Gravitational field intensity g = GM - r² — Hinglish notes

1.2.20 · Physics › Newton's Laws & Dynamics

WHAT it is

"Per unit mass" KYUN? Kyunki test mass par force $m$ ke proportional hoti hai. $m$ se divide karne par visitor ka mass cancel ho jaata hai, aur ek aisi quantity bachti hai jo sirf source aur aap kahan khade hain par depend karti hai. Yahi ek field ka poora trick hai.

HOW we derive $g = \frac{GM}{r ^{2}}$ (from scratch)

Formula Newton's law of gravitation se derivation

Step 1 — Force law se shuru karo. Newton ka universal law do point masses $M$ aur $m$ ke liye jo $r$ se alag hain: $F = \frac{GM m}{r ^{2}}$ Yeh step kyun? Yeh hamaara first principle hai — ek hi assumption. $G = 6.67 \times 1 0^{- 11} N m^{2} kg^{- 2}$ .

Step 2 — Field ki definition apply karo. Field intensity force per unit mass hoti hai: $g = \frac{F}{m} = \frac{1}{m} \cdot \frac{GM m}{r ^{2}}$ Yeh step kyun? Hum test mass ka influence hataa rahe hain taaki sirf us location ko describe kar sakein.

Step 3 — $m$ cancel karo. $g = \frac{GM}{r ^{2}}$ Yeh step kyun? Numerator aur denominator mein $m$ cancel ho jaata hai — yeh prove karta hai ki field test mass se independent hai. ✔

Vector form mein ( $\overset{r}{^}$ ko $M$ se door point karte hue): $g = - \frac{GM}{r ^{2}} \overset{r}{^}$ Minus sign ka matlab hai yeh andar ki taraf, $M$ ki taraf point karta hai (gravity attract karti hai).

Figure — Gravitational field intensity g = GM - r²

Worked examples

Common mistakes (Steel-manned)

Active recall

Recall Forecast-then-Verify: padhne se pehle predict karo

Agar $M$ se apni distance triple karo, toh $g$ ka kya hoga? → (forecast)
$g$ ki units kya hain, aur do equivalent forms kyun hain?
Test mass final formula mein kyun nahi dikhti?

Answers: 1. $g \to g /9$ (inverse-square, $3^{2}$ ). 2. $N kg^{- 1} = m s^{- 2}$ kyunki $g$ free-fall acceleration ke barabar hota hai. 3. Yeh $F / m$ mein cancel ho jaata hai; field source + location ki property hai.

Recall Feynman: ek 12-saal ke bache ko explain karo

Ek bahut badi magnetic ball socho (lekin weight ke liye, metal ke liye nahi). Yeh apne aas-paas invisible "kheenchne wali rays" banata hai. Paas mein, rays crowded hain aur zyada kheenchti hain; door mein yeh ek giant bubble par fail gayi hain, toh kaafi kam kheenchti hain. Yeh kheenchne ki strength hi field hai. Interesting part yeh hai: yeh ek marble aur ek bowling ball ko same strength per kilogram se kheenchti hai — sirf planet aur distance decide karte hain kitni strong hai, na ki kaun visit kar raha hai.

Flashcards

Gravitational field intensity ki definition hai

force per unit mass,

g = F / m

Point/spherical mass

M

se distance

r

par

g

ka formula

g = GM / r^{2}

M

ki taraf directed

Gravitational field intensity ki units

N kg^{- 1}

m s^{- 2}

g

mein se test mass kyun cancel ho jaati hai?

Force

F \propto m

;

m

se divide karne par sirf source-dependent quantity bachti hai

Distance

r

kahan se measure hoti hai?

Source mass ke centre se, isliye altitude

h

par

r = R + h

Distance

2 R

par field surface value ki kitni fraction hogi?

1/4

(inverse-square,

2^{2}

)

Geometrically gravity inverse-square law kyun hai?

Field lines sphere area

4 π r^{2} \propto r^{2}

par spread hoti hain

Field intensity aur free-fall acceleration ka relation

Dono equal hain:

a = F / m = g

Earth–Moon null point par net field

zero (opposite fields magnitude mein equal hain)

\overset{r}{^}

M

se door point karte hue field ka vector form

g = - \frac{GM}{r ^{2}} \overset{r}{^}

Connections

Newton's Law of Universal Gravitation — woh force law jisse hum $g$ derive karte hain
Newton's Second Law (F=ma) — $g$ free-fall acceleration ke kyun barabar hota hai
Gravitational Potential and Potential Energy — $g = - d V / d r$
Gauss's Law for Gravity — $1/ r^{2}$ field-line picture ka formal reason
Variation of g with Altitude and Depth — yeh $r \neq = R$ ke liye extend karta hai
Electric Field Intensity E=kQ/r² — identical structure (force per unit charge)