Foundations — Inertial navigation — accelerometer measures non-gravitational specific force
3.5.13 · D1· Physics › Guidance, Navigation & Control (GNC) › Inertial navigation — accelerometer measures non-gravitation
Parent note padhne se pehle, aapko har woh symbol apna banana hoga jo woh aap par fenkta hai. Hum unhe ek-ek karke build karenge, kuch bhi nahi se, har ek apni jagah earn karta hai agle ke aane se pehle.
1. Direction wali quantity: the vector (bold letters)
Plain number (ek scalar) sirf "paanch" keh sakta hai. Lekin "paanch, upar ki taraf" ke liye ek arrow chahiye. Navigation mein har cheez ki ek direction hoti hai — forces ek taraf kheenchti hain, motion doosri taraf hoti hai — isliye hum plain numbers use nahi kar sakte.
Components. Arithmetic karne ke liye hum ek vector ko numbers ki list ke roop mein likhte hain, ek number per direction (axis): Yahan hai "arrow ka kitna hissa x-axis ke saath point karta hai," aur aage bhi aise hi. Chota subscript letter sirf kaun si axis ka naam batata hai. Isliye parent note jaisi cheezein likhta hai — matlab hai "sideways zero, aage zero, aur neeche z-axis ki taraf point karta hai."
2. "Neeche" kaun si taraf hai? The coordinate axes
Figure dekho: green z-arrow upar point kar raha hai. Gravity (coral arrow) ulti taraf point karti hai, isliye uska component hai, jo deta hai. Sign decoration nahi hai — yahi woh poori wajah hai ki baad mein ek still accelerometer upar ki taraf kyun read karta hai. Jab bhi koi sign aaye, picture ko yaad rakho.
3. Mass aur force
Hum forces ko do families mein separate karte hain, aur yeh split is topic ka dil hai:
| Family | Symbol | Picture | Spring ko touch karta hai? |
|---|---|---|---|
| Contact force | ek spring, ek haath, ek seat jo tumhare upar press kar rahi ho | Haan | |
| Body force (gravity) | Earth har atom ko ek saath kheenchti hai | Nahi |
4. Acceleration vs specific force
"Per unit mass" kyun ( se divide karna)? Kyunki ek bhaari proof mass ko same motion ke liye zyada bada spring push chahiye. se divide karne par mass cancel ho jaata hai, isliye reading sirf situation par depend karti hai, na ki hum ne sensor kitna bada banaya. Isliye ke units hain — force () divided by mass ().
5. Gravity
Kyunki humne mass divide kar diya, ke units aur jaisi hi hain — yahi woh cheez hai jo humein mein unhe cleanly subtract karne deti hai.
6. The inertial frame — jahan Newton's law allow hai
Topic yeh kyun insist karta hai: parent note Newton's law ko "inertial frame mein" likhta hai taaki sach mein true acceleration ho, observer ke khud accelerate hone se bana koi fake wala nahi. Frame galat karo aur extra phantom forces appear ho jaati hain.
7. Newton's second law — woh ek equation jisse sab kuch aata hai
Har term ko se divide karo aur rearrange karo — yahi woh derivation hai jo parent note perform karta hai:
\qquad\Longleftrightarrow\qquad \mathbf{f} = \mathbf{a} - \mathbf{g}$$ Poori statement ke liye [[Newton's Second Law]] dekho; yahan humein sirf yeh ek form chahiye. ![[deepdives/dd-physics-3.5.13-d1-s03.png]] Figure proof mass ko sirf do arrows ke saath dikhata hai: spring ka contact push (lavender) aur gravity ki pull (coral). Newton kehta hai yeh dono, add hone par, $m\mathbf{a}$ ke equal hain. Spring ki push ke liye solve karo aur aapke paas exactly woh hai jo sensor read karta hai. --- ## 8. The **integral** $\int dt$ — acceleration ko position mein badalna > [!definition] Integration $\int \,dt$ > Symbol $\int$ (ek stretched "S," "sum" ke liye) $dt$ ke saath matlab hai "**time ke tiny slices par add up karo**." Acceleration ko integrate karne se velocity milti hai; velocity ko integrate karne se position milti hai. Hazaaron chote "$\text{speed} \times \text{tiny time}$" rectangles ko stack karne ki socho total distance travelled paane ke liye. $$\mathbf{a} \;\xrightarrow{\int dt}\; \mathbf{v} \;\xrightarrow{\int dt}\; \mathbf{r}$$ **Topic ko yeh kyun chahiye:** sensor acceleration-jaisa data deta hai, lekin ek navigator ko **position** $\mathbf{r}$ chahiye. Doh baar integration woh bridge hai. Isliye $\mathbf{a}$ mein early error ek position error mein balloon ho jaata hai jo $\tfrac12 g t^2$ ki tarah badhta hai — [[Dead Reckoning and Error Drift]] dekho. --- ## 9. Symbols ko order mein rakhna ```mermaid graph TD V["Vector = arrow with size and direction"] --> C["Components a_x a_y a_z"] C --> AX["Axes with z pointing up"] AX --> G["Gravity g points down so minus z"] M["Mass m"] --> F["Force F push or pull"] F --> CF["Contact force spring touches"] F --> BF["Body force gravity no touch"] CF --> SF["Specific force f = Fc over m"] BF --> G V --> ACC["Acceleration a rate of velocity change"] IF["Inertial frame fair observer"] --> N2["Newtons second law m a = sum F"] ACC --> N2 SF --> N2 G --> N2 N2 --> EQ["f = a minus g"] EQ --> INT["Integrate twice to get position"] INT --> TOP["Inertial navigation"] ``` Map ko top-down padho: arrows aur axes humein gravity ka sign dete hain; force split humein specific force deta hai; Newton's law ek inertial frame mein unhe $\mathbf{f}=\mathbf{a}-\mathbf{g}$ mein weld karta hai; integration ise position mein badal deta hai. Woh final chain **hi** parent topic hai. --- > [!recall]- Signs par quick self-check > $\mathbf{g}=(0,0,-g)$ mein minus sign kyun hai? > ::: Kyunki humne z ko **upar** pointing choose kiya, aur gravity **neeche** point karti hai — opposite direction — isliye uska z-component negative hai. > [!mnemonic] > **Bold = arrow, plain = number.** Agar koi letter bold hai ($\mathbf{f},\mathbf{a},\mathbf{g}$) toh uski ek direction hai; agar plain hai ($m$, $g$, $t$) toh yeh sirf ek size hai. --- ## Equipment checklist Har sawaal padho, out loud jawab do, phir reveal karo. Agar koi bhi stumps kare, parent note se pehle woh section dobara padho. Ek **bold** letter jaise $\mathbf{a}$ ka kya matlab hai, aur yeh kaisa dikhta hai? ::: Ek vector — size AUR direction wali quantity, arrow ki tarah draw ki gayi. $(a_x,a_y,a_z)$ mein subscripts kya batate hain? ::: Vector ka kitna hissa har axis (x, y, z) ke saath point karta hai. Z upar pointing ke saath, gravity ko components mein kaise likhte hain? ::: $\mathbf{g}=(0,0,-g)$ — minus isliye kyunki neeche $-z$ direction hai. Mass $m$ aur force $\mathbf{F}$ mein kya fark hai? ::: Mass kitna stuff hai (ek plain number, kg); force ek push/pull hai (ek vector, newtons). Spring kaun si forces feel kar sakti hai — contact ya body forces? ::: Sirf contact forces; gravity jaisi body forces har atom ko equally kheenchti hain aur spring ko kabhi touch nahi karti. Specific force $\mathbf{f}$ ko ek line mein define karo. ::: Unit mass per contact force, $\mathbf{f}=\mathbf{F}_{\text{c}}/m$ — jo accelerometer actually read karta hai. $\mathbf{f}$, $\mathbf{a}$, aur $\mathbf{g}$ sab units $\text{m/s}^2$ kyun share karte hain? ::: Kyunki har ek **unit mass per** force (ya acceleration) hai — $m$ se divide karne par $\text{m/s}^2$ milta hai, jo humein unhe subtract karne deta hai. Inertial frame kya hai aur humein iska kyun zaroorat hai? ::: Ek non-accelerating, non-spinning observer; Newton's law $m\mathbf{a}=\sum\mathbf{F}$ sirf wahan exactly sahi hai. Yahan do forces ke saath Newton's second law state karo. ::: $m\mathbf{a}=\mathbf{F}_{\text{c}}+m\mathbf{g}$. Acceleration ko doh baar integrate karne se kya milta hai? ::: Pehle velocity $\mathbf{v}$, phir position $\mathbf{r}$. --- ## Connections - [[Newton's Second Law]] — woh single equation jiske liye yeh poora page taiyaar karta hai. - [[Equivalence Principle]] — gravity ka "body force" hona sensor ke liye use invisible kyun banata hai. - [[Strapdown Inertial Navigation System]] — jahan integration chain rehti hai. - [[Dead Reckoning and Error Drift]] — jab integrate ki gayi $\mathbf{a}$ thodi si galat ho toh kya hota hai. - [[3.5.13 Inertial navigation — accelerometer measures non-gravitational specific force (Hinglish)|Parent topic →]]