Gravitation
Gravity shapes the cosmos. It holds planets in orbit, keeps our feet on the ground, and governs the life and death of stars.
Start with the simplest version: this lesson is about Gravitation. If you can explain the core idea to a friend using everyday language, examples, and one clear reason why it matters, you have moved from memorising to understanding.
Gravity shapes the cosmos. It holds planets in orbit, keeps our feet on the ground, and governs the life and death of stars. Yet gravity is the weakest of nature's forces—so weak that the gravity of a small magnet can lift a paper clip against Earth's entire gravitational pull. Newton's law of universal gravitation reveals that the same force holding you down holds the moon in orbit around Earth.
Kepler's Laws
Before Newton, Kepler observed planetary motion and discovered three laws purely from data:
First Law: Planets orbit the sun in ellipses, with the sun at one focus.
Second Law (areal velocity): The line from the sun to a planet sweeps equal areas in equal times. Equivalently, dA/dt = L/(2m) is constant — a consequence of conservation of angular momentum under a central force.
Third Law: T² ∝ r³ for any orbiting body around the same central mass. Newton later derived this from the inverse-square law.
Newton's Law of Universal Gravitation
F = G m₁m₂ / r², with G ≈ 6.67 × 10⁻¹¹ N·m²/kg². The force is always attractive, acts along the line joining the masses, and obeys Newton's third law. The inverse-square form means doubling r reduces F by a factor of 4.
Gravitational Constant G and Acceleration g
G is small, which is why gravity is weak. At Earth's surface, the gravitational pull on mass m is GMm/R², so the free-fall acceleration is g = GM/R². Above the surface at height h: g(h) = GM/(R+h)² ≈ g(1 − 2h/R) for h ≪ R. Below the surface: g(d) = g(1 − d/R) (for uniform Earth model).
Gravitational Potential and Potential Energy
For two point masses, U(r) = −G m₁m₂ / r, taking U(∞) = 0. Near Earth's surface this reduces to ΔU ≈ mgh. The gravitational potential at a point is V(r) = −GM/r — the work done per unit mass to bring a test mass from infinity to that point.
Orbital and Escape Velocity
For a circular orbit of radius r about a body of mass M, gravity supplies the centripetal force: GMm/r² = mv²/r, so v_orb = √(GM/r). The total mechanical energy is E = −GMm/(2r) (bound orbit, negative).
Escape velocity is set by E = 0: ½mv_e² − GMm/R = 0, giving v_e = √(2GM/R) = √2 · v_orb(R) ≈ 11.2 km/s for Earth.
Satellites and Geostationary Orbits
Kepler's third law for satellites: T² = (4π²/GM) r³. A geostationary satellite has T = 24 h and orbits in the equatorial plane at r ≈ 4.22 × 10⁷ m (height ≈ 36 000 km).
Socratic Questions
- Why is gravitational force so much weaker than electromagnetic force, yet gravity dominates the large-scale structure of the universe?
- If you're orbiting Earth in a spacecraft, you experience "zero gravity". Yet Earth's gravity is still pulling you toward Earth. Why don't you feel it?
- Why does Kepler's second law (equal areas in equal times) follow directly from any central force, not just gravity?
- Escape velocity depends on only the central body's mass and radius, not on the escaping object. Why doesn't the object's mass matter?
- Geostationary satellites orbit above the equator every 24 hours. How would the orbital radius change if you wanted a 12-hour period instead?