Visual walkthrough — Variation of g — with altitude, latitude, depth

1.2.21 · D2 · Physics › Newton's Laws & Dynamics › Variation of g — with altitude, latitude, depth

---

Step 1 — "$g$" ka matlab kya hai: ek ball doosri ball ko kheenchti hai

KYA. Earth ko ek bada circle draw karo aur uski surface pe ek chhota dot rakho (tum, mass $m$). Ek arrow hai jo us dot ko seedha centre ki taraf kheench raha hai. Us arrow ki lambai, per kilogram, wahi hai jise hum $g$ kehte hain.

KYUN yahan se shuru karein. Yeh kehne se pehle ki $g$ kaise badalta hai, humein yeh pakka karna hoga ki $g$ hai kya: yeh kisi point pe pull ki taakat hai, aur Newton's gravitation law ke hisaab se yeh taakat distance ke saath ghatti hai. Is page pe sab kuch ek hi idea hai — distance ya effective pulling mass badlo, aur arrow ki lambai badal jaati hai.

PICTURE.

Yahan $R$ Earth ki radius hai (≈ $6400$ km), kyunki shell theorem humein allow karta hai ki Earth ki saari mass ko ek point pe squeeze mana jaaye centre mein — toh surface ka dot us point se $R$ distance pe baitha hai.

---

Step 2 — UPAR jaana: neeche wala distance badh jaata hai

KYA. Dot ko surface se $h$ height upar uthao. Centre se uski distance ab $R$ nahi rahi — woh $R+h$ ho gayi. Yeh nayi, lambi distance usi law mein daalo.

KYUN. Earth mein kuch nahi badla; sirf teri distance badi. Kyunki distance denominator mein hai (squared), badi distance chhhoti pull banati hai. Altitude ke liye sirf yahi mechanism kaam kar raha hai.

PICTURE.

Numbers plug in karne ki jagah divide kyun karein? Kyunki hume $g$ ki value nahi chahiye, sirf woh fraction chahiye jo woh ban jaata hai. Ratio $h/R$ poori kahani batata hai.

---

Step 3 — Small-height shortcut (aur woh factor 2 kahan se aata hai)

KYA. Kisi pahad ya balloon ke liye, $h$ $R$ ke muqaable mein bahut chhota hai ($32$ km vs $6400$ km). Jab ek chhota number $x=h/R$ ko $(1+x{)}^{-2}$ pe raise karo, toh yeh almost exactly $1-2x$ jaisa behave karta hai. Woh $-2$ guess nahi hai — yeh inverse-square curve ki slope hai $x=0$ ke paas.

KYUN yeh tool (binomial approximation). Hum $(1+x{)}^{-2}\approx 1-2x$ use karte hain kyunki $(1.005{)}^{-2}$ haath se calculate karna ugly hai, lekin straight-line approximation instant hai aur, itne chhote $x$ ke liye, hundredth of a percent se bhi kam galat hai. Yeh tool is sawaal ka jawaab deta hai "jis moment main zameen chhod raha hoon, $g$ kitni tezi se girna shuru karta hai?" — aur jawaab hai do guna utni tezi jitna height fraction.

PICTURE.

---

Step 4 — NEECHE jaana: outer shell kheenchna band kar deta hai

KYA. Ab dot ko surface se $d$ depth neeche le jaao. Earth ko do pieces mein split karo: inner ball jiska radius $R-d$ hai jiske upar tum baithe ho, aur outer shell jo ab tumhare sir ke upar hai. Outer shell ko shaded draw karo.

KYUN. Shell theorem ke hisaab se, ek uniform spherical shell apne andar ki kisi bhi cheez ko kuch nahi kheenchta — har direction ke tugs bilkul exactly cancel ho jaate hain. Toh tumhare upar wala shell zero contribute karta hai. Sirf chhota inner ball tumhe kheenchta hai. Kam pulling mass → kamzor $g$. Yeh altitude se bilkul alag mechanism hai (wahan mass wahi rahi, distance badi; yahan distance ghati lekin mass bhi ghati).

PICTURE.

---

Step 5 — Depth formula ek straight line hai (koi approximation nahi)

KYA. Depth $d$ pe tum $R-d$ radius ke ball ke upar baithe ho. Usi $g\propto (\text{radius})$ rule ko reuse karo, lekin $R$ ki jagah radius $R-d$ ke saath.

KYUN yeh exact hai. Kyunki $g$ vs radius genuinely ek straight line hai (Step 4), approximate karne ke liye koi curve nahi hai — jawaab honest hai, koi $\ll$ nahi chahiye. Isko altitude se contrast karo, jahan inverse-square curve ne humein approximate karne par majboor kiya.

PICTURE.

---

Step 6 — Ek graph pe Upar vs Neeche

KYA. $g$ ko "surface se kitni door ho" ke against plot karo, altitude ke liye right aur depth ke liye left. Do straight lines surface pe milti hain; altitude line do guni steep hai.

KYUN. Dono slopes ko side by side dekhna factor-2 ko unforgettable bana deta hai aur ranges dikhata hai: altitude hamesha zero ki taraf curve karta rehta hai (kabhi pura reach nahi karta), depth centre pe exactly zero reach karta hai.

PICTURE.

---

Step 7 — Sideways: LATITUDE aur spinning merry-go-round

KYA. Earth ghoomti hai. Latitude $\lambda$ (equator se upar ka angle) pe ek point ek circle mein carry hota hai. Us circle ki radius $R$ nahi hai — woh spin axis se distance hai, jo ki $r=R\cos \lambda$ hai. Axis, point, aur woh chhota horizontal circle draw karo jis pe woh ride karta hai.

KYUN circular motion enter karta hai. Jo bhi circle mein jaata hai usse curve karte rehne ke liye ek real inward force chahiye — centripetal force. Spinning Earth pe, gravity ke pull ka kuch hissa us inward force ko supply karne mein "kharcha" ho jaata hai, toh bacha hua pull jo tum actually feel karte ho (tumhara apparent weight, dekho Weight vs Mass) chhhota hota hai.

PICTURE.

---

Step 8 — Har latitude case, extremes samait

KYA. Angle ko equator se pole tak walk karo aur dono ends check karo, plus "agar Earth tezi se ghoomti" limit.

KYUN. Contract yeh hai: koi reader kisi untested case se na mile. Hum $\lambda =0$, $\lambda =90^{\circ }$, aur runaway limit cover karte hain.

PICTURE.

• Equator, $\lambda =0$: ${\cos }^{2}0=1$, toh $g_{\text{eq}}=g-{\omega }^{2}R$ — sabse bada subtraction, $g$ sabse chhota. Equatorial bulge ke saath combine karo (tum wahan centre se bhi zyada door ho) aur $g$ yahan sab jagahon se sabse chhota hai.
• Pole, $\lambda =90^{\circ }$: ${\cos }^{2}90^{\circ }=0$, subtraction gayab ho jaata hai, $g_{\text{pole}}=g$ — sabse bada.
• Tezi se ghoomne ki limit: agar $\omega$ tab tak badhta jab tak equator pe ${\omega }^{2}R=g$ na ho jaaye, $g_{\lambda }\to 0$ aur loose objects uthne lagte (orbital motion ke liye relevant — woh literally ground level pe orbit condition hai). ${\omega }^{2}R=g$ solve karne pe lagbhag $1.4$ ghante ka din milta hai; spin karke $g$ ko negative nahi bana sakte.

---

Ek-picture summary

Recall Feynman retelling — plain words mein walkthrough

Earth ek giant ball hai jo sab kuch apne middle ki taraf kheenchti hai, aur kitni zor se kheenchti hai woh $g$ hai. Pahad pe chadho aur tum middle se zyada door ho; kyunki pull distance ke square ke saath kamzor hoti hai, $g$ tezi se girta hai — har kadam upar ke liye, height fraction se do guna tezi se. Khodo neeche aur kuch sneaky hota hai: ab tumhare sir ke upar wala poora Earth tumhe har direction se kheenchta hai aur khud cancel ho jaata hai, toh sirf chhota ball tumhare neeche tumhe kheenchta hai — $g$ girta hai, lekin sirf aadhi speed pe chadne ke muqaable mein, aur centre pahunchne pe exactly zero ho jaata hai. Aakhir mein, Earth merry-go-round ki tarah ghoomti hai. Equator ke paas tum sabse bade circle mein ghoom rahe ho, aur gravity ko apni kuch taakat sirf tumhe us curve pe rakhne mein lagaani padti hai, toh tum wahan halka feel karte ho; poles pe tum circle mein almost hilte hi nahi, toh gravity ki poori pull feel hoti hai. Poles pe sabse bhaari, mine mein neeche halka, pahad pe upar halka — teen alag wajahaat, ek saada picture.

Recall Quick self-test

Altitude ke liye $h/R$ ke aage factor ::: $2$ (inverse-square law se) Depth ke liye $d/R$ ke aage factor ::: $1$ ($g\propto R$ se, exact aur linear) Latitude $\lambda$ pe do cosines kyun ::: ek circle radius $R\cos \lambda$ se, ek local vertical pe project karne se Earth ke centre pe $g$ ki value ::: $0$ — sab kuch tumhare upar shell hai Spin ka subtraction sabse bada kahan hai ::: equator pe, jahan ${\cos }^2\lambda =1$

Connections

• Newton's Law of Universal Gravitation — Step 1 ka $1/r^2$ law.
• Shell Theorem — depth derivation mein outer shell ko khatam karta hai (Steps 4–5).
• Centripetal Force & Circular Motion — latitude term ke peeche inward force (Step 7).
• Weight vs Mass — "apparent" $g$ kya measure karta hai.
• Oblate Spheroid Earth — shape effect jo equator ke chhote $g$ mein add hota hai.
• Escape Velocity & Orbital Mechanics — spin-faster limit ground-level orbit condition hai.