2.2.26 · D1Dimensional analysis — Buckingham π theorem
Ek physical law ko nahi pata ki tumne kaun si units choose ki hain — pendulum utna hi time leta hai chahe uski rope metres mein mapo ya miles mein. Isliye har sahi equation ko sirf unit-free numbers se likha ja sakta hai, aur poora topic bas yeh count karne ka tarika hai ki ek problem kitne aise numbers allow karti hai.
Yeh page — bilkul zero se — har symbol, word, aur picture build karta hai jis par parent note Buckingham π theorem rely karta hai. Ise upar se neeche padho; neeche koi bhi cheez aisi nahi hai jo upar already build nahi hui ho.
Definition Dimension vs. unitEk dimension woh tarah ki quantity hai jo tum measure kar rahe ho — ek length , ek duration , ek mass . Ek unit uss tarah ke liye ek particular yardstick hai — ek metre, ek second, ek kilogram. "Length" ek dimension hai; "metre" uss dimension ki ek unit hai .
Ek ruler ki picture socho. Yeh fact ki woh distance measure karta hai — woh uski dimension hai. Chhote tick marks (cm, inch) unit hain. Ticks badlo toh number badlega, lekin jo tarah ki cheez tumne measure ki woh nahi badlti.
Topic ko yeh kyun chahiye: theorem ki saari power yahi hai ki laws tab bhi survive karti hain jab tum units swap karo. "Units swap karne" aur "quantity ki kind na badalne" ki baat karne ke liye, humein dimension (kind) aur unit (yardstick) ko alag rakhna hoga. Dekho Fundamental and derived units .
Definition Fundamental dimensionsM , L , T fundamental dimensions chahiye — building blocks jo ek doosre se nahi ban sakte:
M L T
Square-bracket notation [ X ] "X Yeh ek sawaal-jawaab dene wala symbol hai: "kaun si base quantities, aur kis power mein, X
Intuition Brackets kyun, aur powers kyun?Speed hoti hai distance-per-time. Kinds mein likho: ek length divided by ek time. Hum ise record karte hain [ v ] = L T − 1 Exponents (L + 1 T − 1 M , L , T
Worked example Ek formula se dimensions padhna
Area = length × length, toh [ A ] = L 2
Density = mass ÷ volume, toh [ ρ ] = M L − 3 ρ
Force = mass × acceleration, aur acceleration speed-per-time = L T − 2 [ F ] = M L T − 2
Viscosity (symbol μ [ μ ] = M L − 1 T − 1
Topic ko yeh kyun chahiye: theorem mein har variable ko pehle uske [ X ]
Definition Dimensionless quantityEk dimensionless quantity woh hoti hai jiske dimensions M 0 L 0 T 0 = 1 pure number jisme bilkul bhi units nahi hain. Yeh universe mein sabko same padha jaata hai, chahe unke rulers ya clocks koi bhi hon.
ratio units kyun khatam kar deta hai
Ek length ko ek length se divide karo aur L L + 1 ⋅ L − 1 = L 0 = 1 2 10 0.2
Topic ko yeh kyun chahiye: theorem ka output dimensionless groups ka ek set hota hai (π 1 , π 2 , … Reynolds number se, jo sabse famous aisa number hai.
Definition Dimensional homogeneityEk equation dimensionally homogeneous hoti hai jab har term jo add ya equate ki gayi ho uski exactly same dimensions hon. Tum sirf seb ko seb se add kar sakte ho: ek length kabhi ek time ke barabar nahi ho sakti.
Intuition Ek balance scale ki picture socho= M , L , T combination dikhate hain. Agar woh alag hain, toh "law" bakwaas hai chahe koi bhi numbers daalo.
F = 6 π μ r v [ μ ] [ r ] [ v ] = ( M L − 1 T − 1 ) ( L ) ( L T − 1 ) = M L − 1 + 1 + 1 T − 1 − 1 = M L T − 2 [ F ] = M L T − 2 6 π
Topic ko yeh kyun chahiye: homogeneity woh akeli demand hai jisse poora theorem aata hai. Dekho Dimensional homogeneity .
n physical variables ki sankhya (force, speed, density, …).k actually present independent fundamental dimensions ki sankhya (aksar, lekin hamesha nahi, 3 M , L , T p independent dimensionless groups ki sankhya jo tum bana sakte ho, jo p = n − k
Intuition Physics ke "Knobs"Har dimensionless group ko ek machine par ek knob ki tarah socho. n k p
Topic ko yeh kyun chahiye: p = n − k woh formula hai. Baaki sab k p
Definition Dimension matrixD M , L , T n dimension matrix D k n
Intuition Rank = truly independent rows ki sankhyaD rank yeh hai ki uski kitni dimension-rows genuinely naya information carry karti hain — woh rows jo doosron ki copies ya combinations nahi hain. Agar speed, length aur time tumhare aakele variables hain, toh "time" "speed ÷ length" ke andar chhupa hai, toh rows collapse ho jaati hain aur rank girta hai. Sachi k
Topic ko yeh kyun chahiye: yahi wajah hai ki p = n − k k measure karna padega (as a rank), assume nahi karna. Full Navier–Stokes equations ko non-dimensionalise karna woh jagah hai jahan yeh ideas rigorous hote hain.
Definition Repeating variables aurπ Repeating variables woh k k koi dimensionless group nahi bana sakte).
Har ==π π i π na ki 3.14159
Mnemonic "n minus k = π keys"n k p = n − k π keys hain jo law unlock karte hain. Repeaters = woh k
Topic ko yeh kyun chahiye: parent note ki recipe ("repeaters chuno, har leftover ke saath combine karo") produce karti hai π i Drag force and drag coefficient aur Model testing and similarity mein jaate hain.
Dimensionless = pure number
Speed v [ v ] = L T − 1
Dimension aur unit mein fark Dimension = quantity ki tarah (length); unit = uske liye yardstick (metre).
Bracket [ X ] "X M , L , T
Ek quantity dimensionless kab hoti hai? Jab uski dimensions M 0 L 0 T 0 = 1
Do lengths ka ratio units kyun nahi rakhta? L + 1 ⋅ L − 1 = L 0 = 1
Dimensional homogeneity ka rule batao Har term jo equate ya add ki jaaye uski identical dimensions honi chahiye.
[ F ] [ μ r v ] Dono M L T − 2 F = 6 π μ r v
n k p n k p = n − k π
k Agar variables mein hidden dependence ho, toh dimension-rows collapse ho jaate hain; honest count rank ( D )
Do conditions repeating variables par (1) Milke saare k π
Dimensional homogeneity — balance-scale rule jise yeh page build karta haiFundamental and derived units — jahan se M , L , T Reynolds number — sabse famous dimensionless number jab tum π Drag force and drag coefficient — counting ka pehla payoffModel testing and similarity — scales ke across π Navier–Stokes equations — jahan rank/homogeneity rigorous hote hain 
