2}$ jaisi cheezein aati hain, aur ek baar picture dikhne ke baad yeh aasaan ho jaata hai.

Notation $T^{- 1}$ ka matlab sirf "per time" hai (time se ek baar divide). $T^{- 2}$ ka matlab "per time, per time" — time se do baar divide.

Figure mein dekho kaise ek length bar (blue) khud ke saath stack hoti hai area banane ke liye — do blue bars, isliye $L^{2}$ . Neeche, ek length ko ek time se kata jaata hai speed dene ke liye: yellow time denominator mein baithta hai, jo negative exponent record karta hai.

5. " $=$ " ka matlab kya hota hai: same-kind-only addition

Yeh akela restriction Principle of Homogeneity hai, aur yeh poore parent topic ka engine hai. Baaki sab kuch — equations check karna, pendulum law derive karna — sirf yahi rule hai jo carefully apply kiya gaya hai.

Figure mein do sums hain. Left side par, do blue length-bars milkar ek lambi blue bar banaate hain — legal, same colour. Right side par, ek blue length-bar aur ek yellow time-bar ko saath dhakelne ki koshish hoti hai — result ek question mark hai, kyunki koi aisI kind nahi jo dono ho. Woh crossed-out sum exactly wahi hai jise dimensional analysis galat equation mein dhundhta hai.

6. $sin$ , $ln$ , $e^{x}$ ko naked numbers kyun chahiye

Aakhri symbol-family jo parent use karta hai woh functions hain jaise $sin$ , $cos$ , $ln$ , aur exponential $e^{x}$ . Ek khoobsurat wajah hai ki inke inputs pure numbers kyun hone chahiye.

Isliye jab bhi tum physics mein $sin (θ)$ ya $e^{- k t}$ dekhte ho, andar ki cheez guaranteed dimensionless hoti hai. Yeh koi convention nahi hai; yeh usi addition rule se force hota hai.

Prerequisite map

Upar se neeche padho: yeh jaanna ki quantity kya hoti hai humein teen base kinds name karne deta hai; bracket notation aur exponent rules un kinds ko algebra mein badal dete hain; homogeneity ek hi rule hai jo checking aur deriving dono ko power deta hai; aur function-argument rule iska direct corollary hai. Saare arrows parent topic par aake milte hain.

Units and the SI system — woh sizes (units) jo inhe kinds ko realise karti hain.
Significant figures and error propagation — numbers handle karna jab kinds settle ho jaayein.
Vectors — components and addition — quantities jo direction bhi carry karti hain, har component ab bhi ek kind pahne hota hai.
Equations of motion (kinematics) — pehle equations jinhe tum dimensionally check karoge.
Buckingham Pi theorem — "match the powers" derivation trick ka grown-up generalisation.

Equipment checklist

Khud ko test karo — right side cover karo.

Physical quantity minimally kya hoti hai?

Ek number ek kind (label) ke saath; akela bare number incomplete hai.

Mechanics ki teen base kinds aur unke symbols batao.

Mass

[M]

, length

[L]

, time

[T]

Notation

[Q]

ka kya matlab hai?

Q

ki dimension (kind)" — sawaal "yeh kis colour ka hai?"

Unit aur dimension mein kya fark hai?

Unit ek chosen size hoti hai (metre); dimension woh kind khud hoti hai (

L

), size se independent.

Jab do quantities multiply hoti hain, unke exponents ka kya hota hai?

Woh add hote hain:

[A B] = [A] [B]

Jab divide hon?

Exponents subtract hote hain:

[A / B] = [A] / [B]

T^{- 2}

ka words mein kya matlab hai?

"Per time, per time" — time se do baar divide.

Principle of homogeneity ek line mein batao.

Har term jo add ya equate hoti hai uski same dimension honi chahiye.

e^{x}

ka argument dimensionless kyun hona chahiye?

e^{x}

ek sum hai

1 + x + x^{2} /2 + \dots

;

x

x^{2}

se add karne ke liye

x

ka pure number hona zaroori hai.

Ek pure number (ya angle) ki dimension kya hoti hai?

M^{0} L^{0} T^{0}

— dimensionless.