Length contraction — derivation — Hinglish notes

2.3.30 · Physics › Modern Physics

HUM MEASURE KYA KAR RAHE HAIN?

LENGTH MEASURE KARNE KE LIYE SIMULTANEITY KYUN CHAHIYE? Agar ek rod aapke paas se guzar rahi hai, toh uski length paane ke liye aapko ek hi instant mein uske front end aur back end ki positions note karni hongi. Agar aapne front ko time $t_{1}$ par aur back ko baad mein $t_{2}$ par record kiya, toh rod move ho chuki hogi, aur aapko galat result milega. Isliye length = (dono ends ki positions jo simultaneously record ki gayi hoon). Kyunki simultaneity frames ke beech alag hoti hai, length bhi alag hoti hai.

DERIVE KAISE KAREIN — Lorentz transformation se

Hum do frames use karte hain:

$S$ = ground (lab) frame, jahan rod move karti hai speed $v$ se.
$S^{'}$ = rod ka rest frame, jo $S$ ke relative $+ v$ se move karta hai.

$S$ se $S^{'}$ ka Lorentz transformation: $x^{'} = γ (x - v t), γ = \frac{1}{1 - v ^{2} / c ^{2}}$

Step 1 — $S^{'}$ mein proper length define karo

Rod ke apne frame $S^{'}$ mein ends fixed coordinates $x_{1}^{'}$ (back) aur $x_{2}^{'}$ (front) par baithe hain, isliye $L_{0} = x_{2}^{'} - x_{1}^{'} .$ Yeh step kyun? Rod $S^{'}$ mein rest mein hai, isliye uske ends move nahi karte — hum inhe jab chahein read kar sakte hain.

Step 2 — $S$ mein ek hi instant par measure karo

$S$ mein observer dono ends ko same time $t$ par record karta hai: $t_{1} = t_{2} = t .$ Yeh step kyun? Yahi poori physics hai — ek moving rod ki length measurement ke liye measuring frame mein simultaneous endpoint readings zaroori hain.

Step 3 — Har end par transformation apply karo

$x_{1}^{'} = γ (x_{1} - v t), x_{2}^{'} = γ (x_{2} - v t) .$

Step 4 — Subtract karo

$x_{2}^{'} - x_{1}^{'} = γ [(x_{2} - v t) - (x_{1} - v t)] = γ (x_{2} - x_{1}) .$ $v t$ ke terms cancel hote hain sirf isliye kyunki $t_{1} = t_{2}$ — yahan simultaneity kaam karti hai.

Ab $x_{2}^{'} - x_{1}^{'} = L_{0}$ aur $x_{2} - x_{1} = L$ ( $S$ mein length). Toh: $L_{0} = γ L .$

Step 5 — Measured length ke liye solve karo

Figure — Length contraction — derivation

Worked examples

Common mistakes

Recall Feynman: ek 12-saal ke bacche ko explain karo

Socho ek train bahut tezi se tumhare paas se guzar rahi hai. Train ki length measure karne ke liye, tumhe ek photo snap karni hogi jo front aur back ko bilkul ek hi waqt par pakde. Lekin yahan woh ajeeb baat hai jo Einstein ne discover ki: jab koi cheez light speed ke paas move kare, "bilkul ek hi waqt" ka matlab tumhare liye aur train ke logon ke liye alag hota hai. Uss mismatch ki wajah se, tumhari measurement choti nikalta hai jo train ke passengers apni train ke liye measure karte hain. Kuch crush nahi ho raha — bas itna hai ki tez motion "ab" ka matlab bigaad deti hai, aur woh bigaad moving cheezein unki travel direction mein squished dikhata hai.

Active recall

Proper length

L_{0}

kya hai?

Woh length jo us frame mein measure ki jaati hai jahan object rest mein ho; yeh maximum measured length hoti hai.

Length contraction formula batao.

L = L_{0} 1 - v^{2} / c^{2} = L_{0} / γ

Moving rod chota kyun measure hota hai?

Kyunki uski length measure karne ke liye dono ends ko simultaneously record karna padta hai, aur simultaneity frame-dependent hoti hai (relativity of simultaneity).

Derivation mein

v t

ke terms cancel kyun hote hain?

Kyunki dono ends

S

mein same time

t_{1} = t_{2}

par record hote hain, isliye

v t

terms identical hain aur subtract ho jaate hain.

Kya length motion ke perpendicular contract hoti hai?

Nahi — sirf motion ki direction ke along wali dimension contract hoti hai.

Contraction aur dilation ki directions compare karo.

Length

γ

se divide hoti hai (choti hoti hai); time

γ

se multiply hota hai (bada hota hai); proper length max hai, proper time min hai.

v = 0.8 c

par rod kis factor se contract hoti hai?

γ = 5/3

, isliye

L = 0.6 L_{0}

(rest length ka 60%).

Kya length contraction ek real physical squeezing force hai?

Nahi — yeh spacetime/simultaneity ka geometric consequence hai, mechanical compression nahi.

Connections

Lorentz transformation — woh master equations jinse yeh derive hoti hai.
Time dilation — "opposite" twin effect ( $\times γ$ ).
Relativity of simultaneity — contraction ki root cause.
Lorentz factor gamma — universal $γ = 1/ 1 - v^{2} / c^{2}$ .
Muon decay experiment — real-world evidence.
Spacetime interval — woh invariant jo dono effects preserve karte hain.