Gravitational potential energy — U = −GMm - r (not mgh): Foundations

1.2.22 · D1 · Physics › Newton's Laws & Dynamics › Gravitational potential energy — U = −GMm - r (not mgh)

Pehle $U = - \frac{GM m}{r}$ ko padhne se pehle, tumhe isme har ek squiggle aur unke peeche chupi har ek idea ko pehchaannaا chahiye. Hum unhe order mein banate hain — har ek sirf unhi ko use karta hai jo pehle aaye hain.

1. Distance $r$ — woh ek number jis par sab kuch depend karta hai

Pehla figure dekho. Ek bada planet centre mein baitha hai. Ek chota ball usse door float kar raha hai. Do centres ko jodne wali akeli peeli line $r$ hai.

Figure — Gravitational potential energy — U = −GMm - r (not mgh)

$r$ hamesha positive hota hai aur kabhi zero nahi (do centres ek hi point par nahi ho sakte).
Chhota $r$ = paas saath. Bada $r$ = door alag. $r \to \infty$ = infinitely door, "escaped".

2. Mass $M$ aur mass $m$ — do players

Yeh distinction sirf bookkeeping ke liye hai — dono masses asliyat mein ek doosre ko equally khichti hain. Bas hume chhote wale ko bade ke field mein move karte picture karna zyada aasaan lagta hai.

3. Gravitational constant $G$

4. Force $F$ aur arrow jiska matlab "pull" hai

Gravity ki force hamesha ek inward pull hoti hai, bade mass ki taraf. Neeche figure mein ball par lal arrow planet ki taraf wapas point karta hai — kabhi door nahi.

Abhi ke liye, bas yeh pakad ke rakho: gravity ek vector hai jo inward point karta hai, aur uski strength distance $r$ par depend karti hai. Hum abhi poora formula likhne ke liye taiyaar nahi hain — uske do symbols ( $\overset{r}{^}$ aur $r^{2}$ neeche ka meaning) ka abhi koi picture nahi hai. Hum unhe §5 aur §6 mein banate hain, aur tabhi complete law assemble karte hain.

5. Unit vector $\overset{r}{^}$ — ek pure "which way" arrow

6. $r^{2}$ neeche kyun — inverse-square idea

Denominator mein woh $r^{2}$ wajah hai ki gravity ki ek special, non-flat shape hai — aur precisely isliye potential energy $1/ r$ niklegi na ki kuch simpler. Uss thought ko §9 ke liye pakad ke rakho.

Ab har symbol ka ek picture hai, isliye hum finally Newton's Law of Universal Gravitation se complete force law padh sakte hain: $F = - \frac{GM m}{r ^{2}} \overset{r}{^} .$ Piece by piece decode karo: strength hai $GM m / r^{2}$ (constant $G$ × do masses, inverse-square $1/ r^{2}$ se kamzor ki gayi), aur direction hai $- \overset{r}{^}$ (inward, §5 se). Yahaan har symbol upar earn kiya gaya tha.

7. Tiny step $d r$ aur dot product $F \cdot d r$

Parent note $F \cdot d r$ ka integral compute karta hai. Do naye pieces yahaan hain — tiny step $d r$ , aur dot.

8. Work $W$ aur potential energy $U$ — effort ko ek number mein bottle karna

9. Integral $\int$ — ek changing pull ko add karna

Pull ki strength change hoti hai jab $r$ change hota hai (woh hai $1/ r^{2}$ ). Poori journey par total work karne ke liye hum ek baar multiply nahi kar sakte — hum path ko tiny steps $d r$ mein slice karte hain, jinme se har ek par force almost constant hai, aur saare slivers add kar dete hain. Woh "infinitely many tiny pieces add karo" machine hai integral $\int$ .

Tumhe is topic ko padhne ke liye integrals karna nahi aana chahiye — tumhe dekhna aana chahiye ki $\int_{\infty}^{r}$ ka matlab hai "infinitely door se distance $r$ tak saare tiny bits of work sum karo," aur $1/ r^{2}$ sum karne se $- 1/ r$ milta hai.

10. Height $h$ , planet radius $R$ , aur reference point

Prerequisite map

Equipment checklist

Recall Self-test: kya tum padhne se pehle har ek ka jawab de sakte ho?

$r$ kya measure karta hai, aur kahaan se? ::: Do masses ke centres ke beech ki seedhi-line distance (surface se height nahi). $r$ kabhi zero kyun nahi ho sakta, aur $r \to 0$ par $U$ ka kya hota hai? ::: Domain $r > 0$ hai; $r = 0$ dono centres ko ek point par rakh dega. Jab $r \to 0$ , $U = - GM m / r \to - \infty$ — ek singularity (infinitely gehra well). $\overset{r}{^}$ kya hai aur kitna lamba hai? ::: Length bilkul 1 ka ek pure-direction arrow, centre se outward point karta hai. Gravity ki force $F = - \frac{GM m}{r ^{2}} \overset{r}{^}$ mein minus sign kyun carry karta hai? ::: Pull inward ( $- \overset{r}{^}$ ) hai jabki $\overset{r}{^}$ outward point karta hai, isliye direction flip ho jaati hai. $G$ aur $g$ mein difference? ::: $G$ universal constant hai (har jagah same); $g$ ek local field strength hai jo $G$ , ek planet ki mass aur radius se compute ki jaati hai. $d r$ kya hai? ::: Path ke saath ek tiny vector step — ek microscopic footstep jiska ek chota length aur ek direction hai. Full dot-product rule aur yeh kya extract karta hai? ::: $A \cdot B = ∣ A ∣∣ B ∣ cos θ$ ; yeh ek arrow ka woh hissa extract karta hai jo doosre ke saath lie karta hai, $cos θ$ se sign ke saath. $U$ exist karne ke liye gravity conservative kyun honi chahiye? ::: Sirf tabhi work sirf endpoints par depend karta hai, isliye ek single position-number $U (r)$ isse describe kar sakta hai. Plain force $\times$ distance ki jagah integral kyun chahiye? ::: Force $r$ ke saath change hoti hai; integral infinitely many tiny constant-force slivers sum karta hai. $\int r^{- 2} d r$ kya hai aur kyun? ::: $- 1/ r$ , power rule se ( $- 2$ ko $- 1$ tak badhao, $- 1$ se divide karo); checked kyunki $\frac{d}{d r} (- 1/ r) = 1/ r^{2}$ . $- GM m / r$ hamesha negative kyun hai lekin $m g h$ positive ho sakta hai? ::: Alag chosen zero points — pehle ke liye $r = \infty$ (uske neeche sab kuch negative), doosre ke liye ek arbitrary floor. $r = R + h$ ka kya matlab hai? ::: Centre-to-surface distance $R$ plus surface ke upar height $h$ = poora centre-to-centre distance $r$ .

Connections

यह टॉपिक Hinglish में →
Newton's Law of Universal Gravitation — woh force $F = - GM m / r^{2} \overset{r}{^}$ supply karta hai jise yeh symbols assemble karte hain.
Conservative Forces and Potential Energy — woh property jo $U$ ko exist karne deti hai.
Work-Energy Theorem — work, kinetic energy aur $Δ U$ ko link karta hai.
Acceleration due to gravity g — in pieces se bana local $g = GM / R^{2}$ .
Escape Velocity — finished formula ka pehla bada payoff.
Kepler's Laws & Orbital Energy — orbital energy $- GM m /2 r$ use karta hai.