Foundations — Kepler's second law — equal areas in equal times, from angular momentum conservation

3.2.6 · D1 · Physics › Orbital Mechanics & Astrodynamics › Kepler's second law — equal areas in equal times, from angul

Pehle tumhe feel karna hoga ki "equal areas in equal times" kyun sach hai, usse pehle tumhe har woh symbol apna banana hoga jo parent note mein use hota hai. Yeh page har ek cheez ko zero se build karta hai — simple words, ek picture, aur woh reason kyun topic ko uski zaroorat hai. Upar se neeche padho; har block sirf wohi use karta hai jo pehle aa chuka hai.

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1 — Ek vector, aur arrow ka matlab kya hota hai

Arrow ke upar chhota hat, $\stackrel{⃗}{}$, hamaara shorthand hai "yeh ek arrow hai, sirf ek number nahi." Ek plain letter jaise $r$ (bina arrow ke) sirf us arrow ki length ko batata hai — ek single positive number, Sun se planet ki distance.

Topic ko iska kyun zaroorat hai: Kepler's law us line ke baare mein hai jo "planet ko Sun se jodti hai." Woh line hi $\vec{r}$ hai. Sab kuch — area, angle, speed — is ek arrow se measure hota hai.

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2 — Velocity $\vec{v}$: "kahan ja raha hai" ka arrow

Planet ko thodi der baad imagine karo: woh ek chhote arrow se shift ho gaya hai jise hum $d\vec{r}$ kehte hain ("$\vec{r}$ ke change ka ek tiny piece"). Kisi bhi cheez ke aage chhota $d$ ka matlab hai "uski ek tiny sliver." Velocity exactly woh tiny shift hai jo us tiny time se divide ho jisme woh hua:

$\vec{v}=\frac{d\vec{r}}{dt}$

Fraction kyun? $d\vec{r}$ kitna move hua; $dt$ kitna time laga; distance over time is speed — aur unhe arrows ke roop mein rakhne se direction bhi bachi rehti hai.

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3 — Do arrows ke beech angle $\varphi$, aur $\sin \varphi$

Do arrows jo same corner se shuru hote hain unke beech ek angle banta hai. Hum $\vec{r}$ aur $\vec{v}$ ke beech ke angle ko Greek letter $\varphi$ ("phi") se name karte hain.

Topic ko iska kyun zaroorat hai: Sirf velocity ka sideways part hi actually naya area sweep karta hai — woh part jo seedha $\vec{r}$ ke along daudta hai woh sirf arrow ko lamba ya chhota karta hai bina koi slice paint kiye. Isliye "har second mein kitna area" mein hamesha ek chhupa hua $\sin \varphi$ hota hai.

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4 — Cross product $\vec{a}\times \vec{b}$: ek area machine

Yeh ek genuinely naya tool hai. Maano do arrows $\vec{a}$ aur $\vec{b}$ same point se shuru ho ke ek parallelogram (ek tilted rectangle) banate hain. Cross product us parallelogram ka area measure karta hai.

Ise ek recipe ki tarah padho: base length $\times$ height. Ek arrow base hai; $|\vec{b}|\sin \theta$ hai parallelogram kitna uuncha khada hai (phir woh sideways part!). Parallelogram ko diagonal se aadha kato aur tumhe ek triangle milta hai jiska area $\frac{1}{2}|\vec{a}\times \vec{b}|$ hai.

Topic ko exactly yahi tool kyun chahiye aur simple multiplication kyun nahi: Hum do arrows se sweept area chahte hain, aur area is baat pe depend karta hai ki woh kitne slanted hain — do parallel arrows zero area sweep karte hain, do perpendicular arrows sabse zyada. Ordinary multiplication ($|\vec{a}||\vec{b}|$) slant nahi dekh sakta; cross product ka built-in $\sin \theta$ dekh sakta hai. Exactly isliye parent ka pehla step $dA=\frac{1}{2}|\vec{r}\times d\vec{r}|$ likhta hai. Dekho Cross Product and Area.

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5 — Area $A$ aur areal velocity $dA/dt$

Kepler's Second Law, in symbols mein, simply yeh hai: $dA/dt$ kabhi nahi badlta. Sun ke paas mote chhote slices, door mein lambe patले slices — har second same area.

Topic ko iska kyun zaroorat hai: yeh single quantity hi woh law hai. Parent page ki baaki sab cheez is ek number ko constant prove karne ke liye hai.

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6 — Angle $\theta$, aur uski rate $\dot{\theta }$

Hum planet ko angle $\theta$ ("theta") se bhi track karte hain jo $\vec{r}$ ne kisi starting line se measure karke sweep kiya hai. Jaise planet orbit karta hai, $\theta$ badhta hai.

Polar (angle) form mein tiny slice ek patli circle wedge hoti hai: $dA=\frac{1}{2}{r}^{2}d\theta$. $dt$ se divide karo:

$\frac{dA}{dt}=\frac{1}{2}{r}^2\dot{\theta }$

Wedge mein $r^2$ kyun? Radius $r$ aur tiny angle $d\theta$ wali ek pie slice almost ek triangle hai jiska base $rd\theta$ (arc length) aur height $r$ hai, toh area $\approx \frac{1}{2}(rd\theta )(r)=\frac{1}{2}{r}^{2}d\theta$.

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7 — Angular momentum $L$ aur mass $m$

Uski length $L=m{r}^2\dot{\theta }=mrv\sin \varphi$ nikalta hai. Step 6 se compare karo:

$\frac{dA}{dt}=\frac{1}{2}{r}^2\dot{\theta }=\frac{L}{2m}$

Toh geometric area-rate hi physical angular momentum divided by $2m$ hai. Kyunki Angular Momentum Conservation ek central force ke liye $L$ ko fixed rakhta hai, $dA/dt$ fixed hai. Dekho Central Forces and Torque ki kyun ek radial force $L$ ko constant rakhti hai (zero torque).

Topic ko iska kyun zaroorat hai: $L$ woh bridge hai. Yeh ek picture (area) ko ek conserved physics quantity mein convert karta hai, aur conserved quantities woh hain jo tumhe poora orbit solve kiye bina predict karne deti hain.

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Sab kuch law mein kaise jaata hai

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Equipment checklist

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Connections

• Parent: the full topic note
• Cross Product and Area — yahan define ki gayi area machine
• Angular Momentum Conservation — kyun $L$ (aur toh $dA/dt$) fixed hai
• Central Forces and Torque — kyun gravity zero torque deti hai
• Kepler's First Law (Ellipse) · Kepler's Third Law (T² ∝ a³) · Vis-viva Equation