JEE/NEET Physics · Gravitation series · Part 3 of 9 · All parts →
- Rule 1: planets move in slightly squashed circles (ellipses), Sun at one side
- Rule 2: planets move faster when closer to the Sun (like a ball rolling in a bowl)
- Rule 3: T² ∝ r³ — farther orbit = MUCH longer year
- The 15-second trick: r becomes 4× → year becomes 8×
- These 1609 rules still guide NASA today
In 1609, a man with no telescope predicted the motion of every planet using three simple rules. NASA still uses them to plan missions today. Kepler’s laws look like boring history — they’re actually three of the easiest marks in NEET and JEE Main. Part 3 of the Gravitation series, told simply.
- Rule 1: The squashed circle
- Rule 2: Faster near the Sun
- Rule 3: Farther = much longer year
- The 15-second exam trick
- What each letter means
- Solved examples
- Common mistakes
- This physics in your daily life
- Practice set
- Recap
Rule 1: The Squashed Circle
Everyone pictures orbits as perfect circles. They’re not — they’re ellipses: circles gently squashed, like a slightly oval roti. And the Sun doesn’t sit in the middle — it sits slightly off-centre, at a point called a focus.
What this means: a planet’s distance from the Sun keeps changing through the year. Earth is closest in January (147 million km) and farthest in July (152 million km). A perfect circle is just the special case where the squash is zero.
Rule 2: Faster Near the Sun
Picture a ball rolling in a large round bowl. Near the edges (high up) it rolls slowly. Near the centre it whizzes fast. Planets do the same: closer to the Sun = faster; farther = slower.
Earth runs at 30.3 km/s in January and 29.3 km/s in July. The reason (angular momentum) is Part 4‘s story.
Rule 3: Farther = Much Longer Year
| Letter | What it means (plain words) | Value / unit |
|---|---|---|
| T | the time for one complete round — the planet’s ‘year’ | Earth: 1 year; Jupiter: 12 years |
| r | the average distance from the Sun (centre to centre) | Earth: 1.5×10¹¹ m |
| M | mass of the big central body being orbited | Sun: 2×10³⁰ kg |
| G | gravity’s fixed strength number | 6.67 × 10⁻¹¹ |
Why does distance matter SO much? Two reasons stack up: a bigger orbit is a longer track AND the planet moves slower on it (Part 2’s rule). Longer track + slower speed = much, much more time. That’s why the r is raised to the power 1.5.
The 15-Second Exam Trick
Almost every exam question on Rule 3 is a comparison. Don’t compute full years — compare: year ratio = (distance ratio)^1.5. Memorise three: ×4→×8, ×9→×27, ×2→×2.83.
Solved Examples
Apply the trick: 4^1.5 = 4 × √4 = 4 × 2 = 8.
Reality check: Jupiter orbits ~5× farther out and takes 12 years — same pattern. ✔
Answer: 8 Earth-years
Flip the power: distance ratio = 8^(2/3) = (2³)^(2/3) = 2² = 4.
Remember: distance→year uses power 1.5; year→distance uses power 2/3. Flip one, flip the other.
Answer: 4× farther
Rearrange: M = 4π²r³/(GT²).
Piece by piece: r³ = 3.4×10³³; T² = 9.9×10¹⁴; 4π² ≈ 39.5.
Combine: M = 39.5 × 3.4×10³³ ÷ (6.67×10⁻¹¹ × 9.9×10¹⁴) ≈ 2×10³⁰ kg.
Think about this: this IS how the Sun’s mass is known — a timer and a ruler. You just weighed a star with Class 11 maths. ✔
Answer: M ≈ 2×10³⁰ kg
- Comparing across different centres. The trick only works for bodies orbiting the same big mass. Earth vs Mars (both around Sun): fine. A Moon-orbiter vs a Sun-orbiter: not allowed — different M.
- Thinking year grows in proportion to distance. Distance ×2 does NOT mean year ×2 — it means ×2.83. The extra comes from the speed also dropping.
- Plugging full numbers when a comparison works. G and M cancel inside one system. Ratio first, numbers only if forced.
This Physics in Your Daily Life
- Neptune was found on paper in 1846: Uranus’s orbit misbehaved slightly; mathematicians asked ‘what unseen planet’s pull would do this?’, calculated where to look — and Neptune was there within a degree. Maths discovered a planet.
- Over 5,800 known exoplanets were found through timing patterns — Kepler’s rule converts those timing signals into distances, and distances into ‘could this planet hold water?’
- Every space mission’s route — Chandrayaan, Mangalyaan, JWST — starts with these three rules as the skeleton.
- Your birthday is slightly seasonal: Earth moves fastest in January — days tick by measurably faster near perihelion (the effect is tiny, but it’s real and it’s Rule 2).
Practice set (answers hidden — try first)
(NEET-level) Two planets: distance ratio 1:9. Year ratio:
(Concept) A planet moves fastest when it is:
(NEET-level) A planet’s year is 64 years. Its distance from the same star, vs a 1-year planet at 1 AU:
(Concept) Can you compare the Moon’s orbit (around Earth) with Earth’s orbit (around Sun) using T² ∝ r³?
(JEE Main-level) A satellite’s orbital radius is 4× another’s (same planet). Period ratio:
- 🧠 Power 1.5 chant: ‘distance ×4 → year ×8; ×9 → ×27’ — the three perfect cases cover 90% of questions.
- 🧠 Bowl picture: planets roll in a bowl — slow at the edges, fast at the centre. That’s Rule 2 forever.
- 🧠 Same-boss rule: the T²∝r³ trick only compares planets orbiting the SAME boss (same central mass).
- 🏠 Daily: Neptune was found on paper in 1846 — the maths pointed the telescope. Over 5,800 exoplanets found the same way since.
- 🏠 Daily: your birthday in January comes marginally faster — Earth runs quickest at its closest point to the Sun.
- Rule 1: ellipse (squashed circle), Sun off-centre
- Rule 2: faster near the Sun, slower far away
- Rule 3: T² ∝ r³ — same central body only
- exam trick: year ratio = (distance ratio)^1.5; ×4→×8, ×9→×27
- M = 4π²r³/GT² — weigh any central body with a timer and ruler
Quick revision
- Rule 1: planets move in slightly squashed circles (ellipses), Sun at one side
- Rule 2: planets move faster when closer to the Sun (like a ball rolling in a bowl)
- Rule 3: T² ∝ r³ — farther orbit = MUCH longer year
- The 15-second trick: r becomes 4× → year becomes 8×
- These 1609 rules still guide NASA today
- Rule 1: The squashed circle
- 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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