Week 9 — The force that shapes the cosmos
Picture a cannon on a tall mountain. You fire a ball. It falls toward Earth. Fire faster — it goes farther before landing. Fire even faster — it "falls around" the curve of Earth, never touching the ground. That's an orbit!
Newton imagined this exact thought experiment in 1687. A satellite in orbit is constantly falling toward Earth — but its forward speed is so great that the Earth curves away beneath it at the same rate. It never hits the ground.
Every mass attracts every other mass with a force proportional to the product of the masses and inversely proportional to the square of the distance.
The negative sign means gravity is attractive — you need to add energy (positive work) to escape.
For small heights above Earth's surface, PE ≈ mgh (the familiar formula). The −GMm/r form is needed for orbits and large distances.
A satellite orbits Earth at an altitude of 400 km (LEO). What is its orbital speed? (ME = 5.97 × 10²⁴ kg, RE = 6.37 × 10⁶ m)
Step 1 — Orbital radius:
Step 2 — Orbital speed:
✅ Speed ≈ 9.33 km/s (about 33,600 km/h!)
What is the escape velocity from Earth?
✅ Escape velocity ≈ 8.91 km/s (commonly rounded to 11.2 km/s at Earth's surface when accounting for atmospheric drag)
| Law | Statement |
|---|---|
| 1st (Ellipses) | Planets orbit in ellipses with the Sun at one focus |
| 2nd (Equal Areas) | A line from the Sun to a planet sweeps equal areas in equal times |
| 3rd (Periods) | T² = (4π²/GM) × r³ or T² ∝ r³ |
Kepler's laws describe planetary motion. GPS satellites use all three — their orbits are elliptical, they speed up near perigee (2nd law), and their orbital periods relate to their distance from Earth (3rd law). Without correcting for relativity, GPS would drift by ~10 km per day!
Problem 1: If you double the distance between two masses, the gravitational force becomes:
Problem 2: A satellite orbits at radius r. If the radius is quadrupled, the orbital speed becomes:
Problem 3: What is the escape velocity from a planet with half Earth's mass and half Earth's radius?
Q1: Why do astronauts in the ISS appear weightless if gravity is still ~90% as strong up there?
Astronauts aren't truly weightless — they're in free fall. The ISS and everything in it are falling toward Earth at the same rate, so there's no normal force. It's the same sensation you get in a rapidly descending elevator (before it stops). Weightlessness = free fall, not zero gravity!
Q2 (mini-problem): Two 100-kg spheres are 1 meter apart. What is the gravitational force between them?
Answer: 6.67 × 10⁻⁷ N — extremely small! That's why we don't feel gravity from everyday objects.