Kepler's second law — equal areas in equal times, from angular momentum conservation: Question bank

3.2.6 · D5 · Physics › Orbital Mechanics & Astrodynamics › Kepler's second law — equal areas in equal times, from angul

Full derivation ke liye dekho parent note.

True or false — justify karo

True or false: "Equal areas in equal times" ka matlab hai ki planet har jagah equal speed se move karta hai.

False. Sirf area swept per second constant hoti hai; Sun ke paas planet ek short-and-fat triangle mein fast move karta hai, door slow hoke ek long-and-thin mein — same area, bahut alag speed.

True or false: Kepler's second law ke liye inverse-square force zaroori hai.

False. Koi bhi central force (

F ∥ r

) zero torque deta hai aur isliye constant areal velocity; inverse-square sirf orbit ko ellipse banana ke liye chahiye (first law).

True or false: Perfectly circular orbit pe ek planet phir bhi second law follow karta hai.

True, trivially —

r

constant hone ke saath areal rate

\frac{1}{2} r^{2} \dot{θ}

fixed rehti hai kyunki

\dot{θ}

bhi constant hai, to ye equal angles ke saath equal areas sweep karta hai.

True or false: Planet ka angular momentum

L

exactly zero hota hai kyunki gravity seedha andar kheenchti hai.

False. Gravity zero torque deti hai, zero angular momentum nahi;

L

ek fixed nonzero vector hai jab tak planet dead-radially Sun mein fall nahi kar raha.

True or false: Agar Sun suddenly apna mass double kar le, to areal velocity double ho jaayegi.

False.

d A / d t = L /2 m

planet ke angular momentum pe depend karta hai, directly

M

pe nahi;

M

change karne se orbit ki shape aur future

L

change hoga, lekin change ke instant par

L

(aur isliye

d A / d t

) unchanged rehta hai.

True or false: Areal velocity ek scalar hai, lekin ye ek vector cross product se aata hai.

True.

d A / d t = \frac{1}{2} ∣ r \times v ∣

— cross product ek vector hai, lekin hum uska magnitude lete hain taaki (scalar) rate of area swept mile.

True or false: Bahut elongated orbit pe ek comet ke liye,

d A / d t

Sun ke paas bada hota hai aur door chhota.

False. Yehi to poora point hai —

d A / d t

constant hai; comet ki wild speed changes exactly radius changes ko compensate karti hain taaki swept area rate fixed rahe.

True or false: Second law fail ho jaata agar ek chhoti si tangential drag force hoti.

True. Ek tangential force

r

ke parallel nahi hoti, to ye torque produce karta hai,

L

drift karta hai, aur areal velocity slowly change hoti hai — orbits decay hote hain, areas shrink hote hain.

Edge cases

Edge case: Ek radial line par Sun ki taraf seedhe fall karte hue mass ki areal velocity kya hai?

Ye zero hai —

v

r

ke parallel hai, to

r \times v = 0

; degenerate "orbit" ek line hai aur koi area sweep nahi karta (consistent

L = 0

ke saath).

Edge case: Kya ye law ek planet ke paas slingshot / gravity assist ke dauran hold karta hai?

Sun ke baare mein encounter ke dauran nahi — planet ki gravity doosre center ke baare mein ek doosra central force hai, Sun ke baare mein torque add karta hai; Sun ke baare mein

L

change hota hai, aur exactly isi tarah spacecraft energy gain karta hai.

Edge case: Two-body problem mein jahan Sun bhi move karta hai, hum kiska

r

use karein?

Center of mass se vector use karo (ya relative separation), moving Sun ke center se nahi; conserved

L

reduced-mass motion ka hai, aur areal velocity us frame mein constant hai.

Edge case: Agar orbit par kisi jagah

\dot{θ}

momentarily zero ho jaaye, to kya law violate hota hai?

Real bound orbit par

\dot{θ}

kabhi zero nahi hota (uske liye

L = 0

chahiye hoga), to ye actual ellipse ke liye kabhi nahi hota — zero radial fall signal karta hai, orbit nahi.

Edge case: Perihelion se aphelion tak aadhi orbit mein, kya swept area exactly ellipse ki aadhi hoti hai?

Haan — lekin agar perihelion se start karo to half period se zyada time lagta hai, kyunki equal areas equal times mein sirf balanced halves mein hoti hain; peri-to-aphelion (equal time se measure kiya) equal area rate ke bawajood kam angular arc cover karta hai. (Peri↔apo half specifically area ko aadha aur symmetry se time ko bhi aadha karta hai.)

Recall Har trap ki one-line summary

Yahan almost har galti chain ke ek word ko bhool jaane se aati hai Central → Torque-zero → L-constant → Area-rate-constant, ya area rate (constant) ko speed / arc length / angle (sab changing) ke saath confuse karne se.

Connections

Angular Momentum Conservation — har trap ke peeche conserved quantity
Central Forces and Torque — kyun chain zero torque se shuru hoti hai
Kepler's First Law (Ellipse) — kya cheez actually inverse-square maangti hai
Kepler's Third Law (T² ∝ a³) — $π ab = (L /2 m) T$ use karta hai
Vis-viva Equation — changing speed jo constancy nahi todti
Cross Product and Area — $\frac{1}{2} ∣ r \times v ∣$ ki geometry

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

True or false — justify karo

Error dhundo

Why questions

Edge cases

Connections