3.2.2 · D5 · HinglishOrbital Mechanics & Astrodynamics
Question bank — Conservation of energy and angular momentum in gravitational field
3.2.2 · D5· Physics › Orbital Mechanics & Astrodynamics › Conservation of energy and angular momentum in gravitational
True or false — justify
A satellite in an elliptical orbit moves at constant speed.
False. Sirf constant hai. Jaise perihelion ki taraf chhota hota hai, drop karta hai toh (aur speed) badhti hai; speed maximum hoti hai perihelion par, minimum hoti hai aphelion par.
Angular momentum is conserved because gravity is weak.
False. Iska strength se koi lena-dena nahi. conserved hai kyunki gravity central hai (), jisse torque ban jaata hai.
Total mechanical energy of a bound orbit is negative.
True. Ek bound orbit mein hota hai (kyunki ). Negative sign ka matlab hai ki body ek potential well mein baithti hai aur uske paas infinity tak pahunchne ki energy nahi hai.
Two orbits with the same semi-major axis but different eccentricities have different total energies.
False. sirf par depend karta hai; eccentricity kabhi appear nahi karta. Same ⇒ same , chahe ellipse kitna bhi squashed ho.
The orbit is always confined to a single flat plane.
True. Kyunki ek fixed vector hai, aur dono hamesha uske perpendicular rehte hain, isliye motion us plane ko kabhi nahi chhod sakti jo centre se hoke ke perpendicular hai.
Kepler's 2nd law (equal areas in equal times) is a separate law you must add on top of energy and momentum conservation.
False. Yeh asliyat mein angular momentum conservation hi hai: , toh constant directly equal areas in equal times force karta hai.
If you double the speed at a given radius, the total energy doubles.
False. ; ko double karne se term quadruple ho jaata hai jabki unchanged rehta hai, isliye simply double nahi hota.
An orbit with is a circle.
False. ka matlab hai — ek parabola (borderline escape trajectory). Circle ek bound orbit hai jisme hota hai.
Along the whole orbit, is conserved.
False. continuously ke saath trade karta rehta hai. Sirf unka sum conserved hai; perihelion ke paas bada hota hai, aphelion ke paas chhota hota hai.
Spot the error
"As a planet falls toward the Sun, becomes more negative, so energy is being lost."
Error: energy sirf redistribute hoti hai, lost nahi hoti. Jaise drop karta hai, exactly utni hi amount se badhta hai, ko constant rakhte hue.
"The planet speeds up near the Sun, so its angular momentum increases there."
Error: constant hai. Sun ke paas chhota hota hai, isliye compensate karne ke liye badhta hai — tangential rate change hoti hai lekin fixed rehta hai.
"A more eccentric orbit is 'wilder', so it must carry more energy."
Error: energy par depend karti hai, par nahi. Same wale near-circular aur highly elongated orbit ki energy identical hoti hai.
"To find escape speed, set ."
Error: escape ke correspond karta hai, yani , jisse milta hai. set karna ek body ko rest mein describe karta hai, jo bilkul ulti situation hai.
" — both terms are attractive."
Error: term ek repulsive centrifugal barrier hai (positive, par badhta hai); sirf truly attractive hai. Unka yeh opposition potential well minimum create karta hai.
"At perihelion the whole speed is radial, so there."
Error: turning point par hota hai, isliye puri speed tangential hoti hai: . Yahi karan hai ki turning points use karne ki easy jagah hai.
"Vis-viva says ."
Error: correct form hai — galat version mein aur swap hain, jo galat signs bhi deta.
Why questions
Why does a central force guarantee a flat (planar) orbit?
Kyunki conserved hai aur direction mein constant hai; aur hamesha us fixed ke perpendicular rehne chahiye, jo puri motion ko ek hi plane mein pin kar deta hai.
Why is potential energy defined with a minus sign and referenced to infinity?
Hum set karte hain aur define karte hain, yeh woh work hai jo mass ko andar laane ke liye gravity ke against karna padta hai. Kyunki gravity andar ki taraf kheenchti hai, yeh deta hai, ek well jise tumhe climb out karna hoga.
Why does the vis-viva equation contain both and but not or explicitly?
Kyunki yeh ko ke saath combine karne se aata hai; energy aur ko already ke andar hide kar leti hai, isliye eliminate karne ke baad sirf (jahan tum ho) aur (orbit ka size) bachte hain.
Why is energy conserved for gravity but not, say, for a body sliding with friction?
Gravity conservative hai — uska work sirf start aur end positions par depend karta hai, isliye ek potential exist karta hai jisme ho. Friction ka work path length par depend karta hai, isliye aisa koi potential exist nahi karta aur mechanical energy heat ke roop mein leak ho jaati hai.
Why does the effective potential have a minimum, and what does that minimum represent?
Repulsive centrifugal term chhote par dominate karta hai aur attractive bade par dominate karta hai; unka yeh tug-of-war ek dip produce karta hai. Dip ka bottom us ke liye stable circular orbit ka radius hai.
Why can you determine the size of an orbit () from energy alone, without knowing the shape?
Kyunki energy aur ko one-to-one link karta hai; shape (eccentricity) alag se se set hoti hai. Toh size fix karta hai, shape fix karta hai.
Edge cases
What is the total energy and semi-major axis for a body launched at exactly escape speed?
exactly, aur formally (from ). Path ek parabola hai — yeh barely infinity tak pahunchta hai, zero speed ke saath.
What happens to the "orbit" if ?
ke saath centrifugal barrier vanish ho jaata hai aur koi tangential motion nahi hoti; body seedha ek radial line ke saath centre mein gir jaati hai (ya seedha bahar nikalti hai) — ek degenerate, purely radial "orbit."
For a hyperbolic flyby, what are the signs of and ?
(unbound, escape ke liye zyada energy) aur , kyunki tabhi positive ho sakta hai jab negative ho — ek formal bookkeeping value, physical length nahi.
At the turning points of an ellipse, what fraction of the speed is radial?
Zero — perihelion aur aphelion par hota hai, isliye velocity puri tangential hai, . Yeh extremal radius hai, jahan momentarily change karna band kar deta hai.
As on a bound orbit, does the body ever actually reach infinity?
Nahi. Ek bound orbit mein hota hai, isliye ek finite maximum radius (aphelion) hai jahan aage jaane ke liye ko negative jaana padta — jo impossible hai. Body wapas turn kar leti hai.
In the limit , what does the ellipse become, and what stays true?
Yeh ek circle ban jaata hai jisme har jagah hota hai; speed constant ho jaati hai, aur vis-viva collapse karke ban jaata hai. Yahaan bhi, aur abhi bhi hold karte hain.
If a rocket burn changes only the direction of velocity (not speed) at some point, does the total energy change?
Nahi — speed aur radius par depend karta hai, direction par nahi. Fixed aur fixed par velocity redirect karne se unchanged rehta hai lekin generally aur isliye orbit ki shape alter ho jaati hai.
Recall Self-test protocol
Har item par wapas jao aur jinhe tum miss kiye, unke liye governing equation (, , vis-viva, ya ) apni galat reasoning ke saath likhlo. Inhein traps ka ninety percent us moment dissolve ho jaata hai jab tum name karo ki kaunsa conservation law kaam kar raha hai.