Gravitation
Easy Overview
Why does an apple fall to the ground, but the Moon doesn't crash into Earth? Newton figured out it's the same force — gravity. This chapter is about the force that governs everything from falling objects to orbiting planets. It's also where Kepler figured out how planets actually move (spoiler: circles were wrong, ellipses are right).
Universal Law of Gravitation
Every object in the universe attracts every other object with a force. F = G m₁m₂ / r², where G = 6.67 × 10⁻¹¹ N·m²/kg². The force is always attractive, proportional to both masses, and inversely proportional to the square of the distance. It's called 'universal' because it works for apples, planets, and galaxies. Same rule. Everywhere.
Kepler's Laws — How Planets Actually Move
Kepler didn't know WHY planets moved the way they did — he just described it. Law 1: Planets orbit in ellipses with the Sun at one focus. Law 2: A line from Sun to planet sweeps equal areas in equal times (so they move faster when closer). Law 3: T² ∝ r³ — the square of the orbital period is proportional to the cube of the semi-major axis.
Gravitational Acceleration (g)
g = GM/R² — that's acceleration due to gravity at Earth's surface. It's about 9.8 m/s². But g varies: decreases as you go higher (g ∝ 1/r²), decreases as you go inside Earth (only the mass below you pulls), and is less at the equator (Earth bulges) than at poles. That's why your weight changes slightly depending on where you are.
Gravitational Potential Energy
Near Earth's surface, we use PE = mgh. But that's an approximation. The real formula is U = −GMm/r, where r is the distance between centers. It's negative because we set U=0 at infinity. As objects come closer, U becomes more negative (they lose potential energy, gain kinetic). Escape velocity comes from here: v_esc = √(2GM/R).
Satellites and Orbital Motion
A satellite stays in orbit because its centripetal force comes from gravity. GMm/r² = mv²/r, so orbital velocity v = √(GM/r). Geostationary satellites orbit at about 36,000 km above the equator, matching Earth's rotation. They look fixed in the sky — perfect for TV and weather. Polar satellites go over poles and scan the whole Earth.
Key Points
- •Gravitational force: F = G m₁m₂ / r² (always attractive)
- •G = 6.67 × 10⁻¹¹ N·m²/kg²
- •Kepler's 1st: elliptical orbits, 2nd: equal area in equal time, 3rd: T² ∝ r³
- •g varies with altitude, depth, and latitude
- •Escape velocity: v_esc = √(2GM/R) = 11.2 km/s for Earth
- •Orbital velocity: v₀ = √(GM/r)
- •Geostationary satellites have T = 24 hours, orbit above equator
Practice Questions
- Derive the relation between g and G.
- Show that the orbital velocity of a satellite is √2 times the escape velocity.
- State and explain Kepler's laws of planetary motion.
- At what height above Earth's surface does g become half its value on the surface?