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Grade 11-12 (age 16-18)

high school physics Gravitation and Kepler's Laws

The force that makes an apple fall and the force that keeps the Moon around the Earth are the same universal gravitation. Every object with mass attracts every other, and the force is inversely proportional to the square of the distance. Kepler's laws are three: a planet moves on an ellipse with the Sun at one focus, it sweeps out equal areas in equal times, and the square of its period is proportional to the cube of its semi-major axis. All of these laws can be derived from universal gravitation, and the orbital speed of a satellite comes out the same way. In the picture, changing the eccentricity flattens the orbit, and the planet's positions marked at equal time intervals spread farther apart on the side near the Sun.

Intuition for Universal Gravitation

🌍 Newton's Apple and the Moon
①The force that makes an apple fall and the force that holds the Moon are the same kind of force
②Every object with mass attracts every other object; this force is called universal gravitation
③Just as doubling the distance makes the force 1/4 as large, gravitation is inversely proportional to the square of the distance

Kepler's Laws in a Picture

e = 0.40
💡 Kepler's Three Laws
①First law (law of ellipses): a planet moves on an ellipse with the Sun at one focus
②Second law (law of equal areas): equal areas are swept out in equal times, so the planet is fast at perihelion, near the Sun, and slow at aphelion, far from it; that is why the green dots are farther apart on the perihelion side
③Third law (law of periods): the square of the orbital period is proportional to the cube of the semi-major axis (T² ∝ a³)

The Law of Universal Gravitation

Law of Universal Gravitation
F = GMmr²
G = 6.67 × 10⁻¹¹ N·m²/kg², M: mass of the central body, m: mass of the orbiting body, r: distance between their centers
Gravitational Acceleration and Universal Gravitation
g = GMR²
The gravitational acceleration at the surface is GM/R², about 9.8 m/s²

Kepler's Third Law

Kepler's Third Law (Law of Periods)
T² = 4π²GMa³
The square of the period is proportional to the cube of the semi-major axis; setting centripetal force equal to gravitation for a circular orbit gives this equation
🛰️ Applying It to Satellites
①For a satellite in a circular orbit, gravitation provides the centripetal force, so mv²/r = GMm/r²
②The orbital speed v = √(GM/r) depends only on the distance r from the Earth's center, not on the satellite's mass
③A geostationary satellite has a period of 24 hours, so it orbits about 36,000 km above the surface

Worked Examples

Example 1
If the distance between two objects is tripled, by what factor does the gravitational force change?
1
Universal gravitation is F = GMm/r², so it is inversely proportional to the square of the distance.
F = GMmr2
2
If r becomes 3 times larger, r² becomes 9 times larger, so F becomes 1/9 as large.
r→3r ⇒ F ∝ 1(3r)2 = 19F
▸ 1/9
When the distance becomes n times larger, the force becomes 1/n² as large. A common slip is to answer 1/3 when the distance is tripled.
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)
1
From Kepler's third law T² ∝ a³, a ∝ T2/3.
T2 ∝ a3 ⇒ a ∝ T2/3
2
T is 8 times larger, so a is 82/3 = 4 times larger.
aA/aB = 82/3 = (23)2/3 = 4
▸ 4×
Raising the ratio of periods to the 2/3 power gives the ratio of semi-major axes. Since 8 = 2³, 82/3 = 2² = 4 comes out cleanly.

Summary

Gravitation
F = GMmr²
Kepler's Third Law
T² ∝ a³
CSAT-style problem, adapted
If the gravitational acceleration at the surface is g, what is it at a height of 2 Earth radii (a distance of 3R from the center)?
①g2
②g3
③g4
④g9
⑤g6
▸ ④ g9
1
The gravitational acceleration is g = GM/r², inversely proportional to the square of the distance from the Earth's center.
g = GMr2
2
At the surface r = R and at the point in question r = 3R, so g becomes 1/9 as large.
g(3R) = GM(3R)2 = g9
🎯 Exam Points
①Universal gravitation F = GMm/r² is inversely proportional to the square of the distance
②Since the orbital speed is v = √(GM/r), a satellite orbiting higher moves more slowly
③Kepler's first law: a planet moves on an ellipse with the Sun at one focus
④Kepler's second law: the areal speed is constant, so the planet is fast at perihelion and slow at aphelion
⑤Kepler's third law: T² = (4π²/GM)a³, so the square of the period is proportional to the cube of the semi-major axis
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Projectile Motion

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