1.3.12 · D1Spring potential energy — derivation
Ek spring jitna zyada deform karo, utna zyada pushback karta hai — isliye usse stretch karne mein jo work lagti hai woh unevenly pile up hoti hai — shuru mein thodi, end mein zyada. Is idea ko symbols mein store karne ke liye sirf teen cheezein chahiye: ek displacement x stiffness k tiny bits of work add karne ka — aur parent page par sab kuch inheen teeno se bana hai.
Is page par assume kiya gaya hai ki tumne kuch bhi nahi dekha. Hum har symbol se milte hain, uska picture banate hain, aur batate hain ki derivation uske bina kyun aage nahi badh sakti. Upar se neeche padho — har cheez uske upar wali cheez par lean karti hai.
Definition Natural length aur equilibriumEk spring jab bilkul akele chhodi jaaye — na koi khich raha, na koi dhak raha — woh apni natural length par baithe rehti hai. Woh ek point jahan spring koi force nahi lagaati, equilibrium position kehlaata hai. Hum use origin mark karte hain, woh jagah jahan hamaara measuring ruler zero read karta hai.
Ek table par padi coil ki picture socho jiske free end par ek marker laga hai. Untouched, woh marker ek jagah par baitha hai jise hum "0" kahenge. Parent page par baaki sab kuch is jagah se measure hota hai.
Intuition Yahaan se kyun shuru karein?Parent note baar baar warn karta hai "measure x
x x free end equilibrium se kitna door gaya hai yeh batata hai, metres (m x positive hai. Left squash karo → x negative hai. Rest par, x = 0
Picture: "0" mark se lekar end abhi jahan hai waahan tak ek arrow drawn hai. Us arrow ki length batati hai x direction uska sign deta hai.
Intuition Signed number kyun chahiye, sirf "kitna" nahiSpring ki force direction flip karti hai depending on ki tum kis side par ho. Ek plain distance ("0.1 m") left-right nahi bata sakti. Ek signed displacement bata sakta hai, aur wahi sign exactly woh cheez hai jo minus ko agle Hooke's law mein le jaata hai.
Definition Force aur restoring directionForce ek push ya pull hoti hai, newtons (N restoring kehlaati hai kyunki yeh hamesha equilibrium ki taraf waapas point karti hai — tumhare imposed displacement ke opposite.
Dekho Hooke's Law poori story ke liye ki kyun real springs isko achhi approximation se follow karti hain.
Intuition Topic yeh sign kyun skip nahi kar saktaParent page force ko do versions mein split karta hai: spring ki F spring = − k x tumhari applied force F applied = + k x dheere stretch karne ke liye tumhe exactly spring ko cancel karna hai, isliye tumhara push + k x
Definition Spring constantk k spring kitni stiff hai : force ke newtons per metre of stretch. Units: N/m k k
Picture: jab tum force plot karo displacement ke against, toh straight line ki steepness . Stiff spring steep line deta hai; soft spring shallow line deta hai. k hi woh slope hai.
Intuition Variable nahi, constant kyun"Ideal spring" mein ideal ka matlab hai k triangle banati hai na ki koi lumpy blob. Agar k x 2 1 k x 2
Work woh energy transfer hai jab ek force kisi cheez ko distance ke saath push karti hai. Jab force constant hoti hai, work simply hoti hai
W = F × d J
Picture: ek rectangle ka area — height F d
Common mistake Woh trap jo parent page se ladhta haiW = F × d Spring ke liye kyun fail hota hai: isko ek single, constant F k x F Fix (preview): motion ko itna thin slice karo ki har slice par F
Dekho Work done by a variable force general machinery ke liye.
d x ′ d x ′ bahut chhota piece " — itna chhota ki uspar force visibly nahi badalti. Prime (′ x ′ running position hai jab hum 0 x
∫ ∫ 0 x ( something ) d x ′ d x ′ x ′ 0 x ∫ Sum .
Intuition Integral exactly sahi tool kyun haiHumne integral isliye choose kiya kyunki force vary karti hai. Har baal-jitne patal slice par force effectively constant hai, isliye us slice par W = F × d d W = k x ′ d x ′ triangle bharta hai, aur ek triangle ka area 2 1 × base × height = 2 1 x ⋅ k x = 2 1 k x 2
Definition Potential energyU Stored energy jo baad mein release ho sakti hai, joules (J U = 2 1 k x 2
Intuition "Work in = energy stored" kyun allowed haiSirf conservative force ke liye hi work ka har joule recoverable storage mein jaata hai — kuch heat mein leak nahi hota. Spring (idealized) conservative hai, isliye uske against ki gayi work bilkul PE gained ke barabar hai. Yahi promise hai Conservative forces and potential energy ki. Jab spring baad mein release karta hai, woh U Conservation of mechanical energy ke through.
Notice karo x squared appear karta hai: 0.04 m 0.04 m same positive energy store karta hai, kyunki ( − 0.04 ) 2 = ( 0.04 ) 2
Work = F times d for constant F
Slice dx' force nearly constant
Triangle area one half base height
Khud test karo — kya tum reveal karne se pehle har ek ka jawab de sakte ho?
Ek real spring par x = 0 Free end ki resting position par jab koi use touch nahi karta (natural length / equilibrium).
Negative x Spring compressed hai — free end equilibrium ki opposite side par chali gayi.
F = − k x Force hamesha equilibrium ki taraf waapas point karti hai, displacement ke opposite.
k k N/m
Spring ke liye seedha W = F × d Us formula ko ek single constant force chahiye, lekin spring ki force 0 k x
∫ 0 x k x ′ d x ′ k x ′ d x ′ d x ′ x ′ 0 x
Stored energy ek rectangle ki area nahi, balki triangle ki area kyun hai? Kyunki force 0 k x 2 1 ×
Spring ke liye work-in energy-stored ke barabar kyun hoti hai? Spring force conservative hai — koi energy heat mein leak nahi hoti, isliye saari work U
Same amount stretch aur compress karne par equal energy kyun store hoti hai? U = 2 1 k x 2 x
