Conservation of energy and angular momentum in gravitational field

3.2.2 · Physics › Orbital Mechanics & Astrodynamics

1. Ye do quantities conserved KYUN hoti hain?

Angular momentum conserved KYUN hota hai. Central body ke baare mein torque hai $τ = r \times F$ . Central force ke liye, $F ∥ r$ , isliye

$τ = r \times f (r) \overset{r}{^} = f (r) (r \times \overset{r}{^}) = 0 .$

Kyunki $τ = \frac{d L}{d t}$ , hume milta hai $\frac{d L}{d t} = 0 \Rightarrow L = const$ .

Energy conserved KYUN hoti hai. Gravity conservative hai: iska kiya hua kaam sirf start aur end positions pe depend karta hai, path pe nahi. Hum ek potential energy $U (r)$ define kar sakte hain jaise ki $F = - \nabla U$ . Phir motion ke along work–energy theorem deta hai

$\frac{d}{d t} (\frac{1}{2} m v^{2} + U (r)) = 0 \Rightarrow E = \frac{1}{2} m v^{2} + U (r) = const .$

2. Scratch se derivation

2.1 Gravitational potential energy $U (r)$

Minus sign kehta hai tum ek potential well mein ho — escape karne ke liye tumhe energy add karni padegi.

2.2 Speed aur radius ke terms mein total energy

$E = \frac{1}{2} m v^{2} - \frac{GM m}{r}$

2.3 $L$ use karke speed ko split karna (effective potential idea)

Orbital plane mein polar coordinates $(r, θ)$ use karo. Velocity ka ek radial aur ek tangential part hota hai:

$v^{2} = \overset{r}{˙}^{2} + (r \dot{θ})^{2} .$

Angular momentum magnitude hai $L = m r^{2} \dot{θ}$ (kyunki tangential speed $r \dot{θ}$ hai aur lever arm $r$ hai). Solve karo: $\dot{θ} = \frac{L}{m r ^{2}}$ , to

$r^{2} \dot{θ}^{2} = \frac{L ^{2}}{m ^{2} r ^{2}} .$

$E$ mein substitute karo:

$E = \frac{1}{2} m \overset{r}{˙}^{2} + U_{eff} (r) \frac{L ^{2}}{2 m r ^{2}} - \frac{GM m}{r} .$

Figure — Conservation of energy and angular momentum in gravitational field

2.4 Kepler's 2nd law free mein nikal aata hai

Har unit time mein sweep ki gayi area hai $\frac{d A}{d t} = \frac{1}{2} r^{2} \dot{θ} = \frac{L}{2 m} = const$ . Equal areas in equal times — yahi angular momentum conservation disguise mein hai.

3. $E$ ko orbit shape se connect karne wale key relations

Ek bound elliptical orbit ke liye jiska semi-major axis $a$ hai, energy sirf $a$ pe depend karti hai:

$E = - \frac{GM m}{2 a}$

Isse vis-viva equation milti hai (dono boxes combine karo, $E$ eliminate karo):

$v^{2} = GM (\frac{2}{r} - \frac{1}{a})$

Orbit type	$E$	$a$	shape
Circle/Ellipse	$< 0$	finite $> 0$	bound
Parabola	$= 0$	$\infty$	escape (just barely)
Hyperbola	$> 0$	$< 0$	flyby

Escape speed: $E = 0$ set karo radius $r$ pe: $\frac{1}{2} m v_{esc}^{2} = \frac{GM m}{r} \Rightarrow v_{esc} = \frac{2 GM}{r}$ .

4. Worked examples

5. Common mistakes (steel-manned)

Recall Feynman: ek 12-saal ke bachche ko samjhao

Socho tum apne sir ke upar ek rubber band pe ek ball swing kar rahe ho. Koi bhi do cheezein kabhi nahi badalti, chahe ball jaise bhi loop kare: ball kitni tezi se ghoomti hai times kitni door hai (yahi angular momentum hai — andar khicho aur yeh tezi se spin karega), aur uski moving-energy plus uski stretch-energy ka total (yahi total energy hai). Planets bilkul yahi karte hain, rubber band ki jagah gravity ke saath. Jab koi planet Sun ke paas swoosh karta hai toh woh tezi se jaata hai; jab door drift karta hai toh dheere chalata hai — lekin wo do "savings accounts" hamesha same total pe add hote hain.

Flashcards

Gravitational field mein angular momentum conserved kyun hoti hai?

Gravity central hai (

F ∥ r

), to torque

τ = r \times F = 0

, isliye

d L / d t = 0

Constant

L

orbit ki geometry ke baare mein kya imply karta hai?

Motion ek single fixed plane mein rehti hai (orbit flat hoti hai).

Gravitational potential energy derive karo.

U (r) = - \int_{\infty}^{r} (- GM m / r^{'2}) d r^{'} = - GM m / r

Orbiting body ki total mechanical energy likho.

E = \frac{1}{2} m v^{2} - \frac{GM m}{r}

Effective potential kya hai aur uske do parts kya hain?

U_{e f f} = \frac{L ^{2}}{2 m r ^{2}} - \frac{GM m}{r}

: centrifugal barrier (repulsive) + gravity (attractive).

Vis-viva equation state karo.

v^{2} = GM (\frac{2}{r} - \frac{1}{a})

Total orbital energy kis pe depend karta hai?

Sirf semi-major axis pe:

E = - \frac{GM m}{2 a}

(eccentricity se independent).

Ellipse, parabola, hyperbola ke liye

E

ka sign?

E < 0

ellipse (bound),

E = 0

parabola (escape),

E > 0

hyperbola (unbound).

Energy conservation se escape speed derive karo.

E = 0

set karo:

\frac{1}{2} m v^{2} = \frac{GM m}{r} \Rightarrow v_{esc} = 2 GM / r

Kepler's 2nd law ka

L

se kya relation hai?

d A / d t = \frac{L}{2 m} =

const, to equal areas in equal times = constant angular momentum.

Orbit mein speed maximum kahan hoti hai aur kyun?

Perihelion pe;

r

sabse chhota hota hai to

U

sabse zyada negative hai, isliye fixed

E

pe KE (speed) greatest hoti hai.

Turning point pe (

\overset{r}{˙} = 0

), speed ko kaise express karte hain?

v = L / (m r)

kyunki velocity wahan purely tangential hoti hai.

Connections

Kepler's Laws of Planetary Motion — 2nd law = angular momentum conservation; 3rd law $a$ aur period ko link karta hai.
Effective Potential and Orbit Classification — orbit type padhne ke liye $U_{e f f}$ use karta hai.
Vis-Viva Equation — energy conservation ka direct child.
Central Force Problem — general framework jahan se ye laws aate hain.
Escape Velocity and Hohmann Transfers — $E = 0$ aur $a$ change karne ke applications.
Conservative Forces and Potential Energy — $U$ exist kyun karta hai.