Variation of g — with altitude, latitude, depth

1.2.21 · Physics › Newton's Laws & Dynamics

0. Starting point — surface gravity

YE form KYU? Shell theorem ke hisaab se Earth ki saari mass aise act karti hai jaise center pe concentrated ho, isliye surface object jo distance $R$ pe hai, wo $GM / R^{2}$ feel karta hai. Baaki sab is cheez ke corrections hain.

1. ALTITUDE ke saath variation (height $h$ surface se upar)

" $g$ se divide karo" step KYU? Isse $GM$ cancel ho jaata hai, aur ek clean ratio milta hai jo sirf geometry $h / R$ pe depend karta hai — easy hai aur numerically $G$ ya $M$ ki zaroorat nahi padti.

2. DEPTH ke saath variation (depth $d$ surface ke neeche)

Depth ke liye ye EXACT kyun hai (approximate nahi) jabki altitude ke liye binomial chahiye tha: kyunki uniform sphere ke andar $g \propto R$ linear hai, isliye depth formula mein koi approximation nahi hai. Center pe ( $d = R$ ): $g = 0$ ho jaata hai.

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

3. LATITUDE ke saath variation (equator se angle $λ$ )

$cos λ$ nahi balki $cos^{2} λ$ KYU? Ek $cos λ$ circle ke radius $r = R cos λ$ se aata hai; doosra horizontal centripetal vector ko vertical (local "neeche") direction pe project karne se aata hai.

4. Forecast-then-Verify table

Recall Padhne se PEHLE change ki direction predict karo

Action	$g$ badhega ya ghattega?	Formula	Surface ke paas rate
$h$ upar jao	decrease	$g (1 - 2 h / R)$	$- 2/ R$ per metre
$d$ neeche jao	decrease	$g (1 - d / R)$	$- 1/ R$ per metre
Equator→pole move karo	increase	$+ ω^{2} R cos^{2} λ$ restored	up to $+ 0.034$
Key cross-check: altitude, depth se dugna tez girta hai. ✅

Flashcards

Surface gravity in terms of

G, M, R

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

Formula for

g

at altitude

h

(exact)

g_{h} = \frac{GM}{( R + h ) ^{2}} = g (1 + \frac{h}{R})^{- 2}

Approximate

g

at small altitude

g_{h} \approx g (1 - \frac{2 h}{R})

Formula for

g

at depth

d

(uniform Earth)

g_{d} = g (1 - \frac{d}{R})

g = 0

Earth ke center pe kyun hota hai?

Saari mass ab tumhare upar shells mein hai → shell theorem zero net force deta hai.

Altitude, depth ke comparison mein

g

ko dugna tez kyun reduce karta hai surface ke paas?

Altitude inverse-square use karta hai (

- 2 h / R

); depth linear

g \propto R

use karta hai (

- d / R

Effective

g

at latitude

λ

g_{λ} = g - ω^{2} R cos^{2} λ

Rotation ki wajah se

g

sabse bada/chhota kahan hota hai?

Poles pe sabse bada (

cos^{2} 90° = 0

), equator pe sabse chhota (

cos^{2} 0° = 1

Latitude formula mein do cosines kyun hain?

Ek circle radius

r = R cos λ

se, ek local vertical pe project karne se.

Uniform sphere ke andar,

g \propto

g \propto r

(center se distance).

Recall Feynman: 12-saal ke bache ko samjhao

Socho Earth ek giant ball hai jo tumhe apne middle ki taraf kheenchti hai. Agar tum ek unchi pahaad chadho, tum middle se thoda door ho, toh pull thoda kamzor hoga — tum thoda kam weight feel karoge. Agar tum ek gehri khai khodo aur usme khade ho, Earth ka jo hissa tumhare sar ke upar hai, wo tumhe thoda upar kheenchta hai aur khud ko cancel kar leta hai, toh sirf neeche wala chhota ball tumhe kheechta hai — phir se kamzor. Aur kyunki Earth merry-go-round ki tarah ghoomti hai, equator ke paas khade rehna (sabse mota, sabse tez ghumne wala hissa) tumhe thoda bahar fenkta hai, jisse tum poles ke comparison mein halka feel karte ho. Toh tum poles pe sabse bhaaari hote ho, gehri kaan mein halke ho, aur pahaad pe bhi halke ho!

Connections

Newton's Law of Universal Gravitation — parent $1/ r^{2}$ law.
Shell Theorem — justify karta hai ki outer shells zero contribute kyun karte hain (depth case).
Centripetal Force & Circular Motion — rotation (latitude) term ka source.
Weight vs Mass — weight $= m g$ location ke saath badalta hai, mass nahi.
Oblate Spheroid Earth — latitude variation ka geometric part.
Escape Velocity & Orbital Mechanics — usi $GM / r^{2}$ field ko use karta hai.