Gravitation and Kepler's Laws

The force that drops an apple and the force that keeps the Moon in orbit are the same universal gravitation: every mass attracts every other, falling with the square of distance. Kepler's laws say orbits are ellipses, equal areas are swept in equal times, and T² is proportional to a³. From the gravity formula you can also derive satellite orbital speed. Slide eccentricity here to watch the elliptical path and the equal-area rule.

Intuition for Gravitation

🌍 Newton's Apple and the Moon

①The force pulling an apple down and the force keeping the Moon in orbit are the same!

②Every mass attracts every other mass — gravitation

③Greater distance, weaker force (inverse square of distance)

Visualizing Kepler's Laws

Eccentricity e (×100)40%

💡 Kepler's Three Laws

①Law 1 (orbit): planets orbit the Sun on ellipses with the Sun at one focus

②Law 2 (area speed): equal areas swept in equal times (faster at perihelion!)

③Law 3 (harmonic): T² ∝ a³ (period² ∝ semi-major axis³)

Gravitation Formula

Law of Universal Gravitation

F = GMmr²

G = 6.67×10⁻¹¹ N·m²/kg², M: central body mass, m: orbiting mass

Gravity from Universal Gravitation

g = GMR²

Surface gravity = GM/R² ≈ 9.8 m/s²

Kepler's Third Law

Kepler's Third Law (Harmonic)

T² = 4π²GMa³

Period² ∝ semi-major axis³ — derivable from gravitation!

🛰️ Application to Satellites

①Satellite centripetal = gravitational: mv²/r = GMm/r²

②Orbital speed: v = √(GM/r) — depends only on altitude

③Geostationary: T = 24 h → r ≈ 36,000 km altitude

Worked Examples

Example 1

If the distance between two objects is tripled, by what factor does the gravitational force change?

Gravity is F = GMm/r², inversely proportional to the square of the distance.

F = GMmr²

If r triples, r² becomes 9× → F is 1/9.

r→3r ⇒ F ∝ 1(3r)² = 19F

▸ 1/9

Gravity follows the inverse-square law. If distance is n×, the force is 1/n².

Example 2

Planet A has an orbital period 8 times that of planet B. By what factor is A’s semi-major axis larger? (Kepler’s third law)

From Kepler’s third law T² ∝ a³, so a ∝ T⁽2/3).

T² ∝ a³ ⇒ a ∝ T^2/3

Since T is 8×, a is 8⁽2/3) = 4×.

a_A/a_B = 8^2/3 = (2³)^2/3 = 4

▸ 4×

From T² ∝ a³, the axis ratio is the 2/3 power of the period ratio. Since 8 = 2³, 8⁽2/3) = 4 exactly.

Summary

Gravitation

F = GMmr²

Kepler 3rd

T² ∝ a³

CSAT-style

If the surface gravitational acceleration is g, what is the acceleration at a height of two Earth radii (distance 3R from the center)?

①g2

②g3

③g4

④g9

⑤g6

▸ ④ g9

Gravitational acceleration is g = GM/r², inversely proportional to the square of the distance from the center.

g = GMr²

Surface is r = R; the point is r = 3R, so g is 1/9.

g(3R) = GM(3R)² = g9

🎯 Exam Points

①Gravitation: F = GMm/r² (inverse square of distance)

②Orbital speed: v = √(GM/r) — higher orbit, slower

③Kepler 1: elliptical orbit (Sun at focus)

④Kepler 2: equal areas (faster near perihelion)

⑤Kepler 3: T² = (4π²/GM)a³ — period² ∝ a³

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Uniform Circular Motion

Projectile Motion

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