high school physics Gravitation and Kepler's Laws

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. Change the eccentricity and 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 center distance r (not 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

exam-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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