Variation of g — with altitude, latitude, depth: Foundations

1.2.21 · D1 · Physics › Newton's Laws & Dynamics › Variation of g — with altitude, latitude, depth

Parent note ki ek bhi line padhne se pehle, tumhe us har symbol ko earn karna hoga jo woh tumhare saamne pheinkta hai. Neeche, har idea ko kuch nahi se build kiya gaya hai, picture ki tarah draw kiya gaya hai, aur us sawaal se justify kiya gaya hai jiska woh jawaab deta hai. Upar se neeche padho — har item uske upar waale par lean karta hai.

1. Mass $m$ — "kitna stuff hai"

Yeh topic isko kyun use karta hai: yeh poora sawaal ki "kya ek 70 kg insaan equator par kam weigh karta hai?" tabhi sense banata hai jab hum maan lein ki 70 kg (woh stuff) kabhi nahi badlta, sirf us par pull badlti hai. Mass woh unchanging anchor hai. Full split ke liye Weight vs Mass dekho.

2. Force $F$ aur weight — "woh pull"

Figure — Variation of g — with altitude, latitude, depth

Figure mein, wahi brick (mass $m$ , unchanged) do jagah spring se latki hui hai. Brick waisi hi hai; lal arrow — woh pull, uska weight — ek jagah zyada lamba hai. Yahi difference is poore topic ka subject hai.

Yeh topic isko kyun use karta hai: parent note $F = m g$ likhta hai. Woh sentence kehta hai "pull $F$ barabar hai mass $m$ gunaah pull-per-kilogram $g$ ." Isko padhne ke liye, tumhe pehle se pata hona chahiye ki $F$ woh pull hai aur $m$ woh stuff hai.

3. Gravitational acceleration $g$ — "pull per kilogram"

Yeh topic isko kyun use karta hai: poora topic ka title hai "Variation of $g$ ." $g$ is ka star hai. Dhyan do ki $g$ ek ratio hai — mass $m$ ko divide kar diya gaya hai — yehi reason hai ki $g$ jagah describe karta hai, object ko nahi. Ek feather aur ek boulder ek hi spot par same $g$ feel karte hain.

4. Inverse-square distance rule $1/ r^{2}$

Yeh woh tool hai jis par topic sabse zyada lean karta hai, isliye hum ise dhyan se build karte hain.

Figure dekho: fixed size ka lal patch paas se bahut saare arrows pakadta hai, lekin door se usi size ka patch bahut kam pakadta hai, kyunki arrows ek bade sphere par fan out ho gaye hain.

Yeh topic isko kyun use karta hai: altitude section ( $h$ se upar jao) kuch nahi balki " $r$ bada ho gaya, toh $1/ r^{2}$ chhota ho gaya" hai. 2 ka factor jo $g_{h} \approx g (1 - 2 h / R)$ mein hai woh seedha square in $r^{2}$ ka fingerprint hai. Agar tumhe feel nahi hota ki yeh square kyun hai, toh woh mysterious 2 hamesha arbitrary lagega.

5. Constants $G$ , $M$ , $R$ , aur height/depth symbols $h$ , $d$

Ab hum item 4 ke do generic masses ko specialise karte hain: unme se ek Earth hai, toh ab se hum use $M$ naam dete hain. Doosra tumhara object $m$ hai items 1–2 se.

Definition Geometry symbols

$G$ — gravitational constant, nature ka ek fixed tiny number jo gravity ki strength set karta hai: $G \approx 6.67 \times 1 0^{- 11} m^{3} kg^{- 1} s^{- 2}$ (padho: "cubic metres per kilogram per second-squared"). Universe mein har jagah same.
$M$ — Earth ka total mass (lagbhag $6 \times 1 0^{24}$ kg); yeh item 4 ka $M_{1}$ hai, ab ise ek naam diya gaya hai.
$R$ — Earth ka radius (lagbhag $6400$ km), Earth ke centre se uski surface tak ki distance.
$h$ — woh height jo tum SURFACE SE UPAR chadhe (ek mountaintop, ek balloon). Hamesha surface se seedha upar measure kiya jaata hai.
$d$ — woh depth jitna tum SURFACE SE NEECHE utre (ek mine shaft). Hamesha surface se seedha neeche measure kiya jaata hai, isliye depth $d$ par tum centre se $R - d$ ki doori par ho.
$r$ — Earth ke centre se us jagah ki doori jahaan tum actually khadey ho. Surface par $r = R$ ; upar $r = R + h$ ; neeche effective $r = R - d$ .

Figure mein Earth ek circle dikhta hai centre $O$ ke saath. Lal line $r$ hai — centre se tumhari actual distance. Dhyan do ki jab tum height $h$ chadh jaate ho toh yeh $R + h$ tak stretch hoti hai, aur jab tum depth $d$ utarte ho toh $R - d$ tak shrink hoti hai.

Yeh topic split $r = R \pm x$ kyun use karta hai: parent ka har section bas yahi formula hai jismein $r$ ko thoda nudge kiya gaya hai. Altitude $r$ ko $h$ se upar nudge karta hai, depth effective $r$ ko $d$ se neeche nudge karta hai, aur dono ko surface value $GM / R^{2}$ se compare kiya jaata hai. Yeh pehchanna ki $h$ , $d$ , $R$ mein se kaun kaun sa hai, chapter ke har algebra slip ko rokta hai.

6. Shell Theorem — kyun tumhare upar ki chhat tumhe nahi kheenchti

Density se pehle, hume woh fact earn karna hoga jis par item 5 lean karta tha: underground, tumhare sar ke upar ki Earth zero net force se pull karti hai. Yeh Shell Theorem hai, aur yeh apni picture deserve karta hai.

Figure mein tum lal dot par off-centre khade ho. Near patch (chhota arrow, chhota patch) aur far patch (lamba arrow — lekin arrows equal length hain kyunki bada far patch apni distance ko exactly compensate karta hai) opposite directions mein pull karte hain aur cancel ho jaate hain.

Yeh topic isko kyun use karta hai: jab tum depth $d$ par khode jaate ho, toh Earth ko (a) radius $R - d$ ki inner ball tumhare neeche aur (b) tumhare upar ke saare shells mein divide karo. Upar ke shells exactly "tum ek shell ke andar ho" waala case hain — woh zero contribute karte hain. Isliye sirf inner ball pull karti hai. Yeh ek akela fact hai jo depth formula (item 7) ko possible banata hai.

7. Density $ρ$ , mass = density × volume, aur depth formula

Ab hum items 5, 6 aur is item ko milakaar parent note ka depth result build kar sakte hain. Maano Earth ka uniform density $ρ$ hai (har jagah same).

Yeh topic isko kyun use karta hai: density ke bina tum inner-ball mass nahi likh sakte, aur Shell Theorem ke bina tum outer shells nahi hata sakte — dono milke yeh exact, approximation-free depth law dete hain. Ise altitude ke $1 - 2 h / R$ se compare karo: depth linear hai (2 ka factor nahi) exactly isliye kyunki ek uniform sphere ke andar $g \propto R$ hai.

8. Angular speed $ω$ aur woh circle jo ek spinning point trace karta hai

Figure mein vertical line spin axis hai. ==Latitude $λ$ == par ek point (equator se upar ka angle) ek aise circle par baitta hai jiska lal radius $r = R cos λ$ hai. Equator par ( $λ = 0$ ) woh radius poora $R$ hai; pole par ( $λ = 9 0^{\circ}$ ) woh zero tak shrink ho jaata hai — ek pole point bas wahi spin karta rehta hai.

Yeh topic isko kyun use karta hai: latitude section $g$ se ek term $ω^{2} R cos^{2} λ$ subtract karta hai. Tum us term ko tab tak nahi samajh sakte jab tak tum nahi dekhte ki (a) $ω$ spin rate hai, (b) spin-circle radius $R cos λ$ hai, ek cosine deta hai, aur (c) us horizontal effect ko "down" par project karne se doosra cosine aata hai — isliye $cos^{2} λ$ . Yeh Centripetal Force & Circular Motion se connect hota hai.

9. Centripetal acceleration $ω^{2} r$ — kyun spinning gravity "churaati" hai

Ab woh geometry jo doosra cosine produce karti hai. Centripetal acceleration spin axis ki taraf point karta hai (horizontally, chhote circle ke radius ke saath), lekin $g$ Earth ke centre ki taraf point karta hai (local "down" ke saath). Latitude $λ$ par yeh dono directions same nahi hain — unke beech ka angle exactly $λ$ hai.

Figure mein, lal arrow $ω^{2} R cos λ$ horizontally axis ki taraf point karta hai. Yeh dekhne ke liye ki iska kitna hissa "down" (Earth ke centre ki line) ke saath hai, uski shadow us line par daalo — us projection mein $cos λ$ ka factor lagta hai. Toh vertical $g$ se jo amount subtract hota hai woh hai $(ω^{2} R cos λ) \times cos λ = ω^{2} R cos^{2} λ .$ Pehla $cos λ$ circle ke radius $R cos λ$ (item 8) se aaya; doosra $cos λ$ is vertical par projection se aaya. Do alag cosines, do alag reasons — yeh $cos^{2} λ$ ke peeche ki poori kahaani hai.

10. Binomial shortcut $(1 + x)^{n} \approx 1 + n x$

Yeh topic isko kyun use karta hai: exact altitude ratio hai $(1 + \frac{h}{R})^{- 2}$ . Kyunki mountain ki height $h$ Earth ke radius $R$ ke saamne tiny hai, hum $x = h / R$ set karte hain aur clean $g_{h} \approx g (1 - 2 h / R)$ milta hai. Woh surviving 2 (exponent $- 2$ se) yehi reason hai ki altitude $g$ ko depth ki tulna mein do guna tezi se kamzor karta hai.

11. Ratios — har jagah use ki jaane waali trick

Topic ka example: $\frac{g _{h}}{g} = \frac{R ^{2}}{( R + h ) ^{2}}$ — $GM$ gayab ho gaya, aur sirf shapes bacha.

Prerequisite map

(Alt text / reading order: Mass → Force/weight → g as a ratio. Alag se, inverse-square rule g ke saath milkar Newton's law aur surface gravity deta hai. Height h aur depth d altitude aur depth cases ko feed karte hain; Shell Theorem plus density depth case build karte hain; angular speed ω aur centripetal acceleration latitude case build karte hain; binomial shortcut altitude case build karta hai. Teeno cases topic "Variation of g" ko feed karte hain.)

Equipment checklist

Daayein side cover karo aur dekho ki kya tum har ek ko memory se bata sakte ho.

Symbol

m

ka kya matlab hai, aur kya yeh location ke saath badlta hai?

Stuff ki miqdar (kg); yeh location ke saath kabhi nahi badlta.

Weight kya hai, aur kya yeh badal sakta hai?

Earth ki gravity ka kisi object par pull (ek force); haan, yeh jagah ke saath badal sakta hai.

g

ki definition ek ratio ke roop mein do.

g = F / m

— gravitational pull per kilogram, N/kg = m/s² mein.

Distance law

1/ r^{2}

kyun hai aur

1/ r

kyun nahi?

Gravity area

4 π r^{2}

waale sphere par phailti hai, jo

r

ke square ke saath badhta hai.

Do masses ke liye Newton's law of gravitation likho.

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

G

M

R

kya hain, aur

G

ke units kya hain?

G

= universal gravitational constant, units

m^{3} kg^{- 1} s^{- 2}

;

M

= Earth ka mass;

R

= Earth ka radius.

h

aur

d

define karo, aur dono ke liye

r

do.

h

= surface se upar height,

r = R + h

;

d

= surface se neeche depth, effective

r = R - d

Underground

r = R - d

GM / r^{2}

mein simply plug kyon nahi kar sakte?

Tumhare upar ka shell pull karna band kar deta hai (Shell Theorem), isliye mass bhi badal jaata hai —

M

ab sahi mass nahi raha.

Surface gravity ko

G, M, R

ke terms mein likho.

g = \frac{GM}{R ^{2}}

Shell Theorem ek line mein batao, aur kyun hold karta hai.

Ek uniform shell ke andar net pull zero hai; near-small aur far-large patches perfectly cancel ho jaate hain.

Density

ρ

kya hai aur ek sphere ka mass kya hota hai?

ρ = M / V

; sphere ke liye

M = ρ \cdot \frac{4}{3} π R^{3}

Uniform Earth ke liye depth

d

par

g

derive/state karo.

g_{d} = \frac{4}{3} π Gρ (R - d) = g (1 - \frac{d}{R})

; centre par zero.

ω

kya hai aur latitude

λ

par spin-circle radius kya hai?

ω

= turning rate (rad/s); circle radius

R cos λ

hai.

Centripetal acceleration kya hai aur spinning ki pull kahin jaati hai?

a_{c} = ω^{2} r

, inward directed; gravity ka kuch hissa use provide karne mein "kharcha" ho jaata hai, effective

g

kam kar deta hai.

Latitude term mein do cosines kyun hain?

Ek circle radius

R cos λ

se; ek horizontal centripetal vector ko local "down" par project karne se (unke beech angle

λ

hai).

(1 + x)^{- 2}

ke liye binomial shortcut batao jab

x

tiny ho.

(1 + x)^{- 2} \approx 1 - 2 x

Derivations mein

g_{h}

g

se kyun divide karte hain?

Yeh

GM

cancel karta hai, sirf ek clean geometry-only ratio chhod deta hai.

Connections

Newton's Law of Universal Gravitation — woh $1/ r^{2}$ law jis par yeh sab kuch rests karta hai.
Shell Theorem — kyun outer shells zero pull dete hain (depth ke liye zaroori).
Centripetal Force & Circular Motion — jahaan se $ω^{2} r$ term aata hai.
Weight vs Mass — mass-vs-weight split items 1–2 mein build ki gayi.
Oblate Spheroid Earth — woh shape effect jo latitude case mein rotation ke saath add hota hai.
Escape Velocity & Orbital Mechanics — same $GM / r^{2}$ machinery, Earth chhodne par apply ki gayi.
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