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

1.2.19 · Physics › Newton's Laws & Dynamics

YE LAW KYUN EXIST KARTA HAI?

FORCE dono masses par kyun depend karti hai? Newton ke 3rd law se, body A ka B par pull, B ka A par pull ke barabar hota hai. Agar force sirf $m_{1}$ par depend karti, toh unequal masses ke liye dono pulls barabar nahi ho sakti. $F_{12} = F_{21}$ banana ka symmetric tarika yeh hai ki ise product $m_{1} m_{2}$ par depend karaya jaaye.

Inverse-square kyun? Socho ki gravitational "influence" radius $r$ ke expanding sphere ki surface par ek point source se phail rahi hai. Surface area hota hai $4 π r^{2}$ . Wahi total influence ek bade area par dilute ho jaata hai, toh uski intensity $\propto 1/ r^{2}$ . Yeh wahi geometry hai jo light intensity aur sound ke peeche hai.

Formula ko scratch se banana

Step 1 — Force har mass ke saath badhti hai. Kyun? Zyada matter = zyada "kheenchne wali cheez." $m_{1}$ double karo toh har attracting bit double ho jaata hai, isliye $F \propto m_{1}$ . $m_{2}$ ke liye bhi yehi. Milake: $F \propto m_{1} m_{2}$

Step 2 — Force geometrically dilute hoti hai. Kyun? Influence $4 π r^{2}$ area wale sphere par phailti hai: $F \propto \frac{1}{r ^{2}}$

Step 3 — Combine karo aur ek constant daalo. Constant kyun? Proportionalities ko ek unit-fixing number chahiye jo experiment se milta hai (Cavendish): $F = \frac{G m _{1} m _{2}}{r ^{2}}$

Step 4 — Ise vector banao (direction matter karta hai). Mass 2 ki wajah se mass 1 par force 1 se 2 ki taraf point karti hai. $\overset{r}{^}_{12}$ ko 1 se 2 ki taraf point karne do: $F_{12} = + \frac{G m _{1} m _{2}}{r ^{2}} \overset{r}{^}_{12}$ $\overset{r}{^}_{12}$ ka doosre mass ki taraf point karna aur $+$ sign — yeh attraction ko encode karta hai.

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

"Action at a distance" kya hai?

Field reformulation (80/20 upgrade): $g = \frac{F}{m _{test}} = \frac{GM}{r ^{2}} (toward M)$ Ab field space mein exist karta hai; ek test mass bas ise locally sample karta hai. Yeh conceptually "instant action" ki chinta khatam kar deta hai.

Worked Examples

Common Mistakes (Steel-manned)

Active Recall

Recall Quick self-test (answers cover karo)

Product $m_{1} m_{2}$ kyun, sum $m_{1} + m_{2}$ kyun nahi? → Newton's 3rd law symmetry.
Inverse-square kyun? → Influence sphere area $4 π r^{2}$ par dilute hoti hai.
Saare objects same $g$ se kyun girte hain? → $a = GM / r^{2}$ mein mass cancel ho jaata hai.
"Action at a distance" kya hai? → Empty space mein force, koi contact nahi.
$g$ aur $G$ mein farq? → $G$ universal constant hai; $g$ local field strength hai.

Recall Feynman: 12-saal ke bachche ko explain karo

Socho ki har cheez jisme stuff hai woh ek chhota magnet hai jo sirf kheenchta hai (kabhi push nahi karta). Badi cheezein zyada kheenchti hain. Aur jitna door jaao, pull utni hi tezi se weak hoti jaati hai — do guna door jaao, pull chaar guna weak ho jaati hai. Earth itni badi hai ki uski pull tumhe zameen se chipkaye rakhti hai, Chaand ko hmare around ghumaati hai, aur Earth ko Sun ke around ghumaati hai. Sabse spooky baat: kuch touch nahi karta — pull empty space ke paar pahunch jaati hai, ek aisi invisible rope ki tarah jo tum dekh ya kaat nahi sakte.

Connections

Newton's Third Law — $m_{1} m_{2}$ symmetry aur equal-and-opposite pulls deta hai.
Gravitational Field & Potential — action-at-a-distance ka field solution.
Kepler's Laws — inverse-square gravity + circular/elliptical motion se milte hain.
Circular Motion & Centripetal Force — "falling Moon" check mein use hota hai.
Shell Theorem — spheres ke liye $r$ centre-to-centre kyun hota hai.
General Relativity — instant action ko speed $c$ par curved spacetime se replace karta hai.
Weight vs Mass — $W = m g$ , aur $g = GM / R^{2}$ .

Newton's law of gravitation ka formula?

F = \frac{G m _{1} m _{2}}{r ^{2}}

, attractive, masses ko milane waali line ke saath.

F

masses ke product par depend karta hai, sum par kyun nahi?

Newton's 3rd law satisfy karne ke liye: symmetric form

m_{1} m_{2}

automatically

F_{12} = F_{21}

bana deta hai.

Inverse-square kyun, inverse-

r

kyun nahi?

Influence

4 π r^{2}

surface area waale sphere par phailti hai, isliye intensity

1/ r^{2}

se girti hai.

G

ki value aur units?

6.674 \times 1 0^{- 11} N m^{2} k g^{- 2}

, har jagah/har waqt universal.

g

aur

G

mein farq?

G

universal constant hai;

g = GM / r^{2}

local field strength hai (planet aur altitude ke saath change hoti hai).

Saare objects same acceleration se kyun girte hain?

a = F / m = GM / r^{2}

— girne waale object ka mass cancel ho jaata hai.

Earth ki surface par

g

derive karo.

m g = G M_{\oplus} m / R_{\oplus}^{2} \Rightarrow g = G M_{\oplus} / R_{\oplus}^{2} \approx 9.8

m/s².

"Action at a distance" kya hai?

Empty space mein bina kisi contact ke force lagti hai aur (Newton ke model mein) instantaneously.

Do spheres ke liye

r

kya hota hai?

Unke centres ke beech ki doori (shell theorem).

Newton ka "falling Moon" argument kya tha?

Chaand ka orbital acceleration

\approx g /6 0^{2}

hai jo

4 π^{2} r / T^{2}

se match karta hai, yeh prove karta hai ki apple-force = Moon-force hai.