1.3.11
Ek spring apni natural length par rehna chahti hai. Usse stretch karo, toh woh wapas kheenchti hai. Compress karo, toh woh bahar dhakkelti hai. Jitna zyada tum usse rest se door karo, utna hi zyada woh resist karti hai — aur yeh resist karna proportion mein hota hai jitna tumne usse displace kiya hai. Yahi proportionality, ek minus sign ke saath jo kehta hai "ghar ki taraf wapas," Hooke's law HAI.
Ek ideal spring ke liye, restoring force directly proportional hoti hai natural (unstretched) length se displacement ke, aur us displacement ke opposite direction mein point karti hai:
F = − k x
F dwara exert ki gayi force (N)x k spring constant (N/m), stiffness ka ek measure
Sabse important word hai restoring — force hamesha equilibrium (x = 0
Intuition Sign kyun matter karta haix > 0 left kheenchti hai, toh F < 0 x < 0 right dhakkelti hai, toh F > 0 Dono cases mein F x opposite signs hain — exactly wahi jo F = − k x
Worked example Taylor expansion se derivation (kyun koi bhi spring near rest linear hoti hai)Claim: Equilibrium ke paas, har elastic system Hooke's law follow karta hai. Yahan reason hai.
Ek spring potential energy U ( x ) F = − d x d U
Yeh step kyun? Conservative forces hamesha ek potential se aati hain: ek system lower energy ki taraf roll karta hai.
Ab U ( x ) x = 0 U ( x ) = U ( 0 ) + U ′ ( 0 ) x + 2 1 U ′′ ( 0 ) x 2 + ⋯
Yeh step kyun? Koi bhi smooth energy function ko ek point ke paas polynomial se approximate kiya ja sakta hai.
U ( 0 ) U ′ ( 0 ) = 0 x = 0 minimum hai (yehi "equilibrium" ka matlab hai: zero slope).
Yeh step kyun? Ek stable rest point par force zero hoti hai, aur F = − U ′ U ′ ( 0 ) = 0
Toh leading surviving term hai:
U ( x ) ≈ 2 1 U ′′ ( 0 ) x 2 = 2 1 k x 2 , k ≡ U ′′ ( 0 )
Differentiate karo:
F = − d x d U = − k x ✓
Punchline: Hooke's law metal coils ki koi special property nahi hai — yeh kisi bhi stable system ka generic small-displacement behavior hai. Isliye springs, atomic bonds, aur guitar strings sab near rest par F = − k x
Worked example Example 1 — Force nikaloEk spring jisme k = 200 N/m x = 0.05 m F = − k x = − ( 200 ) ( 0.05 ) = − 10 N Minus kyun? Right ki taraf stretch kiya ⇒ spring left ki taraf kheenchti hai (magnitude 10 N). Sign tumhe direction batata hai.
Worked example Example 2 — Stored energySame spring, same stretch. Stored energy kitni hai?
U = 2 1 k x 2 = 2 1 ( 200 ) ( 0.05 ) 2 = 2 1 ( 200 ) ( 0.0025 ) = 0.25 J Yeh step kyun? Humein F x area chahiye, yaani 2 1 k x 2 F ⋅ x = 0.5
Worked example Example 3 — Hanging mass (k 0.5 kg 0.10 m k x = m g k = x m g = 0.10 ( 0.5 ) ( 9.8 ) = 49 N/m Yeh step kyun? "Rest par" ⇒ net force zero ⇒ restoring force ki magnitude weight ke barabar hai.
Worked example Example 4 — Pehle forecast karo, phir verify karoForecast: Agar main stretch double karun, toh kya stored energy double hogi?
Verify: U ∝ x 2 x 2 2 = 4 × Force double hoti hai, energy quadruple ho jaati hai. Yeh springs ke baare mein sabse zyada miss hone wala fact hai.
F ⋅ x = k x 2 Kyun sahi lagta hai: Work = force × distance humein bahut drill kiya gaya hai. Kyun galat hai: force jaise tum stretch karte ho 0 se k x — yeh constant nahi hai. Sahi work average force 2 1 k x x 2 1 k x 2 Fix: integrate karo, ya F x
Common mistake "Minus sign ka matlab hai force negative hai"Kyun sahi lagta hai: F = − k x Kyun galat hai: agar x < 0 − k x > 0 positive force! Fix: F x
k Kyun sahi lagta hai: jaise tum stretch karte ho, force badlti hai. Kyun galat hai: k F x slope hai — spring ki ek fixed property (material, thickness, coils). Force badlti hai; stiffness k Fix: elastic limit se aage, Hooke's law bilkul fail ho jaata hai — spring deform ho jaati hai.
Recall Feynman: ek 12-saal ke bachche ko explain karoEk slinky imagine karo. Jab woh sirf wahan baitha hota hai, woh khush hota hai. Agar tum usse kheencho, woh wapas kheenchta hai taaki phir se chhota ho jaye. Agar tum usse squish karo, woh wapas dhakkelata hai taaki phir se lamba ho jaye. Aur trick yeh hai: jitna zyada tum usse mess karo, utna hi zyada woh wapas push ya pull karta hai — twice as far, twice as hard. Formula mein chhota minus sign spring ka tarika hai yeh kehne ka "rukho, wapas aao!" — woh hamesha tumhe wahan dhakkelata hai jahan se woh shuru hua tha.
"F = minus k x" → "Force Mostly Kicks back eXactly opposite."
Aur energy ke liye: "Half-k-x-squared" — spring tumhari effort ka sirf aadha rakhti hai, kyunki force zero se shuru hui thi.
F = − k x Spring PE 2 1 k x 2 k x 2
Kyun har stable system equilibrium ke paas ek spring jaisi dikhti hai?
Hooke's law kya kehta hai? Ek ideal spring ki restoring force displacement ke proportional aur opposite hoti hai: F = − k x
F = − k x Force hamesha displacement ke opposite point karti hai, yaani equilibrium ki taraf wapas (restoring).
Spring constant k Newtons per metre (N/m).
Ek stretched spring mein stored potential energy kya hoti hai? U = 2 1 k x 2
Spring PE 2 1 k x 2 k x 2 Force linearly 0 se k x triangle ka area hai (average force 2 1 k x x
Agar tum stretch x Woh double ho jaati hai (F ∝ x
Agar tum stretch x Woh quadruple ho jaati hai (U ∝ x 2
Kyun koi bhi stable system small displacements ke liye spring jaisa behave karta hai? Uski energy minimum ko Taylor-expand karne par, leading term 2 1 U ′′ ( 0 ) x 2 F = − U ′′ ( 0 ) x = − k x
Force aur potential energy mein kya relation hai? F = − d x d U
F x k Line ka slope (stiffness); steeper line matlab stiffer spring.
Hooke's law kab break down karta hai? Elastic limit se aage, jahan spring permanently deform ho jaati hai aur F x
Potential energy U = half k x squared
Generic near rest behaviour
