Foundations — Orbital velocity for circular orbit — derivation

1.2.24 · D1 · Physics › Newton's Laws & Dynamics › Orbital velocity for circular orbit — derivation

Isse pehle ki tum parent page par derivation enjoy karo, the topic note, tumhe har letter aur har picture ko pehchanna hoga jo woh silently rely karta hai. Yeh page har ek cheez ko kuch nahi se build karti hai, us order mein jisme har naya piece pichle pe lean karta hai.

1. "Vector" aur "speed" ka matlab yahan kya hai

Figure mein amber arrow dekho. Uski length speed hai; uska pointing direction hai. Jaise-jaise satellite circle ke around jaata hai, arrow ki length same rehti hai lekin woh swing karta rehta hai — wahi swinging "accelerating" ka matlab hoga ek moment mein.

2. Circle: radius $r$, aur hum ise kahan measure karte hain

Figure mein cyan line $R$ planet ke centre se uski surface tak jaati hai (planet ka apna radius). White line $h$ altitude hai — satellite ground se kitna upar float kar raha hai. Full radius dono ko add karke milti hai:

$r=R+h$

Humein $r$ ki zaroorat kyun hai? Kyunki gravity ki strength aur turn ki sharpness dono depend karte hain ki tum kitne door ho. Yeh puri topic mein sabse important dial hai.

3. Acceleration — woh arrow jo inward point karta hai

Is inward acceleration ki size ka ek naam aur formula hai jise hum simply use karenge (iski full derivation Centripetal Force and Uniform Circular Motion mein hai):

4. Force aur mass — Newton's Second Law

Toh mass $m$ ko turn karke rakhne ke liye inward force chahiye:

$F_c=m{a}_c=\frac{m{v}^2}{r}$

5. Gravity — woh pull jo bending karti hai

Aao har symbol samjhein:

• $G$ — gravitational constant, nature ka ek fixed number ($6.67\times 10^{-11}$ SI units mein). Yeh universe mein gravity ki overall strength set karta hai.
• $M$ — bada mass (planet). Bada $M$ $\Rightarrow$ zyada strong pull.
• $m$ — chhota mass (satellite).
• $r^2$ — distance squared neeche. Yeh inverse-square law hai: do baar door jao, gravity quarter ho jaati hai.

6. Surface gravity $g$ aur shortcut $GM=g{R}^2$

7. Square root  ​ — square ko undo karna

Derivation $v_o^2=\frac{GM}{r}$ par khatam hoti hai. Hamare paas $v_o^2$ hai lekin $v_o$ chahiye, toh hum dono sides ka square root lete hain "square ko peel off" karne ke liye. Isliye root appear hota hai — yeh woh tool hai jo us squaring ko undo karta hai jo circular motion ($v^2$) ne hum par force ki.

Prerequisite map

Equipment checklist

Apne aap ko test karo — tum derivation ke liye ready ho jab tum bina dekhe har sawaal ka jawab de sako.

Connections

• Orbital velocity for circular orbit — derivation — woh parent jiske liye yeh page tumhe prepare karta hai.
• Centripetal Force and Uniform Circular Motion
• Newton's Law of Universal Gravitation
• Acceleration due to gravity g and GM = gR²
• Escape Velocity
• Kepler's Third Law
• Geostationary Orbit