Foundations — Hooke's law — spring force F = −kx

1.3.11 · D1 · Physics › Work, Energy & Power › Hooke's law — spring force F = −kx

Pehle se $F=-kx$ padhne ke liye tumhare paas aath choti ideas honi chahiye. Yeh page unhe ek ek karke bilkul zero se banata hai — pehle simple words mein, phir ek picture, phir yeh kyon zaroori hai. Upar se neeche padho; har idea uske upar wale par tikaa hai.

1. Position aur displacement — number $x$

"Signed" word yahan kaam kar raha hai. Hum ek direction ko positive maante hain:

• $x>0$ → tumne spring ko bahar kheencha (stretch kiya).
• $x<0$ → tumne spring ko andar dhakela (compress kiya).
• $x=0$ → ghar. Kuch bhi nahi kheencha ya dhakela.

Figure mein number line dekho: ghar beech mein hai. Daayein chaloge toh $x$ positive badhega; baayein chaloge toh negative. Isliye $x$ sirf "distance" nahi hai — ek plain distance hamesha positive hoti hai aur spring ko yeh kabhi nahi bata sakti ki woh kis taraf resistance kare. $x$ ki sign direction carry karti hai, aur baad mein poora minus-sign wala story isi par depend karta hai.

2. Force — number $F$ aur uski sign

• $F>0$ → force positive (daayein) taraf push/pull karti hai.
• $F<0$ → force negative (baayein) taraf point karti hai.

Sabse important aadat: $x$ aur $F$ ek hi "positive" arrow share karte hain. Agar position ke liye daayein positive hai, toh force ke liye bhi daayein positive hai. In do conventions ko mix karna hi zyaatar sign mistakes ki wajah hai.

Figure mein, spring daayein stretch hai ($x>0$, pale-yellow arrow) aur uski produce ki hui force baayein point karti hai ($F<0$, pink arrow). Same axis, opposite directions — yahi opposition step 6 mein mile minus sign ka beej hai.

3. Proportionality — "directly proportional" phrase

Hum ise $F\propto x$ likhte hain, padha jaata hai "$F$, $x$ ke proportional hai." Symbol $\propto$ ka matlab sirf yeh hai "ek fixed rate par saath saath badhna."

Figure mein chalk-blue line seedhi hai aur $(0,0)$ se guzarti hai. Straight-through-origin proportionality ka visual signature hai: koi bend nahi (woh rate change ko indicate karta) aur koi offset nahi ($x=0$ par koi force nahi, kyunki ghar matlab koi push nahi).

4. Constant of proportionality — number $k$

Slope jawaab deti hai "stretch ke har extra metre par force ke kitne extra newtons?" $k=\frac{\text{change in force}}{\text{change in displacement}}$

• Bada $k$ → steep line → stiff spring (car suspension).
• Chhota $k$ → shallow line → floppy spring (slinky).

$k$ spring ki apni property hai — uska metal, thickness, coils ki sankhya. Yeh change nahi hota jab tum stretch karte ho (jab tak tum zyada stretch karke use kharab na kar do). Force change hoti hai; slope nahi.

5. Equilibrium — special point $x=0$

Do baatein is point ko special banati hain:

1. Yeh woh jagah hai jahan spring rehna chahti hai — akela chhod do aur yeh wahan ruki rahegi.
2. Yeh woh jagah hai jahan hum poora coordinate anchor karte hain: displacement yahan se measure hoti hai.

Agar tum spring ko thoda bhi disturb karo toh yeh equilibrium ki taraf wapas push karti hai — yahi ise stable banata hai, jaise bowl ke bottom mein rakhा marble, hilltop par balance karne ki bajay.

6. Restoring force aur minus sign

Ab steps 1, 2 aur 5 combine karo. $x$ ki sign jo bhi ho, force opposite taraf point karti hai:

Table padho: $F$ aur $x$ hamesha opposite signs carry karte hain (except ghar par, jahan dono zero hain). Mathematics mein minus sign exactly yahi karta hai — woh sign flip karta hai. Toh $F=-kx$ likhna sirf "force $k$ times stretch hai, doosri taraf point karte hue" kehne ka compact tarika hai. Minus sign ka matlab "force negative hai" nahi hai; iska matlab hai "force displacement ke opposite hai."

7. Potential energy $U$ aur "force is downhill"

$U$ ko position $x$ ke against plot ki gayi landscape ki height ki tarah socho. Equilibrium valley ke bilkul bottom par hai. Dhaal par rakha ball neeche ki taraf roll karta hai — lower energy ki taraf. Woh "dhaal se neeche roll karna" hi force hai. Symbols mein parent note ise likhta hai $F=-\frac{dU}{dx}$ jo padha jaata hai: "force minus the steepness of the energy hill ke barabar hai." Dhaal jitni steep, push utni strong; minus kehta hai "lower ground ki taraf push karo." ($\frac{dU}{dx}$ notation — energy curve ki slope — Conservative forces and potential energy mein explain ki gayi hai; yahan bas valley wall ki slope socho.)

Figure energy valley $U=\frac{1}{2}k{x}^2$ dikhata hai, ek smooth U-shaped bowl. Bottom par ($x=0$) ground flat hai — zero slope matlab zero force, jo exactly isliye hai kyunki equilibrium woh jagah hai jahan kuch push nahi karta. Dono walls par slope bottom ki taraf wapas point karti hai, jo step 6 ki restoring force deti hai.

8. Graph ke neeche area, aur integral $\int$

Kyunki spring force $F=kx$ (magnitude) straight line mein $0$ se $kx$ tak badhti hai, total added-up area simply ek triangle ka area hai: half base times height.

Figure mein shaded chalk-blue triangle $F=kx$ line ke neeche hai. Base $=x$, height $=kx$, toh $\text{area}=\frac{1}{2}\times \text{base}\times \text{height}=\frac{1}{2}x(kx)=\frac{1}{2}k{x}^2.$ Woh area stored energy $U=\frac{1}{2}k{x}^2$ hai. Isliye answer half $k{x}^2$ hai aur $k{x}^2$ nahi: tum triangle le rahe ho, pura rectangle nahi — force zero se shuru hui, toh average par woh apni final value ki sirf half thi. (Formal machinery Work done by a variable force mein hai.)

Yeh topic ko kaise feed karte hain

Baayein sab kuch foundation hai; daayein ke do boxes khud parent topic hain, Hooke's law.

Equipment checklist

Connections

• Parent: Hooke's law F = −kx — woh topic jiske liye yeh foundations bani hain.
• Work done by a variable force — "slivers jodo" integral poori detail mein.
• Conservative forces and potential energy — $F=-dU/dx$ landscape idea.
• Elastic potential energy — $\frac{1}{2}k{x}^2$ reservoir.
• Simple Harmonic Motion — $F=-kx$ ek mass ko kya karta hai.
• Conservation of mechanical energy — spring stored aur moving energy kaise swap karta hai.
• Interatomic forces — tiny bonds jo apni valleys ke paas springs hain.