Orbital velocity for circular orbit — derivation

1.2.24 · Physics › Newton's Laws & Dynamics

HUM KYA DHUNDH RAHE HAIN?

"Circular" zaroori hai: speed magnitude mein constant rehti hai, sirf direction badlti hai.
$r$ planet ke centre se measure hoti hai, surface se nahi: $r = R + h$ jahan $R$ planet ki radius hai aur $h$ altitude.

YEH KAAM KYUN KARTA HAI? (physical logic)

Do facts ek saath milte hain:

Circular motion ke liye Newton's 2nd law. Jo bhi cheez radius $r$ ke circle mein speed $v$ se move kar rahi ho, usse andar ki taraf (centre ki taraf) $a_{c} = \frac{v ^{2}}{r}$ magnitude se accelerate karna zaroori hai. Koi cheez force $F = m a_{c}$ provide karni chahiye.
Newton's law of gravitation. Earth satellite ko $F_{g} = \frac{GM m}{r ^{2}}$ se andar ki taraf kheenchti hai.

Agar sirf ek hi force gravity hai, toh gravity centripetal force ke barabar honi chahiye. Dono ko equal karo — woh ek equation speed pin down kar deti hai.

DERIVE KAISE KARTE HAIN (first principles se)

Surface par $g = \frac{GM}{R ^{2}} \Rightarrow GM = g R^{2}$ . Substitute karne par: $v_{o} = \frac{g R ^{2}}{r} h = 0, r = R v_{o} = g R$ Kyun useful hai? Ho sakta hai aapko $G$ aur $M$ alag-alag na pata hon, lekin $g$ aur $R$ toh hamesha pata hote hain.

Figure — Orbital velocity for circular orbit — derivation

Common mistakes (Steel-manned)

Active recall

Recall Answers cover karo aur khud test karo

Orbit mein centripetal force kaunsi force provide karti hai? ⇒ Sirf Gravity.
$v_{o}$ ko $G, M, r$ ke terms mein likho. ⇒ $v_{o} = GM / r$ .
Satellite mass kyun appear nahi karta? ⇒ $m$ cancel ho jaata hai — $F_{g}$ aur $F_{c} \propto m$ dono.
$r$ badhne par $v_{o}$ kaise change hoti hai? ⇒ Ghatti hai, $1/ r$ ke roop mein.
Altitude $h$ ke liye $r$ kya hai? ⇒ $r = R + h$ (Earth ke centre se).
Escape velocity se kya relation hai? ⇒ $v_{esc} = 2 v_{o}$ .

Recall Feynman: ek 12-saal ke bachche ko samjhao

Socho ek ball ko ek unche pahaad se bahut, bahut tezi se sideways phenka. Gravity hamesha use neeche zameen ki taraf kheenchti hai. Aaram se phenko toh paas mein giregi. Zyada tezi se phenko, aur door giregi. Insanely tezi se phenko — har second lagbhag 8 km — aur jab woh giregi, toh goli Earth neeche utni hi curve karti jaayegi. Toh woh "ground miss" karti rehti hai hamesha ke liye aur poora chakkar laga leti hai! Usse koi cheez upar nahi rakh rahi; woh sirf itni tezi se sideways gir rahi hai ki kabhi land nahi karti. Badi ya chhoti ball, koi farak nahi — magic speed same hoti hai, kyunki gravity bhaari aur halki dono cheezein ek hi tarah se kheenchti hai.

Flashcards

Orbital velocity formula (circular)

v_{o} = \frac{GM}{r}

Satellite ke liye centripetal force kaun provide karta hai?

Gravity, aur sirf gravity.

Derivation ke core mein set ki gayi equation

\frac{GM m}{r ^{2}} = \frac{m v _{o}^{2}}{r}

v_{o}

mein satellite mass kyun appear nahi karta?

Woh cancel ho jaata hai — gravitational aur required centripetal dono forces

m

ke proportional hain.

Orbital speed orbital radius ke saath kaise vary karti hai?

v_{o} \propto 1/ r

— badi orbit ⇒ dhheemi.

Surface gravity use karke orbital velocity (surface ke paas)

v_{o} = g R \approx 7.9

km/s.

Altitude

h

par satellite ke liye

r

kya hai?

r = R + h

, Earth ke centre se measure kiya.

Orbital aur escape velocity mein kya relation hai?

v_{esc} = 2 v_{o}

Kya gravity circular orbit mein satellite par koi work karti hai?

Nahi — woh velocity ke perpendicular hai, isliye speed constant rehti hai.

LEO orbital speed approximate value

~7.9 km/s (≈ 28,000 km/h).

Connections

Newton's Law of Universal Gravitation
Centripetal Force and Uniform Circular Motion
Escape Velocity
Kepler's Third Law — $v = 2 π r / T$ ko $v_{o}$ ke saath combine karo taaki $T^{2} \propto r^{3}$ mile.
Acceleration due to gravity g and GM = gR²
Geostationary Orbit