Dimensional analysis — checking equations, deriving relations

1.1.3 · Physics › Measurement, Vectors & Kinematics

DIMENSIONS kya hote hain?

Examples jo pehle principles se bane hain (hamesha quantity ki definition pe wapas jao):

Quantity	Defined as	Dimension
Velocity	displacement / time	$[L T^{- 1}]$
Acceleration	velocity / time	$[L T^{- 2}]$
Force	mass × acceleration	$[M L T^{- 2}]$
Work / Energy	force × distance	$[M L^{2} T^{- 2}]$
Pressure	force / area	$[M L^{- 1} T^{- 2}]$
Power	energy / time	$[M L^{2} T^{- 3}]$

Principle of Homogeneity (engine)

YEH RULE KYU EXIST KARTA HAI? Kyunki addition sirf identical kinds ke beech mein hi sense banata hai. "5 metres + 3 seconds" ka koi matlab nahi — koi aisa number nahi jo dono ho ek saath. Toh agar koi formula do terms add karta hai, nature unhe same dimension ka hone par majboor karti hai.

Corollaries (lines nahi, consequences yaad karo):

$sin, cos, ln, e^{x}$ ke arguments dimensionless hone chahiye (tum "2 metres" ka sine nahi le sakte).
Exponents aur angles dimensionless hote hain.
Ek pure number ka dimension $[M^{0} L^{0} T^{0}]$ hota hai.

USE 1 — Equation check karna

USE 2 — Relation derive karna (powerful trick)

Limitations (boundaries jaano)

80/20 — woh cheezein jo marks dilaati hain

Flashcards

Kisi quantity ka dimension kya batata hai?

Woh base quantities

M, L, T

se kaise bana hai — uski tarah, chune hue unit se independent.

Principle of homogeneity state karo.

Kisi valid equation mein har add ya equate ki gayi term ke dimensions identical hone chahiye.

Force ka dimension?

[M L T^{- 2}]

(

F = ma

se).

Energy/work ka dimension?

[M L^{2} T^{- 2}]

(force × distance).

Pressure ka dimension?

[M L^{- 1} T^{- 2}]

(force / area).

Power ka dimension?

[M L^{2} T^{- 3}]

(energy / time).

sin

e^{x}

ka argument dimensionless kyun hona chahiye?

Yeh functions apne argument ki powers ke sums ke barabar hote hain;

x

x^{2}

mein add karne ke liye

x

ka pure number hona zaroori hai.

Kya dimensional analysis kisi equation ko correct prove kar sakti hai?

Nahi — yeh pure numbers (jaise

\frac{1}{2}

), sums mein signs, aur trig/exp forms ke liye andhi hai; yeh sirf disprove kar sakti hai ya fail to disprove.

Failed dimensional check ka matlab?

Equation definitely galat hai.

Derive karo ki pendulum period

ℓ

aur

g

ke saath kaise scale karta hai.

T = k ℓ^{a} g^{c}

;

L, T

match karne se

a = \frac{1}{2}, c = - \frac{1}{2}

milta hai, toh

T \propto ℓ / g

Pendulum period mass se independent kyun hai?

M

ki power match karne se

b = 0

milta hai kyunki

g

aur

ℓ

mein koi mass nahi.

Derivation method ki teen failures?

Numeric constants nahi de sakti; fail hoti hai jab unknowns > base dimensions; multi-term (sum) formulas nahi bana sakti.

1 N ko dyne mein dimensions use karke convert karo.

[F] = M L T^{- 2} \Rightarrow 1 N = 1 0^{3} g \cdot 1 0^{2} c m \cdot s^{- 2} = 1 0^{5}

dyne.

Recall Feynman: ek 12-saal ke bachche ko samjhao

Socho ki har measurement ek colored shirt pehni hoti hai: red length ke liye, blue time ke liye, green weight ke liye. Tumhe sirf same shirt wali cheezein add karne ki permission hai — do red lengths ek red length dete hain, theek hai; lekin ek red length plus ek blue time bakwaas hai. Jab bade log ek physics equation likhte hain, tum ise dono sides ki shirts dekh kar check kar sakte ho: agar match karti hain, equation obviously broken nahi hai; agar clash karti hain, toh definitely galat hai. Cool bonus: agar tumhe pata hai ki ingredients kaunsi shirts pehnte hain, toh tum aksar recipe guess kar sakte ho — jaise guess karna ki ek jhule ka hilne ka time uski length ke saath badhta hai aur zyada gravity se ghatta hai — sirf shirts match karake.

Connections

Units and the SI system — dimensions vs. woh units jo unhe realize karti hain.
Significant figures and error propagation — measurement ka numerical side.
Equations of motion (kinematics) — har result dimensionally checkable hai.
Buckingham Pi theorem — derivation trick ke peeche ka general formalism.
Vectors — components and addition — components vector ka dimension share karte hain.