Gravitational field intensity g = GM - r²: Foundations

1.2.20 · D1 · Physics › Newton's Laws & Dynamics › Gravitational field intensity g = GM - r²

Parent note Gravitational Field Intensity par trust karne se pehle, tumhe use padhna aana chahiye. Neeche har symbol usi note mein appear hota hai. Hum har ek ko plain words mein define karte hain, draw karte hain, aur batate hain ki topic uske bina kyon nahi chal sakta. Inhe is order mein rakha gaya hai ki har ek apne pehle wale par lean kare.

1. Mass $M$ aur mass $m$ — "kitna stuff hai" ke numbers

Picture: socho ek bowling ball ( $M$ ) table par rakha hai aur ek marble ( $m$ ) jo tum uske paas pakad rahe ho. Bowling ball "boss" hai; marble "probe" hai.

Topic ko dono kyun chahiye: ek field ka poora point yeh hai ki describe karo ki source $M$ space ke saath kya karta hai, bina iske ki kaunsa marble $m$ tum test karne ke liye use kar rahe ho. Dono letters ko alag rakhna hi woh cheez hai jo baad mein hume $m$ cancel karne aur sirf source-only quantity ke saath bachne deti hai.

2. Distance $r$ — CENTRE se naapa jaata hai

Yeh woh symbol hai jise students aksar galat karte hain, isliye hum ise dhyan se draw karte hain.

Picture (figure dekho): teal dot planet ka centre hai. Orange arrow jis par $r$ likha hai woh us centre se seedha wahan tak jaata hai jahan marble baitha hai — surface se nahin. Agar tum zameen se $h$ altitude par ho, toh

$r = R + h$

jahan $R$ (capital R) planet ki radius hai. Figure mein plum bracket $h$ ko surface se shuru hote dikhata hai, jabki $r$ gehra, core se shuru hota hai.

Topic ko yeh kyun chahiye: gravity jitni door ho utni kamzor hoti hai. "Kitni door" kehne ke liye, tumhe agree karna hoga ki kahan se — aur gravity aise kaam karti hai jaise sphere ka saara mass uske centre par ho. Isliye $r$ hamesha centre-to-you hota hai.

3. Force $F$ aur uske upar arrow — vectors vs plain numbers

Picture: ek scalar ruler par likhi length hai; ek vector ek arrow hai — uski length size hai, uska direction heading hai.

Figure mein, burnt-orange arrow force $F$ hai: uski ek length hai (kitna hard) aur ek direction hai (kaunsi taraf — yahan, planet ke centre ki taraf seedha). Uske paas grey number, " $5$ ", woh hai jo bachta hai agar tum direction bhool jao — woh scalar $F$ hai (koi arrow nahin).

Topic ko yeh kyun chahiye: gravity hamesha source ki taraf kheenchti hai. "Taraf" ek direction hai, isliye force ek vector hona chahiye. Baad mein, jab Earth aur Moon dono ek hi point par opposite sides se pull karein, toh hum unke pulls ko sirf tab sahi combine kar sakte hain jab hum direction track karein — isliye parent note do fields ko add karne ki jagah subtract karta hai.

4. Unit vector $\overset{r}{^}$ — "bahar kaunsi taraf hai?"

Picture: apne point par khade ho aur seedha planet ke centre ki taraf dekho. Ab 180° mudo. Woh direction jis taraf tum ab face kar rahe ho — $M$ se seedha door — woh $\overset{r}{^}$ hai.

Topic ko yeh kyun chahiye: field ko vector ki tarah likhne ke liye, hum ise "kitna bada" aur "kaunsi taraf" mein split karte hain. Size $\frac{GM}{r ^{2}}$ hai; direction $\overset{r}{^}$ carry karta hai. Kyunki gravity andar kheenchti hai lekin $\overset{r}{^}$ bahar ki taraf point karta hai, hum saamne ek minus sign lagate hain:

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

Minus literally yeh phrase hai "opposite to $\overset{r}{^}$ " = "andar, $M$ ki taraf".

5. Gravitational constant $G$ — prakriti ka fixed strength dial

Picture: $G$ ko gravity ke liye master volume knob samjho, jagah par weld hua. Yeh bahut chhota hai, isliye tum khud ko apne dost ko room ke across kheechchte feel nahin karte.

Topic ko yeh kyun chahiye: force law ko koi number chahiye "kilograms aur metres" ko "newtons of pull" mein badalne ke liye. $G$ woh conversion factor hai. Uske bina, $\frac{M m}{r ^{2}}$ sirf ek shape hoga jiska koi physical size nahin.

6. Exponent " $^{2}$ " aur fraction bar — falloff ki shape

Picture: distance double karo aur field half nahin hoti — yeh quarter ho jaati hai, kyunki $2^{2} = 4$ . Teen guna karo, aur yeh ninth ho jaati hai ( $3^{2} = 9$ ).

Figure dekho: teal curve $g \propto 1/ r^{2}$ hai. Notice karo ki yeh planet ke paas kitni steeply girta hai aur door ek lambi, faint tail mein kaise flat ho jaata hai — yeh zero ke paas aata hai lekin kabhi poora touch nahin karta. Dotted markers surface value dikhate hain, phir $2 R$ par $1/4$ , phir $3 R$ par $1/9$ .

$r$ ya $r^{3}$ nahin, $r^{2}$ kyun? Yahan woh geometric reason hai jo parent note deta hai, draw kiya hua: imagine karo same number of "pull-lines" $M$ se sabhi directions mein nikal rahi hain. Distance $r$ par woh ek sphere ki surface par spread ho jaati hain. Sphere ka area $4 π r^{2}$ hota hai — yeh $r$ ke square ke saath badhta hai. Isliye lines exactly $1/ r^{2}$ ki tarah dilute ho jaati hain. Isliye exponent 2 hai: yeh sphere se aata hai.

7. Symbols ko saath rakhna — formula ko sentence ki tarah padhna

Ab $g = \frac{GM}{r ^{2}}$ ka har piece ka ek meaning aur ek picture hai:

Unit $N kg^{- 1}$ literally hai "newtons of force, visitor ke har kilogram ke liye" — jo exactly definition hai "force per unit mass". Yeh $m s^{- 2}$ ke barabar bhi hai, acceleration ki units, isliye $g$ free-fall acceleration nikalta hai (woh link Newton's Second Law (F=ma) se aata hai).

Yeh foundations topic ko kaise feed karte hain

Baaye taraf har foundation ek symbol hai jo ab tumhara hai. Saath mein woh daaye taraf ke parent formula mein assemble ho jaate hain.

Equipment checklist

Khud test karo — daaya side cover karo aur reveal karne se pehle jawab do.

M

(bada) kiske liye khada hai, aur

m

(chhota)?

M

= source mass jo pull kar rahi hai;

m

= chhoti test mass jo hum field feel karne ke liye use karte hain.

Distance

r

kahan se shuru hoti hai?

Source mass ke centre se, isliye altitude

h

par tum

r = R + h

use karte ho.

Scalar aur vector mein kya fark hai?

Scalar sirf size rakhta hai; vector mein size aur direction dono hote hain (arrow ki tarah draw kiya,

F

likha).

\overset{r}{^}

mein hat ka kya matlab hai, aur yeh kis taraf point karta hai?

Yeh length 1 ka unit vector hai jo bahar ki taraf,

M

se door point karta hai;

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

mein minus sign ise andar flip karta hai.

G

kya hai aur hume ise kyun chahiye?

Gravitational constant

6.67 \times 1 0^{- 11} N m^{2} kg^{- 2}

; yeh masses aur distances ko actual newtons of force mein convert karta hai.

Law inverse-square (

1/ r^{2}

) kyun hai,

1/ r

nahin?

Field lines

4 π r^{2}

area wale sphere par spread ho jaati hain, isliye strength distance ke square ki tarah dilute hoti hai.

Agar tum

r

double karo, toh

g

ka kya hoga?

Yeh

\frac{1}{4}

ho jaata hai (kyunki

2^{2} = 4

g

ki units kya hain aur unke do meanings kya hain?

N kg^{- 1} = m s^{- 2}

— force per kilogram, free-fall acceleration ke barabar.

Connections

Parent topic — Gravitational Field Intensity — woh note jise yeh foundations unlock karte hain
Newton's Law of Universal Gravitation — jahan force law $F = GM m / r^{2}$ aata hai
Newton's Second Law (F=ma) — $g$ free-fall acceleration kyun ban jaata hai
Gravitational Potential and Potential Energy — in symbols par built agla layer
Gauss's Law for Gravity — sphere-area argument ka formal statement
Variation of g with Altitude and Depth — $r = R + h$ seedha use karta hai
Electric Field Intensity E=kQ/r² — same symbol machinery, force per unit charge