Visual walkthrough — Inertial navigation — accelerometer measures non-gravitational specific force

3.5.13 · D2 · Physics › Guidance, Navigation & Control (GNC) › Inertial navigation — accelerometer measures non-gravitation

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Step 1 — Ek hi gadget: box ke andar spring par mass

KYA HAI. Ek accelerometer ek choti si ball (proof mass) hai jo ek sealed box ke andar spring se bandhi hoti hai. Yeh box jis cheez ko hum track karna chahte hain — car, rocket, elevator — uspe bolt hoti hai. Sensor sirf ek hi cheez report karta hai: spring ball ko kitne zor se push ya pull kar rahi hai.

YE KYUN shuru karte hain. Har wo symbol jo hum milenge, woh sirf isi ball par act karne wala ek arrow hai. Agar hum arrows sahi se draw kar lein, toh algebra khud ban jaata hai. Kuch bhi yaad karne ki zarurat nahi.

PICTURE. Ball beech mein tiki hui hai. Do cheezein usp act kar sakti hain, aur hum unhe agli steps mein ek-ek karke dekhenge.

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Step 2 — Arrow #1: gravity sab cheez ko neeche kheenchti hai

KYA HAI. Gravity ball ko seedha neeche kheenchti hai. Is pull ko hum $m\mathbf{g}$ likhte hain.

Har symbol ko samajhte hain:

• $\mathbf{g}$ (bold letter) gravity arrow hai: yeh neeche point karta hai aur iska length $g\approx 9.81$ hai (${\text{m/s}}^2$ mein). Bold ka matlab hai "yeh ek arrow hai — iska ek direction hai," jabki plain $g$ sirf uski length hai.
• $m\mathbf{g}$ ka matlab hai "gravity arrow ko mass $m$ se stretch karo." Zyada bhaari ball zyada kheenchi jaati hai — $m$ se multiply karne ka poora matlab yahi hai.

ALAG kyun naam dete hain. Kyunki gravity ki ek secret property hai jise hum baad mein use karenge: yeh ball ke har atom ko ek saath kheenchti hai, andar se. Gravity ko act karne ke liye ball ko kuch touch nahi karna padta. Yeh baat dhyan mein rakho.

PICTURE. Ek magenta arrow, seedha neeche, label $m\mathbf{g}$.

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Step 3 — Arrow #2: spring, jo sirf ek cheez hum padh sakte hain

KYA HAI. Agar ball centre se drift karne ki koshish kare, toh spring use wapas push karti hai. Is push ko hum contact force ${\mathbf{F}}_{\text{c}}$ kehte hain ("c" for contact — koi cheez physically ball ko touch karke push karti hai).

• ${\mathbf{F}}_{\text{c}}$ (bold, toh yeh ek arrow hai) woh kuch bhi hai jo spring ko ball ko box mein centred rakhne ke liye supply karna padta hai.
• Gravity ke unlike, yeh force touch karke act karti hai — spring physically ball se connected hai. Bas yahi fark sensor ka poora secret hai.

YE STAR kyun hai. Sensor gravity arrow ko dekh nahi sakta. Woh sirf spring ki push ${\mathbf{F}}_{\text{c}}$ measure kar sakta hai. Toh jo bhi equation hum banayenge, readable quantity ${\mathbf{F}}_{\text{c}}$ hi honi chahiye.

PICTURE. Violet spring-arrow ${\mathbf{F}}_{\text{c}}$ magenta gravity arrow ke saath. Do arrows, do alag origins: ek andar se (gravity), ek touch se (spring).

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Step 4 — Newton's law se arrows ko add karo

KYA HAI. Newton's Second Law kehta hai: ball par total force uske mass aur true acceleration ke product ke barabar hoti hai. "True acceleration" $\mathbf{a}$ woh hai ki ball ki velocity sach mein kitni change ho rahi hai, jaise bahar space mein floating koi observer dekhe (ek inertial observer). Hum apne do arrows add karte hain:

$\underset{\begin{array}{c}\text{mass}\times \text{true}\\ \text{acceleration}\end{array}}{\underbrace{m\mathbf{a}}}=\underset{\begin{array}{c}\text{spring}\\ \text{push}\end{array}}{\underbrace{{\mathbf{F}}_{\text{c}}}}+\underset{\begin{array}{c}\text{gravity}\\ \text{pull}\end{array}}{\underbrace{m\mathbf{g}}}$

Term by term:

• $m\mathbf{a}$ — net result: mass times motion mein real change.
• ${\mathbf{F}}_{\text{c}}$ — readable spring push (Step 3).
• $m\mathbf{g}$ — invisible gravity pull (Step 2).

ADD kyun karte hain, subtract nahi? Kyunki forces arrows hain jo head-to-tail place hote hain: net arrow $m\mathbf{a}$ simply do input arrows ka sum hai. Newton's law bas yahi hai: "arrows add hokar net bante hain."

PICTURE. Do arrows head-to-tail draw kiye; unka sum green net arrow $m\mathbf{a}$ hai.

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Step 5 — Spring kya read karti hai solve karo

KYA HAI. Hum spring ka arrow akela chahte hain, kyunki sensor sirf wahi jaanta hai. $m\mathbf{g}$ ko doosri side le jaao, phir $m$ se divide karo:

$m\mathbf{a}-m\mathbf{g}={\mathbf{F}}_{\text{c}}⟹\frac{{\mathbf{F}}_{\text{c}}}{m}=\mathbf{a}-\mathbf{g}$

Left side ko specific force $\mathbf{f}$ naam do (force per unit mass):

$\boxed{\mathbf{f}=\mathbf{a}-\mathbf{g}}$

Term by term:

• $\mathbf{f}$ — sensor reading (spring push per kilogram).
• $\mathbf{a}$ — true acceleration jo hume sach mein navigation ke liye chahiye.
• $-\mathbf{g}$ — gravity arrow, subtract hua. Yeh reading se tab gayab ho jaata hai kyunki usne kabhi spring ko touch nahi kiya.

YE POORA POINT kyun hai. Sensor hume $\mathbf{f}$ deta hai. True acceleration paane ke liye hume subtraction undo karna hoga: $\mathbf{a}=\mathbf{f}+\mathbf{g}$. Kisi ko (navigation computer ko) gravity yaad rakhni hogi aur haath se add karni hogi.

PICTURE. Rearranged arrows: $\mathbf{f}$ ko $\mathbf{a}$ se gravity arrow hatakar dikhaya gaya.

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Step 6 — Rest par case: yeh $+g$ upar ki taraf kyun read karta hai

KYA HAI. Box table par rakha hai. Ball hil nahi rahi, toh true acceleration zero hai: $\mathbf{a}=0$. $\mathbf{g}=(0,0,-g)$ ke saath,

$\mathbf{f}=\mathbf{a}-\mathbf{g}=0-(0,0,-g)=(0,0,+g)$

ZERO kyun nahi? Bhale hi kuch move nahi ho raha, spring ko phir bhi upar push karna padta hai taaki ball box ke floor par na dhayi ho. Woh upward push ek real reading hai: $+g$ sky ki taraf point karta hua.

PICTURE. Ball rest par; spring upar $g$ length se push karti hai, gravity neeche $g$ length se kheenchti hai, dono cancel hote hain toh ball still hai — lekin spring reading woh up-arrow hai.

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Step 7 — Free fall case: yeh exactly zero kyun read karta hai

KYA HAI. Poora box drop karo. Ab sirf gravity hai, toh ball exactly $\mathbf{g}$ se accelerate karti hai: yaani $\mathbf{a}=\mathbf{g}$. Tab

$\mathbf{f}=\mathbf{a}-\mathbf{g}=\mathbf{g}-\mathbf{g}=0$

ZERO kyun. Ball, spring, aur box sab ek saath same rate se girte hain. Ball kabhi centre se drift nahi karti, isliye spring ko kabhi push nahi karna padta. Koi push nahi ⇒ zero reading. Yahi weightlessness hai, aur yeh sensor form mein Equivalence Principle hai: free-fall aur deep space mein float karna indistinguishable hai.

PICTURE. Sab ek saath gir rahe hain; spring relaxed hai, ball centred hai, reading $=0$.

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Step 8 — Accelerating case: upar jaata elevator

KYA HAI. Elevator (z-axis upar) $2{\text{m/s}}^2$ se upar accelerate karta hai, toh $\mathbf{a}=(0,0,+2)$. Gravity unchanged hai, $\mathbf{g}=(0,0,-9.81)$:

$f_z=a_z-g_z=2-(-9.81)=11.81{\text{m/s}}^2$

Term by term:

• $a_z=+2$ — extra upward acceleration jo elevator ball par force karta hai.
• $-g_z=+9.81$ — woh support jo spring rest par pehle se supply kar rahi thi (Step 6).
• Sum $=11.81$ — spring zyada kaam karti hai: tum "heavier feel" karte ho.

YE ADD kyun hota hai. Spring ko ab do kaam karne hain: ball ko gravity ke against upar rokna ($9.81$) aur use elevator ke acceleration ke saath upar dhakela ($2$). Do kaam ⇒ do contributions add hue.

PICTURE. Ek lamba spring-arrow ($11.81$) rest wale arrow ($9.81$) ke comparison mein; difference bar "felt extra heaviness" hai.

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One-picture summary

Ek strip mein teeno cases: rest par reading $+g$ hai, upar accelerate karte waqt $g+a$ hai, free fall mein $0$ hai. Har case ke neeche wahi equation $\mathbf{f}=\mathbf{a}-\mathbf{g}$ number produce karti hai — gravity arrow hamesha same hai; sirf $\mathbf{a}$ badalta hai.

Recall Poore walkthrough ki Feynman-style retelling

Ek shoebox ke andar spring se tika ball imagine karo. Gravity ball ko andar se kheenchti hai — ball ke har bit ko ek saath, toh spring directly usse feel nahi karti. Spring tabhi jaagti hai jab ball centre se drift karne ki koshish kare, aur tab woh wapas push karti hai; woh push hi akeli cheez hai jo box report kar sakta hai.

Box ko table par rakho: gravity ball ko neeche le jaayegi, toh spring upar squeeze karti hai usse rokne ke liye. Box kehta hai "$+g$, upward!" — bhale hi kuch move nahi ho raha. Elevator mein upar jaao: ab spring ko ball ko upar rokna bhi hai aur saath dhakela bhi — toh woh zyada push karti hai — tum heavier feel karte ho, aur box $g$ plus extra read karta hai. Poora box drop karo: ball, spring, aur box sab ek saath girte hain, ball bilkul centre mein rehti hai, spring slack ho jaati hai — box kehta hai "$0$, mujhe kuch feel nahi ho raha," jabki woh floor ki taraf screaming hai. Isliye astronauts float karte hain.

Ek lesson: box kabhi bhi gravity feel nahi kar sakta. Toh true acceleration paane ke liye, computer ko hamesha gravity yaad rakhni hogi aur add karni hogi: $\mathbf{a}=\mathbf{f}+\mathbf{g}$. Bhool jaao, aur har second pretend-position $\frac{1}{2}g{t}^2$ ki tarah bhaag jaati hai — dekho Dead Reckoning and Error Drift.

Recall

Ek still accelerometer (z upar) kya read karta hai? ::: $+g\approx 9.81{\text{m/s}}^2$ upward. Ek free-falling accelerometer kya read karta hai? ::: $0$ (weightlessness). Elevator $2{\text{m/s}}^2$ upar accelerate kare toh z par kya read karega? ::: $11.81{\text{m/s}}^2$. Computer ko $\mathbf{f}$ ke saath integrate karne se pehle kya karna chahiye? ::: $\mathbf{g}(\mathbf{r})$ add karo: $\mathbf{a}=\mathbf{f}+\mathbf{g}$.

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Connections

• Newton's Second Law — Step 4 ka starting arrow-sum.
• Equivalence Principle — Step 7, free-fall zero read karta hai.
• Strapdown Inertial Navigation System — jahan $\mathbf{a}=\mathbf{f}+\mathbf{g}$ integrate hota hai.
• Gravity Model (WGS-84 / J2) — woh $\mathbf{g}$ supply karta hai jo hum wapas add karte hain.
• Dead Reckoning and Error Drift — kya hota hai agar tum bhool jaao.
• Gyroscope and Attitude Determination — $\mathbf{f}$ ko sahi frame mein rotate karne ke liye orientation provide karta hai.