In one line: gravitational potential energy: JEE/NEET Physics · Gravitation series · Part 5 of 8 · All parts →✪ Key points — the 30-second versionU = −GMm/r with U = 0.
JEE/NEET Physics · Gravitation series · Part 5 of 9 · All parts →
- U = −GMm/r — the minus sign means ‘trapped’ (you owe energy to escape)
- Zero is defined at infinite distance — that’s why values go negative
- mgh is just the small-distance shadow of this big formula
- Escape velocity is hiding inside this formula (½mv² = GMm/R)
- Potential (V = −GM/r) is per kilogram; energy (U) is for the whole mass
There’s a minus sign in gravity’s energy formula, and it decides marks. Remove it and every answer goes wrong. But the minus isn’t there to torture you — it has a beautifully simple meaning: negative energy = trapped. Part 5 of the Gravitation series, explained like a bank account.
- The bank account picture
- What each letter means
- Why the minus sign is forced, not chosen
- mgh: the honest small-distance shadow
- Solved examples
- Common mistakes
- This physics in your daily life
- Practice set
- Recap
The Bank Account Picture
Potential energy always needs a ‘zero’ chosen by us. Near the ground, you choose the ground as zero — that’s why mgh is positive when you climb. But for planets, ‘the ground’ is a bad choice (every planet has different ground!). So physics chooses the only universal zero: infinite distance apart = zero energy.
Now think like a bank account. Far away (at infinity): balance zero. Bring a satellite closer to Earth — gravity pulls it in, doing work for free — like receiving free money, your balance goes below zero (into debt). Closer = deeper debt. That debt is the minus sign:
And here’s the elegant connection: to escape, your throwing energy must exactly pay off the debt: ½mv² = GMm/R — which rearranges to v = √(2GM/R). Part 1’s escape velocity was a debt repayment all along!
What Each Letter Means
| Letter | What it means (plain words) | Value / unit |
|---|---|---|
| U | gravitational potential energy — the ‘debt’ stored in the pair of masses | unit: joules (J); negative by design |
| G | gravity’s fixed strength number | 6.67 × 10⁻¹¹ |
| M | mass of the big body (planet) | kg |
| m | mass of the small body (satellite) | kg |
| r | distance between the two centres | m |
Why the Minus Sign Is Forced, Not Chosen
Follow the steps: (1) We declared U = 0 at infinite distance. (2) Gravity is attractive — it pulls masses together for free, doing positive work as r shrinks. (3) Energy conservation then forces stored energy to drop below its zero mark as they come together. Result: at any finite distance, U is negative. The minus sign is bookkeeping for ‘gravity already paid; you’re in debt.’ Closer = more negative = deeper trap.
mgh: The Honest Small-Distance Shadow
Is mgh wrong? No — it’s a local approximation. Near the surface, the big formula expands to: U ≈ (a big negative constant) + mgh. The ‘big constant’ is invisible to us (we only measure changes), so we see just mgh. That’s why mgh works for a shelf and fails for a satellite: for a 400-km orbit, the full −GMm/r must be used.
Solved Examples
Apply U = −GMm/r with r = 2R: the debt halves.
Common-sense check: farther away = shallower trap = less negative. ✔
Answer: −U₀/2
Think: energy needed = final debt minus initial debt = (−GMm/2R) − (−GMm/R) = +GMm/2R.
Numbers: GMm/R = 6.67×10⁻¹¹ × 6×10²⁴ × 100 ÷ 6.4×10⁶ ≈ 6.25×10⁹ J → answer is half.
Common-sense check: ≈3 GJ ≈ 870 kWh — the right order for launch-scale budgets. ✔
Answer: ≈ 3.1 × 10⁹ J
Build each: KE = GMm/2r; |PE| = GMm/r; total = −GMm/2r.
The pattern: the debt is always exactly 2× the speed-energy, so the total is negative — trapped, with the debt winning by exactly 2:1.
This −2:+1 pattern is the whole story of Part 6. ✔
Answer: |PE| : KE : |total| = 2 : 1 : 1
- Assuming energy at the surface is zero. Zero lives at infinity! The surface sits at −GMm/R (deep debt). Every escape-energy question dies on this.
- Dropping minus signs mid-calculation. Write both debts with their signs, THEN subtract. One lost minus flips the whole answer.
- Using mgh at satellite heights. mgh assumes gravity never weakens — true for a shelf, false at 400 km.
- Mixing up U (joules, for the whole mass) with V (joules per kg, per unit mass). NEET swaps them in options to catch you.
This Physics in Your Daily Life
- The price of a rocket launch (₹3–5 lakh per kg to orbit) is literally this card: you’re paying cash to clear a gravitational debt of ~63 million joules per kilogram.
- Voyager 2’s free speed boost at Jupiter: it stole a tiny sliver of Jupiter’s orbital energy — transfers between gravity ‘bank accounts’ that mission designers trade like currency.
- Why the ISS slowly falls: air drag bleeds its energy — the debt deepens, the orbit shrinks. Re-supply craft re-pay the debt monthly.
- A black hole, in debt language: as r shrinks toward its critical radius, the debt goes to infinity — no amount of payment frees you. Part 8 tells that story.
Practice set (answers hidden — try first)
(NEET-level) At the surface, U = −63 MJ/kg. Energy per kg to escape:
(Concept) Two masses at distance r have U = −U₀. Pulled apart to 2r, U becomes:
(JEE Main-level) Energy to shift a satellite from orbit r to orbit 2r:
(Concept) Why is a satellite’s total energy negative?
(NEET-level) The potential (per kg) at a planet’s surface is −V₀. At height R above:
- 🧠 Bank picture: zero balance at infinity; coming closer puts you in DEBT (negative). Escape = pay the debt.
- 🧠 Sign rule: ‘negative = trapped, zero = free’ — say it before every energy question.
- 🧠 mgh is the shadow: fine for a shelf, wrong for a satellite.
- 🏠 Daily: launch prices (₹3–5 lakh/kg) are this debt in rupees — rockets are debt-collection machines.
- 🏠 Daily: Voyager 2’s free Jupiter boost was a transfer between two gravity bank accounts.
- U = −GMm/r; zero defined at infinite distance
- minus sign = trapped: pay +GMm/r to escape
- escape velocity lives inside: ½mv² = GMm/R → Part 1 reborn
- mgh = the small-distance shadow (works for shelves, not satellites)
- V = −GM/r is per kilogram; U is for the whole mass — exams swap them
Quick revision
- U = −GMm/r — the minus sign means ‘trapped’ (you owe energy to escape)
- Zero is defined at infinite distance — that’s why values go negative
- mgh is just the small-distance shadow of this big formula
- Escape velocity is hiding inside this formula (½mv² = GMm/R)
- Potential (V = −GM/r) is per kilogram; energy (U) is for the whole mass
- Why the minus sign is forced, not chosen
- 1Escape Velocity: The Speed That Ends Gravity’s Grip
- 2Orbital Velocity: Why the ISS Never Falls
- 3Kepler’s Laws: The 1609 Prediction Machine NASA Still Uses
- 4Angular Momentum: Gravity Can Pull, It Cannot Twist
- 5Gravitational Potential Energy: Why the Minus Sign Matters
- 6Satellite Energy: Why Total Energy Is Negative KE Over Two
- 7Variation of g: Why You Weigh Less at the Equator
- 8Black Holes, LIGO and Lagrange Points: Gravitation’s Research Frontier
- 9Gravitation Bonus: Field Intensity, Shell Theorem, Weightlessness and GEO Satellites
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