Foundations — Newton's law of gravitation — universal, action at distance

1.2.19 · D1 · Physics › Newton's Laws & Dynamics › Newton's law of gravitation — universal, action at distance

Yeh page ek "toolbox kholo" wali page hai. Parent note (the topic) mein $m$, $r$, $\vec{F}$, $\hat{r}$, $G$, $g$, $r^2$ jaise exponents, aur "vector" aur "field" jaise words freely use hote hain. Agar inme se koi bhi fuzzy lage, toh poora topic collapse ho jaata hai. Isliye hum ek-ek cheez ko bilkul zero se build karenge, aur aise order mein jahan har item sirf pehle se defined cheezein use kare.

1. Mass — "stuff" ki matra

Picture karo. Do identical boxes socho. Ek mein feathers bharo aur ek mein lead. Lead box mein zyada matter hai → bada $m$. Mass size ke baare mein nahi hai ya uthane mein kaisa lagta hai — yeh underlying "stuff" ka count hai.

Topic ko yeh kyun chahiye. Gravity ek kheench hai do piles of matter ke beech. Agar tumhare paas "kitna matter" ka number nahi hai, toh shuru bhi nahi kar sakte. Isliye formula mein DO mass symbols hain: ek har object ke liye.

$m_1$ aur $m_2$ mein chhote numbers subscripts hain — bas name-tags. $m_1$ ka matlab hai "object number 1 ka mass"; yeh $m$ ko $1$ se multiply nahi kar raha.

2. Distance aur $r$ ka matlab

Figure dekho. Do balls alag-alag baith e hain. Green line $r$ hai — yeh ek ball ke centre se doosri ball ke centre tak jaati hai, surface-to-surface nahi. Letter $r$ isliye choose kiya gaya kyunki yeh distance ek imaginary sphere ke radius jaisi hai jo ek object ke centre par hai (agle section mein hum woh sphere use karenge).

Topic ko yeh kyun chahiye. Gravity doori ke saath weak hoti hai. Yeh kehne ke liye ki "kitni weak", hume "kitni door hain" ke liye ek single number chahiye — woh number $r$ hai.

3. Powers aur exponents — $r^2$ actually kya kehta hai

Picture karo. $r^2$ literally ek square ka area hai jiska side $r$ hai. Agar $r=3\text{m}$ hai, toh $r^2=9{\text{m}}^2$ — ek $3\times 3$ tile.

"$÷r^2$" kyun aur "$÷r$" kyun nahi? Kyunki gravity ka influence ek sphere ki surface par phailta hai, aur woh surface area $4\pi r^2$ hai — ek area, isliye square. Agla section woh picture dikhata hai.

4. Area $=4\pi r^2$ kyun — woh sphere jo gravity ko dilute karti hai

Figure dekho. Do concentric spheres. Orange arrows ki same number (the "influence") dono ko pierce karti hain. Chhoti sphere par woh crowded hain; badi sphere par woh sparse hain. Radius $r$ wali sphere ka surface area $4\pi r^2$ hai, isliye influence per unit area $1/r^2$ ki tarah girti hai.

Topic ko yeh kyun chahiye. Yahi reason hai ki exponent exactly $2$ hai — koi guess nahi, balki geometry. Yahi picture explain karta hai ki light aur sound bhi $1/r^2$ ki tarah kyun fade hoti hain.

Yahan $\pi$ (pi) bas woh fixed number hai $\approx 3.14159$ jo circle ki size ko uski width se connect karta hai — yeh hamesha tab aata hai jab spheres aur circles appear hote hain.

5. Proportionality — "$\propto$" arrow

Picture karo. Graph par origin se guzarti seedhi line: horizontal double karo, vertical double ho jaata hai. Koi bend nahi, koi offset nahi.

Topic ko yeh kyun chahiye. Parent note law ko pieces mein build karta hai: pehle $F\propto m_1{m}_2$ (force masses follow kari hai), phir $F\propto 1/r^2$ (force dilution follow kari hai). Proportionality se hum har sach relationship ko tabhi state kar sakte hain jab exact numerical constant pata na ho.

6. Constant $G$ — "$\propto$" ko "$=$" mein badalna

Yeh kyun exist karta hai. $F\propto \frac{m_1{m}_2}{r^2}$ humein law ki shape batata hai lekin scale nahi. Nature scale choose karti hai; hum ise measure karte hain (Cavendish ka experiment). $G$ se multiply karo aur $\propto$ arrow equals sign ban jaata hai: $F=\frac{G{m}_1{m}_2}{r^2}$

Chhota exponent $10^{-11}$ ka matlab hai "decimal point 11 places left karo" — ek absurdly chhota number, isliye gravity everyday objects ke beech weak lagti hai.

7. Vectors — force ko direction dena

Figure dekho. Ball 1 par force ek arrow hai jo ball 2 ki taraf point karta hai (gravity inhe ek saath kheenchti hai). Ball 2 par force ek equal-length arrow hai jo ball 1 ki taraf point karta hai. Same length (equal strength), opposite directions — yeh equal-and-opposite pairing Newton's Third Law hai.

Topic ko yeh kyun chahiye. "Earth apple ko kheenchti hai" incomplete hai — kaunsi taraf kheenchti hai? Neechay, Earth ke centre ki taraf. Direction ke bina force motion predict karne ke liye meaningless hai, isliye humein vectors chahiye.

Plain $F$ (koi arrow nahi) matlab sirf magnitude — arrow ki length, ek positive number.

8. Unit vectors — pure "kaunsi taraf" arrow $\hat{r}$

Picture karo. Ek compass needle socho: yeh batata hai kaunsi taraf lekin kitni door nahi. Ek poora force vector banane ke liye hum direction (${\hat{r}}_{12}$) lete hain aur use magnitude ($\frac{G{m}_1{m}_2}{r^2}$) se stretch karte hain: ${\vec{F}}_{12}=\underset{\text{how strong}}{\underbrace{\frac{G{m}_1{m}_2}{r^2}}}\underset{\text{which way}}{\underbrace{{\hat{r}}_{12}}}$

Topic ko yeh kyun chahiye. Yeh force ko cleanly "kitna hard" times "kaunsa direction" mein split karta hai, toh attractive nature purely encode hota hai $\hat{r}$ ko doosre mass ki taraf point karake.

9. Local field strength $g$ — woh gravity jo tum feel kar sakte ho

$m$ ka cancel hona kyun matter karta hai. Kyunki $m$ gayab ho jaata hai, har object same $g$ se accelerate karta hai — ek feather aur ek hammer, air ignore karte hue. Yahi Galileo ka tower experiment algebra se nikalta hai.

Topic ko $g$ kyun chahiye. $G$ feel karne ke liye bahut tiny aur abstract hai. $g$ everyday, on-a-planet version hai — woh number jo kehta hai "9.8 metres-per-second fast harta har second jab tum kuch drop karo." Poori field idea ke liye Gravitational Field & Potential dekho aur $W=mg$ ke liye Weight vs Mass dekho.

Symbol $R_{\oplus }$ (⊕ Earth-circle ke saath) ka matlab hai "Earth ki radius"; $M_{\oplus }$ ka matlab hai "Earth ka mass".

10. Circular motion words ("falling Moon" check ke liye)

Topic ko yeh kyun chahiye. Newton ka universality ka proof Moon ki gravity-acceleration ($g/60^2$) ko uski circular-motion acceleration ($4{\pi }^{2}r/T^2$) se compare karta hai. Yeh match karte hain — girti apple wali force HI orbiting Moon wali force hai. Circular-motion machinery Circular Motion & Centripetal Force mein hai, aur jo orbits yeh produce karta hai woh Kepler's Laws hain.

Prerequisite map

Equipment checklist

Right side cover karo aur khud se test karo — tum parent note ke liye ready ho sirf tab jab har ek instant ho.

Connections

• Parent topic — in sabhi symbols ko poore law mein assemble karta hai.
• Newton's Third Law — vector figure mein equal-and-opposite arrows.
• Shell Theorem — kyun spheres ke liye $r$ centre-to-centre hai.
• Gravitational Field & Potential — $g$ ka field view.
• Weight vs Mass — stuff-count ko pull se alag karta hai.
• Circular Motion & Centripetal Force — falling-Moon check ke liye $T$ aur $a$ supply karta hai.
• Kepler's Laws — inverse-square gravity jo orbits produce karti hai.
• General Relativity — "instant action at a distance" ko curved spacetime se replace karta hai.