Gravitation
Yaar, soch — subah Noida Metro mein khade ho, chai ka cup haath mein. Cup thoda hilta hai toh chai neeche girti hai, upar nahi jaati. Cricket mein ball upar phenko toh wapas aati hai. Cycle se utro toh neeche pair zameen pe aata hai. Ye sab kyun? Kyunki Earth hum sabko kheench rahi hai.
Par sirf Earth nahi — tum bhi Earth ko kheench rahe ho! Tumhari mummy kitchen mein laddu bana rahi hai, laddu bhi tumhe kheenchta hai, bas bahut halka. Har mass har doosre mass ko kheenchta hai. Isko hi gravitation kehte hain.
Is chapter mein hum seekhenge: Planets kaise ghoomte hain? G kya hai? g upar neeche jaane pe kyun badalta hai? Rocket ko kitni speed chahiye taaki wo wapas na aaye? Satellite kaise tikta hai upar bina gire? Poora khel ek hi force ka hai, bas samajhna hai.
1. Kepler's Laws - Planets Ka Traffic Rule
Socho tum Swiggy se pizza mangate ho, delivery boy ko map pe ghoomna padta hai. Planets bhi Sun ke around ghoomte hain, par unka map 2000 saal pehle bana tha. Pehle log sochte the sab gol gol perfect circle mein ghoomte hain, geocentric — Earth centre mein. Fir Copernicus bola Sun centre mein hai. Fir ek uncle Tycho Brahe ne bina telescope ke 20 saal tak aankhon se planets ke position likhe, diary bhari. Unke assistant Kepler ne us data se 3 laws nikale.
Law 1 - Orbit Law: Saare planets ellipse mein ghoomte hain, Sun ek focus pe hota hai. Ellipse kya? Do pins lo, dhaage se pencil ghoomao toh jo closed anda-shaped curve banta hai wo ellipse. Circle iska special case hai jab dono foci ek hi point ban jaye. Perihelion woh point jab planet Sun ke sabse kareeb hota hai, aphelion sabse door. Earth ke liye farak chhota hai isliye lagta hai circle hai, par b/a = 0.99986.
Law 2 - Area Law: Sun se planet ko join karne wali line equal time mein equal area sweep karti hai. Matlab jab planet Sun ke paas hota hai toh tez bhagta hai, door hota hai toh dheere. Kyun? Kyunki gravitation central force hai — Sun ki taraf. Central force pe angular momentum L = r × p conserved rehta hai. Aur area sweep rate dA/dt = L/2m constant, isliye law. Example socho ice skater jaise — haath andar tez spin, ye bhi waisa hi conservation hai.
Law 3 - Period Law: T² ∝ a³. Matlab jitna bada orbit, utna zyada time. Agar a double toh T 2.8 times badhega. Formula T² = (4π²/GM_sun) a³. Isi se hum Mars ka saal nikal sakte hain: Earth ka a=1, Mars ka 1.52 times, toh T_Mars = (1.52)^1.5 × 365 ≈ 684 din. Jo table mein diya hai, T²/a³ sab planets ke liye same Q ≈ 3×10⁻³⁴ y²/m³.
First-timer confusion: Kepler's laws sirf planets ke liye nahi, satellites ke liye bhi same. Bas M Sun ki jagah Earth mass le lo.
2. Universal Law of Gravitation - Har Cheez Kheenchti Hai
Newton ne socha — apple neeche girta hai, toh Moon bhi toh gir raha hai Earth ki taraf, par girte girte side mein bhaag jaata hai isliye orbit banta hai. Usne bola har do point masses m1, m2 distance r pe ek dusre ko F = G m1 m2 / r² se kheenchti hain. G universal constant hai, sabke liye same, chahe loha ho ya panner.
Vector mein: F12 = -G m1 m2 / r² * r̂12. Minus matlab attractive — direction r ke opposite. Force m1 pe m2 ki wajah se, F21 = -F12 Newton's third law. Ye weak force hai — tum aur tumhare dost ke beech bhi force hai, par mass chhota hai isliye feel nahi hota. Earth bada hai isliye feel hota hai.
Superposition: Agar 3 masses hain toh ek mass pe net force = baki sab ke forces ka vector sum. Jaise 3 friends tumhe alag direction mein kheench rahe ho, tumhara net kahin beech mein. Triangle mein 3 equal masses ke centroid pe rakha 2m pe net zero kyunki symmetry. Lekin agar ek mass double kar do toh net us taraf shift.
Shell theorem — sabse important: 1) Hollow uniform shell ke bahar rakha point mass sochta hai jaise poora mass centre pe rakha ho. Matlab Earth ko hum point mass samajh sakte hain bahar ke liye. 2) Hollow shell ke andar kahin bhi rakho toh net force zero! Kyunki har direction ke pieces cancel. Mom ki gol steel ki dabbi socho, andar makhi ko chaaron taraf se equal kheench, net zero.
Isse Earth ke andar ka scene samajh aayega. Earth ko concentric shells se bana socho. Bahar wale point ke liye saare shells centre pe. Andar wale point ke liye sirf us radius se chhoti wali sphere kaam karegi, bahar wali shell ka zero.
3. Gravitational Constant G & g - Cavendish Ne Earth Taula
G kaise measure hua? Newton ke 100 saal baad Henry Cavendish ne 1798 mein ek jugaad lagaya. Do chhote lead balls ek rod pe latkaye fine wire se, paas mein do bade lead balls laye. Chhote balls bade balls ki taraf kheeche, rod thoda twist hua. Wire ka restoring torque τθ = F×L. Angle θ measure kiya, τ pata tha, F = G M m / d² se G nikala.
Value aaya G = 6.67 ×10⁻¹¹ N m²/kg². Bahut chhota — isliye hum ek dusre ko kheench feel nahi karte. Is value se hi Earth ka mass nikaal liye. g=GM/R², g=9.8 measurable, R=6370 km known, toh M = g R² / G = 5.97×10²⁴ kg. Isliye kehte hain Cavendish ne Earth ko taula.
Ab g kya hai? Earth ke surface pe m mass pe force F = G M_E m / R_E². Newton's second se F=mg, toh g = G M_E / R_E². Ye 9.81 m/s². Yehi wajah hai saare bodies same acceleration se girte hain mass se independent — Galileo ne Pisa se bola tha.
Trick: GM = g R² = 9.8×(6.37e6)² ≈ 4×10¹⁴ m³/s². Bahut kaam aata hai escape aur orbital formulas mein GM replace karne ke liye.
4. g Ka Variation - Upar Jaao Ya Neeche Jaao, g Ghatta Hai
Socho tum Noida se Mussoorie jaate ho, upar pahad pe weight kam lagta hai? Sach mein g kam hota hai height pe. Formula: height h pe g(h)= GM/(R+h)². Surface pe g=GM/R². Toh ratio g(h)/g = R²/(R+h)² = (1+h/R)⁻². Agar h << R (jaise 10 km vs 6370 km) toh binomial: (1+x)⁻² ≈ 1-2x, so g(h) ≈ g(1-2h/R). Matlab har km upar ~0.3% g kam.
Ab neeche mine mein jaao, jaise mom kitchen mein aata ka dabba neeche se khaali hota hai. Depth d pe tum centre se distance (R-d) pe ho. Upar wali shell thickness d ka force zero (shell theorem). Sirf radius (R-d) wali chhoti sphere kaam karegi. Uska mass Ms = ME * (R-d)³ / R³ (density uniform maan ke). Toh g(d)= G Ms/(R-d)² = GM/R³ * (R-d)= g(1-d/R). Matlab depth mein linear ghatta, centre pe zero!
Graph yaad rakho: surface pe max g=9.8, upar jaao toh 1/(R+h)² curve se ghatta, neeche jaao toh straight line se zero tak. Isliye g maximum surface pe hai, upar neeche dono jagah kam. Par daily life mein h=10m pe farak negligible: 2h/R=2*10/6.37e6=0.000003.
First-timer doubt: Kya g negative ho sakta hai? Nahi, magnitude kam hota hai, direction hamesha centre ki taraf. Depth mein zero matlab weightless, jaise centre pe khade ho toh chaaron taraf se equal pull, net zero.
5. Gravitational Potential Energy - Negative Kyun?
Tum gym mein 5 kg dumbbell zameen se utha ke table pe rakhte ho, kaam karte ho mgh. Ye potential energy mgh hum near surface ke liye bolte hain, jab g constant maano. Par asli picture mein g constant nahi, distance se badalta hai.
General case: m mass ko infinity se distance r tak laane mein gravity khud kaam karti hai positive, par hum potential energy negative define karte hain kyunki bound system hai. Formula W(r)= -GMm/r + W1. Infinity pe zero set karte hain W1=0, toh U(r)= -GMm/r. Negative ka matlab kya? System bound hai, isko todne ke liye energy deni padegi. Jaise tumhari dosti — todne ke liye mehnat lage, isliye negative bolte hain attached ho.
Do masses ke liye U= -G m1 m2 / r. 4 masses square ke corners pe ho toh total U = sum over all unique pairs: 4 sides l + 2 diagonals √2 l. Toh U_total = -4 G m²/l -2 G m²/(√2 l)= -5.41 G m²/l. Negative hi aayega.
Potential V: Potential energy per unit mass, V = U/m = -GM/r. Unit J/kg. Scalar field hai, direction nahi. Iska matlab har point pe kitna PE per kg hai. Centre of square pe 4 masses ka V = -4√2 G m / l.
Near surface connection: r = R+h, h<
6. Escape Speed - Kitni Tez Bhaago Toh Wapas Na Aana Pade
Tum cricket ball upar phenkte ho, 20 m/s se — wapas aati hai. 100 m/s se phenko, aur upar jaati hai par wapas. Kya kabhi itni tez phenk sakte ho ki kabhi wapas na aaye, infinity chali jaye? Haan, us minimum speed ko escape speed kehte hain.
Energy conservation se: Surface pe total energy = KE + PE = ½ m ve² - GMm/R. Infinity pe PE=0, KE minimum zero hona chahiye (bas pahunch jaye). Toh ½ m ve² = GMm/R => ve = √(2GM/R). g=GM/R² se ve=√(2gR). Earth ke liye R=6.37e6, g=9.81 => ve=√(2*9.81*6.37e6)=11.2 km/s. Matlab ~40320 km/h! Bullet se 30 guna tez.
Moon pe ve=2.3 km/s hi hai kyunki chhota hai. Isliye Moon pe atmosphere nahi hai — gas molecules ka thermal speed 2.3 se zyada ho jaata hai dhoop mein, bhag jaate hain space mein. Earth pe gas ka speed 11.2 se kam hai isliye atmosphere tika hai.
Orbital speed alag hai: Orbit mein tikne ke liye centripetal = gravity: m vo² / r = GMm / r² => vo = √(GM/r). Surface ke paas r≈R toh vo=√(gR)=7.9 km/s (first cosmic speed). Escape is √2 times orbital! ve=√2 vo. Isliye pehle orbit mein daalo (7.9), fir thoda aur tez (11.2) toh escape.
Common confusion: Escape mass pe depend nahi karta! m cancel ho jata hai. Chahe cycle ho ya rocket, speed same. Mass sirf energy badhayega, speed nahi.
7. Satellites - Energy, Geostationary vs Polar
Moon ek natural satellite hai, 27.3 din mein Earth ka chakkar. Humne artificial banaye 1957 se. Satellite ka motion planet jaisa hi — Kepler laws lagte hain.
Speed toh nikal liya vo=√(GM/(R+h)). Time period T = circumference / speed = 2π(R+h)/vo = 2π √((R+h)³/GM). Ye Kepler's third law hi hai: T² = (4π²/GM)(R+h)³ = k (R+h)³. Low orbit h≈0 pe T≈84 min (~1.4 hr) 85 min aata hai calculation se. ISS 400 km pe ~92 min.
Energy ka jadoo: KE = ½ m vo² = GMm/2r positive, PE= -GMm/r negative, Total E = KE+PE = -GMm/2r negative! Negative matlab bound — satellite bandha hua hai Earth se. Jitna r bada, total E less negative (0 ki taraf), matlab zyada energy chahiye upar bhejne ke liye. Transfer ke liye ΔE = GMm/2 (1/r1 -1/r2) positive dena padega.
Binding energy = |E| = GMm/2r — itna energy dena pade toh satellite free ho jaye (total zero). Escape ke liye total E ≥0.
Geostationary: Equator ke upar 36000 km height pe aisa satellite jiska period 24h, Earth ke saath ghoome, ek jagah pe stationary dikhe aasman mein. TV broadcast, weather ke liye. Ek hi dish fixed rakh ke signal lo. 3 satellites se poori Earth cover.
Polar / Low Earth: 500-800 km, north-south pole ke upar se ghoomta hai, Earth neeche ghoomti hai toh poori surface scan ho jati hai time pe. Mapping, spy, weather, climate ke liye. Low orbit mein friction se dheere dheere neeche aata hai, isliye thruster chahiye.
Weightlessness? Satellite mein gravity zero nahi hai! g(h) ~8.5 at 400 km, kam par hai. Weightless isliye lagta hai kyunki satellite free fall kar raha hai — tum lift mein cable tootne pe jaise weightless feel karte ho. Centripetal hi free fall hai.
| Concept | Formula (Correct) | Daily Example |
|---|---|---|
| Universal F | F=G m1 m2 /r², F_vector=-G m1 m2 r̂/r², G=6.67e-11 N m²/kg² | Apple fall, you-Earth pull |
| Shell theorem | Outside: F=GMm/r² point centre, Inside hollow: F=0 | Inside steel dabba net zero |
| g surface | g=GM/R²=9.81 m/s², GM=gR²=4e14 | All bodies fall same |
| g height | g(h)=GM/(R+h)²=g(1-2h/R) h<| Mussoorie weight slightly less | |
| g depth | g(d)=g(1-d/R), 0 at centre, max at surface | Mine mein halka |
| Kepler 1-3 | 1) Ellipse Sun focus 2) dA/dt=L/2m const vP rP=vA rA 3) T²=4π²a³/GM ∝a³ | Mars year 684 days |
| Potential Energy | U(r)=-GMm/r 0 at ∞ negative bound, System Σ -G mi mj/rij | Bound dosti - todne energy |
| Potential V | V=-GM/r J/kg, near surface ΔU=mgh | mgh small h approximation |
| Escape | ve=√(2GM/R)=√(2gR)=11.2 km/s Earth, 2.3 Moon mass independent | Rocket to infinity, Moon no atm |
| Orbital speed | vo=√(GM/r)=√(gR²/r) 7.9 km/s at surface, ve=√2 vo | ISS 7.6 km/s 400 km |
| Satellite period & Energy | T=2π√(r³/GM) 85 min low, KE=GMm/2r PE=-GMm/r Total=-GMm/2r Binding=GMm/2r | GEO 24h 36000km TV |
| Types | GEO 36000km equatorial 24h comm, Polar LEO 500-800km scan | Weather, mapping |