Foundations — Specific angular momentum h = √(GMp)

3.2.12 · D1 · Physics › Orbital Mechanics & Astrodynamics › Specific angular momentum h = √(GMp)

Is page ko kaise padhen

Neeche, har foundation hai: plain words → picture → topic ko iske zarurat kyun hai. Inhe is order mein rakha gaya hai ki har naya symbol sirf upar se already built ideas use kare. Kuch bhi assume nahi kiya gaya — koi bhi symbol use nahi hota apne section se pehle jo use define karta hai.

1. Ek vector $\vec{r}$ — length aur direction wala arrow

Figure s01 dekho. Kala dot heavy central body hai. Lal arrow $\vec{r}$ hai: yeh planet ki taraf point karta hai, aur iska length distance $r$ hai (koi arrow nahi — plain $r$ matlab sirf length, direction bhool ke; hum yeh precisely next section mein karte hain).

Topic ko iske zarurat kyun hai: orbit woh saari positions $\vec{r}$ ka set hai jo planet visit karta hai. Is page par baad ki har idea — velocity arrow, spin-making cross product, orbit ki shape — is ek arrow ke upar built hai.

2. Length $r$ aur unit arrow $\hat{r}$

Picture: lal arrow lo, use same way point karte raho, lekin use tab tak squash karo jab tak exactly ek unit lamba na ho jaye. Woh stub hai $\hat{r}$. Toh koi bhi position split hoti hai $\vec{r}=r\hat{r}$ = (kitna door) $\times$ (kaunsa rasta).

Topic ko iske zarurat kyun hai: poora page "kitna door" ($r$) aur "kaunsa rasta" ($\hat{r}$) mein distinguish karta hai. Gravity $r$ se set strength ke saath aur $\hat{r}$ se set direction ke saath kheenchti hai; inhe abhi alag karna baad ke har statement ko "outward" versus "inward" ke baare mein exact banata hai.

3. Velocity $\vec{v}$ — motion ka arrow

Figure s02 mein, lal arrow $\vec{v}$ orbit path ke tangent par drawn hai — yeh hamesha travel ke direction mein point karta hai. Yeh generally $\vec{r}$ ke saath aligned nahi hota: $\vec{r}$ centre se outward point karta hai, jabki $\vec{v}$ curve ke along point karta hai.

Topic ko iske zarurat kyun hai: spin amount dono se built hota hai — tum kahan ho ($\vec{r}$) aur tum kahan ja rahe ho ($\vec{v}$). Dono mein se ek ke bina orbit ke rotation ke baare mein kuch nahi pata chalta.

4. $\vec{r}$ aur $\vec{v}$ ke beech ka angle $\varphi$ (flight-path angle)

Figure s02 dekho phir se: chhota wedge $\varphi$ mark karta hai. Yahan har size ka $\varphi$ physically kaisa dikhta hai (hum abhi motion words mein describe karte hain; exact rate symbols §8 mein aate hain):

Topic ko iske zarurat kyun hai: $\vec{r}$ aur $\vec{v}$ ke beech ka angle exactly wahi hai jo §5 mein cross product measure karta hai, aur yahi woh hai jo simple picture "distance $\times$ speed" ko sirf special points par work karta hai. Note karo ki $\sin \varphi$ same value leta hai $\varphi$ aur $180^{\circ }-\varphi$ par, toh outbound half par ek point aur inbound half par uska mirror same value share karte hain sideways ingredient ka — ek conserved quantity ka pehla hint. Flight-path angle dekho.

5. Cross product $\vec{r}\times \vec{v}$ — "spin-area" machine

Yeh poore topic ka dil hai, isliye hum ise dheere build karte hain.

Figure s03 dikhata hai kyun length woh parallelogram area hai. Do arrows ek slanted box sweep karte hain; uska area hai base $\times$ height $=|\vec{a}|\times (|\vec{b}|\sin \theta )$. Jab arrows same way point karte hain ($\theta =0$) box flat hota hai — zero area — toh $\vec{a}\times \vec{a}=0$.

Topic ko iske zarurat kyun hai: $\vec{h}=\vec{r}\times \vec{v}$ specific angular momentum ki definition hai. Iska length (twice) woh rate hai jis par position arrow area sweep karta hai — Kepler's Second Law se connection. Iska direction fixed hai, isliye orbit ek plane mein rehta hai. Kyunki $h=rv\sin \varphi$, simple $h=rv$ sirf apsides par hold karta hai jahan $\varphi =90^{\circ }$ (yeh parent ka "$h=rv$ everywhere" mistake hai).

6. $G$, $M$, aur product $GM$ (parameter $\mu$)

Ab hum finally woh gravity arrow padh sakte hain jo parent note use karta hai: planet ki acceleration hai $\vec{a}=-\frac{GM}{r^2}\hat{r}$. Har piece ab defined hai — $GM$ (strength), $r^2$ (distance ke saath kamzor, length §2 se), aur $-\hat{r}$ (direction: inward, §2 se).

Topic ko iske zarurat kyun hai: kyunki small mass orbital motion se cancel ho jaati hai (parent ka "why divide by $m$" dekho), sirf woh mass matter karti hai jo central $M$ hai, aur yeh hamesha $G$ ke saath glued appear hoti hai. Toh $GM=\mu$ natural single knob hai. Yeh $h=\sqrt{GMp}$ ke andar baitha hai aur gravity $\vec{a}=-\frac{GM}{r^2}\hat{r}$ ke andar bhi.

7. Conic orbit aur uske shape symbols $p$, $e$, $a$, $\theta$

Chaar shape symbols, har ek ek picture (figure s04):

Topic ko iske zarurat kyun hai: $h=\sqrt{GMp}$ conserved spin $h$ ko shape symbol $p$ se link karta hai. $p$ ke bina (aur uske cousins $e,a,\theta$ ke bina) formula ka koi geometric matlab nahi hota. Yeh Two-body problem se nikalta hai.

8. Derivatives aur dot notation $\dot{r},\ddot{r}$

Plain picture: agar $r$ centre se tumhari distance hai, $\dot{r}$ tumhari outward/inward speed hai ($\dot{r}>0$ outward, $\dot{r}<0$ inward), aur $\ddot{r}$ yeh hai ki woh speed speed up ho rahi hai ya nahi. Zero $\dot{r}$ matlab momentarily na paas aa raha na door ja raha — exactly apsides, same $\varphi =90^{\circ }$ points §4 se (ab ek rate ke roop mein stated: $\dot{r}=0$).

Topic ko iske zarurat kyun hai: conservation proof use karta hai $\frac{d\vec{h}}{dt}=0$, aur orbit derivation use karta hai $\dot{r},\ddot{r},\dot{\theta }$ polar form mein. Parent note mein har woh step jisme dot hai ya $\frac{d}{dt}$ hai, woh is idea par khada hai.

Prerequisite map

Equipment checklist

Har line padho, dimag mein jawab socho, phir reveal karo.

Connections

• Parent: h = √(GMp) — yeh page uske har symbol supply karta hai.
• Two-body problem — jahan conic orbit equation (aur $p,e,a,\theta$) janam leti hai.
• Kepler's Second Law — $\vec{h}$ ka cross-product area meaning seedha equal areas in equal times tak le jaata hai.
• Flight-path angle — section 4 mein built angle $\varphi$.
• Eccentricity vector — yahan introduce ki gayi eccentricity $e$ ko gehra karta hai.
• Vis-viva equation aur Specific orbital energy — $h$ ke energy partners.
• Hinglish version →