Gravitational field intensity g = GM - r²: Visual walkthrough

1.2.20 · D2 · Physics › Newton's Laws & Dynamics › Gravitational field intensity g = GM - r²

Step 1 — Do masses aur unke beech ka arrow

KYA. Hum ek bada blob of stuff rakhte hain — uski "stuff" ki quantity ko $M$ (source mass, kilograms mein) bolte hain — ek point par. Kuch doori par hum ek tiny speck rakhte hain, quantity $m$ (test mass). Unke centres ke beech ek straight line draw karte hain aur uski length $r$ (metres mein) measure karte hain.

KYO. Kisi bhi formula se pehle, gravity bas yahi hai: blob speck ko kitni zyada kheenchta hai? Hum teen cheezon ko naam dena chahte hain jo matter kar sakti hain — kitna stuff kheeench raha hai ( $M$ ), kitna stuff kheencha ja raha hai ( $m$ ), aur woh kitni doori par hain ( $r$ ). Baaki sab kuch inheen teeno se bana hai.

PICTURE. Left par blue blob $M$ aur right par small orange dot $m$ dekho. Gray line jis par $r$ likha hai, woh doori hai centre to centre — surface to surface nahi. Yeh detail baad mein waapas aayegi.

Step 2 — Newton humein force deta hai

KYA. Newton's Law of Universal Gravitation kehta hai ki do masses ke beech pulling force $F$ (newtons, N mein) yeh hai: $F = fixed number of nature G \cdot more stuff, closer = stronger pull \frac{M \cdot m}{r ^{2}}$

Term by term padhte hain:

Upar $M \cdot m$ — kisi bhi mass ko double karo, force double ho jaati hai. Kisi bhi end par zyada stuff matlab bada tug.
Neeche $r^{2}$ — unhe door karo, force jaldi kamzor ho jaati hai (Step 6 mein "squared" ko visually prove karenge).
$G = 6.67 \times 1 0^{- 11} N m^{2} kg^{- 2}$ — gravitational constant. Yeh bas woh conversion factor hai jo "kilograms aur metres" ko "newtons" mein badalta hai. Yeh kabhi nahi badalta.

KYO. Yeh hamara ek hi starting assumption hai — woh ek law jise hum faith par lete hain (yeh measure kiya gaya hai, derive nahi). Dekho Newton's Law of Universal Gravitation. Is page par sab kuch is ek equation se nikalta hai.

PICTURE. Orange arrow speck par force dikhata hai: yeh blob ki taraf waapas point karta hai (gravity hamesha attract karti hai). Uski length force ki strength hai.

Step 3 — Problem: answer baar baar badalta rehta hai

KYA. Maan lo main blob $M$ aur distance $r$ fixed rakhta hun, lekin speck ko ek bhaari se swap karta hun — $m$ double karta hun. Step 2 se, force $F$ double ho jaati hai. Teen guna speck daalo, force teen guna ho jaati hai.

KYO. Yeh annoying hai. Force is baat par depend karta hai ki kaun visit kar raha hai. Agar main location describe karna chahun — "yeh jagah gravitationally kitni khatarnak hai?" — toh main nahi chahta ki answer har baar badal jaaye jab koi alag object aaye. Main ek aisa number chahta hun jo space khud ka ho.

PICTURE. Teen test masses ( $m$ , $2 m$ , $3 m$ ) usi distance $r$ par baithte hain. Unke force arrows ki lengths alag hain (1×, 2×, 3×). Jagah nahi badi — sirf visitor badla. Yahi woh cheez hai jo hum fix karni chahte hain.

Step 4 — Trick: force per kilogram

KYA. Force ko test mass se divide karo. Hum ek naya quantity invent karte hain, gravitational field intensity $g$ : $g = \frac{F}{m} (newtons per kilogram, N kg^{- 1})$

KYO. Step 3 mein force $m$ ke proportional thi: $m$ double karo, $F$ double. Toh ratio $F / m$ har visitor ke liye same rehta hai. $m$ se divide karna exactly woh operation hai jo visitor ko cancel karta hai aur ek location ki property chhodta hai. Yahi "field" ka poora idea hai: objects par forces describe karna band karo, space describe karna shuru karo.

PICTURE. Step 3 ke same teen test masses, lekin ab har force arrow uski apni mass se scale down ho jaata hai. Teeno shrink hokar same length par aa jaate hain — field $g$ ek arrow hai, jo wahan khade chahे koi bhi ho, us spot ka share karta hai.

Step 5 — Substitute karo aur cancel karo

KYA. Newton ki force (Step 2) ko field ki definition (Step 4) mein daalo: $g = \frac{F}{m} = \frac{1}{m} \cdot \frac{G M m}{r ^{2}} = \frac{G M m}{r ^{2} m} = \frac{GM}{r ^{2}}$

Final result mein term by term:

$G$ — nature ka constant, abhi bhi saath hai.
$M$ — source mass. Sirf blob ka mass upar bachta hai.
$r^{2}$ — centre-to-centre distance, squared, neeche.
$m$ — gaya. Yeh ek baar upar aaya (Newton ke $F$ se) aur ek baar neeche ( $m$ se divide karne se), isliye perfectly cancel ho gaya.

KYO. $m$ ka cancellation Step 4 ka payoff hai. Yeh prove karta hai ki field visitor ki parwah nahi karta — ek feather aur ek bowling ball same point par identical $g$ feel karte hain.

PICTURE. Numerator mein $m$ aur denominator mein $m$ ko ek balance beam par draw kiya gaya hai, jo ek doosre ko strike out kar rahe hain, aur $GM / r^{2}$ akela khada reh jaata hai.

Step 6 — Square kyun? Spreading-sphere picture

KYA. Ab hum dekhte hain ki $r^{2}$ kahan se aata hai, sirf accept nahi karte. Socho ki $M$ se "field lines" ka ek fixed number saari directions mein bahar nikalti hain. Har line jo $M$ se nikalti hai, use iske aas-paas kheechi har sphere ko pierce karni padti hai — lines ki count fixed hai. Field strength in lines ki crowding hai: har square metre mein kitni pierce karti hain.

Radius $r$ wale sphere ka surface area hai $A = 4 π r^{2} .$ Same lines, bada area, toh density (= field) hai $g \propto \frac{fixed lines}{4 π r ^{2}} \propto \frac{1}{r ^{2}} .$

$r^{2}$ kyun aur $r$ ya $r^{3}$ kyun nahi? Kyunki hum teen dimensions mein rehte hain aur sphere ka area uske radius ke square ke saath badhta hai. $r$ double karo → area ×4 → lines chaar guna spread out → field quarter ho jaaye. Exponent 2 literally "area of a surface" mein "2" hai. Yeh Gauss's Law for Gravity ka geometric core hai.

PICTURE. Do nested spheres, radii $r$ aur $2 r$ . Same lines dono ko cross karti hain, lekin outer sphere par woh chaar guna sparse hain — arrows ek-chauthai dense draw kiye gaye hain.

Step 7 — Edge cases: extremes par kya hota hai?

KYA. Formula tabhi trustworthy hai jab hum uski boundaries check karein.

$r \to \infty$ (bahut door): $g = GM / r^{2} \to 0$ . Pull fade hokar kuch nahi reh jaati lekin kabhi exactly zero nahi hoti — gravity ki reach infinite hai, bas vanishingly weak.
$r \to 0$ (point mass ke paas jaana): $g = GM / r^{2} \to \infty$ . Formula blow up kar jaata hai. Yeh idealised point mass ke liye theek hai, lekin ek real planet ka size hota hai — uske andar, tumhe Variation of g with Altitude and Depth use karna padega, kyunki $r < R$ hone par story badal jaati hai.
Ek real planet ki surface par ( $r = R$ ): yahan $r$ planet ke radius $R$ ke barabar hai, toh $g = GM / R^{2}$ . Yahi roz ka $9.8 N kg^{- 1}$ hai.
Altitude $h$ par: centre se distance $r = R + h$ hai, $h$ nahi. Yeh sabse common slip hai.

KYO. Har scenario jo ek reader ko mil sakta hai — deep space, surface, pahaad par, mathematical singularity — sab cover kiya gaya hai taaki koi bhi case tumhe surprise na kar sake.

PICTURE. $g$ versus $r$ ka curve: $r \to 0$ par infinity ki taraf shoot up karta hai, $r = R$ par surface value se guzarta hai, aur $r$ badhne par zero ki taraf decay karta hai. Surface, altitude, aur far-field points sab mark hain.

Step 8 — Jab do sources ek saath kheenchein (vectors)

KYA. Fields vectors ki tarah add hoti hain — direction ke saath, plain numbers ki tarah nahi. Agar ek point par Earth ki field $g_{E}$ left point kar rahi hai aur Moon ki $g_{M}$ right, toh net field hai $g_{net} = g_{E} - g_{M} (opposite directions subtract) .$ Jahan $g_{E} = g_{M}$ , woh completely cancel hoti hain: null point, jahan $g_{net} = 0$ .

KYO. Kyunki $g = F / m$ force ki arrow-nature inherit karta hai. Do arrows head-to-tail resultant dete hain. Same direction → add; opposite → subtract; right angles par → Pythagoras, $g = g_{1}^{2} + g_{2}^{2}$ .

PICTURE. Earth (blue) aur Moon (gray) apne opposing field arrows ke saath. Unke beech, ek point jahan arrows equal hain aur cancel ho jaate hain — null point ke roop mein red mein mark kiya gaya.

Ek-picture summary

Upar sab kuch ek frame mein compress: Newton ka force law definition $g = F / m$ ko feed karta hai; visitor mass $m$ cancel ho jaata hai; $g = GM / r^{2}$ bahar aata hai; spreading sphere square explain karta hai; aur $g$ -vs- $r$ curve ek saath har regime dikhata hai.

Recall Feynman retelling — saara walkthrough simple words mein

Ek bhaari ball aur uske paas ek tiny bead ki picture banao (Step 1). Newton humein bataata hai woh kitna zyada tug karte hain: zyada stuff ya paas distance matlab zyada strong pull, ek fixed nature-number $G$ scale set karta hai (Step 2). Problem: heavier bead daalo aur pull badhti hai — toh "pull" bead ko describe karta hai, spot ko nahi (Step 3). Fix: pull ko bead ke apne kilogram-weight se divide karo. Ab har bead — feather ya brick — us spot par same answer deti hai; humne space capture kar liya, visitor ko nahi (Step 4). Division karo aur bead ka mass literally top-and-bottom cancel ho jaata hai, $GM / r^{2}$ bacha rehta hai (Step 5). Neeche square kyun? Kyunki bhaari ball ki "pulling rays" ek sphere par spread hoti hain, aur sphere ka area radius-squared ke saath badhta hai — do guna door jao, rays chaar guna patli hain (Step 6). Phir hum ends sanity-check karte hain: infinitely far nearly zero hai, point par baithna infinite hai, aur altitude centre se measure hoti hai (Step 7). Aakhir mein, ek saath do balls arrows ki tarah add hoti hain, aur jahan unke arrows match karte hain woh cancel ho jaate hain — null point (Step 8).

Recall Pehle predict karo, phir check karo

$r$ teen guna karo. $g$ ka kya hoga?
$m$ final formula se kyun gayab ho gaya?
Do guna door sphere par, field half kyun nahi balki quarter hai?

Answers: 1. $g \to g /9$ (inverse-square, $3^{2}$ ). 2. Yeh ek baar $F$ se upar aaya aur ek baar $m$ se divide karne se neeche; woh cancel ho jaate hain. 3. Sphere area $= 4 π r^{2}$ $r^{2}$ ke saath scale karta hai, toh $r$ double karo → area ×4 → field ÷4.

Connections

Newton's Law of Universal Gravitation — woh ek first principle jisse humne shuru kiya (Step 2)
Newton's Second Law (F=ma) — kyun $g$ free-fall acceleration bhi hai
Gauss's Law for Gravity — spreading-sphere picture ke peeche formal engine (Step 6)
Variation of g with Altitude and Depth — jab $r \neq = R$ ho tab kya karna hai (Step 7)
Gravitational Potential and Potential Energy — field ka "height map," $g = - d V / d r$
Electric Field Intensity E=kQ/r² — identical algebra, mass ki jagah charge (Step 5)