Foundations — Conservation of energy and angular momentum in gravitational field

3.2.2 · D1 · Physics › Orbital Mechanics & Astrodynamics › Conservation of energy and angular momentum in gravitational

Pehle hum is idea ko symbols mein state bhi kar sakein, uske liye har letter aur squiggle ko earn karna hoga jo parent note mein diya hua hai. Ye page ek dictionary hai jo tum khud banate ho, sahi order mein, taaki koi bhi symbol tab tak saamne na aaye jab tak tumhe uska picture pata na ho.

0. Stage: do bodies aur unke beech ek line

Sab kuch ek bade body (Sun, ya Earth) ke beech hota hai jo centre mein still baitha hai, aur ek chote body (planet, satellite, probe) ke beech jo uske around move karta hai. Hum centre se chote body ki taraf ek arrow kheenchte hain. Wahi single arrow hai jahan lagbhag har symbol rehta hai.

Ek arrow ko teen symbols mein kyun split karein? Kyunki kabhi hum sirf care karte hain kitna dur ($r$), kabhi sirf kis taraf ($\hat{r}$), aur kabhi dono ($\vec{r}$). Relationship bas yeh hai:

$\vec{r}=r\hat{r}(\text{length}\times \text{pure direction}).$

1. Vectors: arrows jinhe add aur turn kar sako

Ek vector ek arrow hai: isme length aur direction hoti hai. Hum ise over-arrow ke saath likhte hain, $\vec{v}$. Ek plain number (length, mass $m$, time) ki koi direction nahi hoti — hum ise scalar kehte hain.

Velocity ko vector ke roop mein chahiye kyunki ek orbit mein body ka motion partly baahir/andar aur partly sideways hota hai — do alag directions jo hum jald hi alag karenge.

1.1 Cross product $\vec{a}\times \vec{b}$ — "kitna swirl"

Parent note angular momentum ko $\vec{L}=\vec{r}\times (m\vec{v})$ (jahan $m$ woh choti mass hai jo humne abhi define ki) aur torque ko $\vec{\tau }=\vec{r}\times \vec{F}$ likhta hai. Woh $\times$ cross product hai. Yahan yeh hai ki iska matlab kya hai, sirf calculate kaise karte hain nahi.

Length woh area of the parallelogram hai jo do arrows span karte hain. Agar do arrows same direction mein point karein ($\varphi =0$), $\sin 0=0$ — parallelogram squash hokar flat ho jaata hai, area zero, cross product zero vector $\vec{0}$ hai.

1.2 Dot product $\vec{a}\cdot \vec{b}$ — "kitna along"

Parent ki energy derivation mein $\vec{F}\cdot d\vec{r}$ hai.

Topic ko yeh kyun chahiye: work (transferred energy) sirf force along motion count karta hai. Sideways motion koi work nahi karta. Dot product exactly "woh part hai jo line up karta hai," isliye $\vec{F}\cdot d\vec{r}$ woh thodi si energy hai jo gravity ek tiny step $d\vec{r}$ par deti hai.

2. Motion ko split karna: polar coordinates $(r,\theta )$

Kyunki orbit flat hai (angular momentum ka tohfa), hum position ko teen ki jagah sirf do numbers se describe karte hain.

Koi bhi motion phir do independent moves ka mix hota hai:

• Radial: arrow ka longer ya shorter hona — body ka baahir/andar move karna.
• Tangential: arrow ka around sweep karna — body ka sideways move karna.

Yeh dot ek derivative hai — calculus se slope/rate tool. Topic ko yeh chahiye kyunki orbits sab change ke baare mein hain: speeding up, sweeping angle, andar girna.

2.1 Speed kyun split hoti hai: $v^2={\dot{r}}^2+(r\dot{\theta }{)}^2$ build karna

Figure s04 dekho. Time ke ek tiny slice $dt$ mein body do independent little moves karta hai jo ek right angle par milte hain:

1. Outward move — arrow $\vec{r}$ $\dot{r}dt$ se badhta hai. Yeh body ko $\hat{r}$ ke seedhe baahir slide karta hai. Distance covered: $\dot{r}dt$.
2. Sideways move — arrow ek tiny angle $\dot{\theta }dt$ se swing karta hai. Radius $r$ par ek point $\dot{\theta }dt$ angle se swing karta hua arc distance $r\times (\dot{\theta }dt)$ travel karta hai — arc length radius times angle hoti hai.

Kyunki do little steps perpendicular hain, hum unhe Pythagoras se jodte hain (right-triangle rule: hypotenuse squared do legs squared ka sum hai). Total little step $ds$ hypotenuse hai:

$d{s}^2=\underset{\text{outward leg}}{\underbrace{(\dot{r}dt{)}^2}}+\underset{\text{sideways leg}}{\underbrace{(r\dot{\theta }dt{)}^2}}.$

Speed distance-per-time hai, $v=ds/dt$, isliye $d{t}^2$ se divide karo aur total lo:

Ek turning point par (perihelion — closest, ya aphelion — farthest), arrow thodi der ke liye longer ya shorter hona band kar deta hai, isliye $\dot{r}=0$. Wahaan sari speed sideways hoti hai: $v=r\dot{\theta }$. Parent yahi trick use karta hai.

3. Force, potential energy, aur minus signs

$U=-GMm/r$ negative kyun hai? "Door, rest mein" ($r\to \infty$) ko zero level lo. Isse closer baithne ke liye, body pehle se "andar gir chuka hai," energy release karke — isliye ab baahir nikalne ke liye energy chahiye. Debt mein hona = negative. Jitna gehra (chota $r$), utna zyada negative.

4. Do conserved quantities, finally named

5. Foundations topic ko kaise feed karte hain

Yeh ek map mein pura build hai. Ise teen streams mein padho:

• Left stream (cross product): arrows → cross product → angular momentum $\vec{L}$, plus yeh fact ki torque zero hai → $\vec{L}$ frozen hai → orbit flat hai aur equal areas sweep hote hain (Kepler 2).
• Middle stream (dot product): arrows → dot product → work aur energy → total energy $E$ frozen hai.
• Right stream (geometry): arrows → polar coordinates → speed split karna, jo energy bookkeeping ko feed karta hai.

Do frozen quantities ($\vec{L}$ aur $E$) phir bottom par milte hain orbit ki size $a$ aur shape $e$ pin down karne ke liye.

6. Ye aage kahan le jaate hain

• Conservation of energy and angular momentum in gravitational field — parent topic jise ye tools power karte hain.
• Conservative Forces and Potential Energy — kyun $U(r)$ exist karta hai.
• Central Force Problem — "force along the line" ka general setting.
• Effective Potential and Orbit Classification — $E$ aur $L$ ko ek graph mein combine karna.
• Vis-Viva Equation, Kepler's Laws of Planetary Motion, Escape Velocity and Hohmann Transfers — direct payoffs.

Equipment checklist

Answers cover karo. Agar tum har blank cold fill kar sako, tum parent note ke liye ready ho.