Playground
Two masses with gravitational field lines and inverse-square force arrow scaling.
Variables
| Symbol | Name | SI | Dimension | Range |
|---|---|---|---|---|
| Gravitational Forceoutput Attractive force between two masses | N | MLT^-2 | 0 – 100 | |
| Gravitational Constant Universal gravitational constant | m³·kg⁻¹·s⁻² | L³M⁻¹T⁻² | 6.674e-11 – 6.674e-11 | |
| Mass 1 Mass of the first body (e.g., planet) | kg | M | 1 – 1e+30 | |
| Mass 2 Mass of the second body (e.g., person) | kg | M | 0.1 – 1000 | |
| Distance Distance between the centers of the two masses | m | L | 1 – 100000000 |
Deep dive
Derivation
Newton postulated that the force causing an apple to fall and the force keeping the Moon in orbit are the same, proportional to both masses and inversely proportional to the square of the distance. Confirmed by showing the Moon's centripetal acceleration matches g/r² scaling from Earth's surface.
Experimental verification
Cavendish experiment (1798) measured G using a torsion balance. Modern measurements use atom interferometry.
Common misconceptions
- Gravity does not require contact — it acts at a distance
- Astronauts in orbit are not weightless — they are in freefall
- Gravity is not 'turned off' in space — it just weakens with distance
Real-world applications
- Satellite orbit calculations
- Tidal force predictions
- Planetary mass estimation
- GPS relativistic corrections
Worked examples
Your weight from first principles
Given:
- G:
- 6.674e-11
- M:
- 5.97e+24
- m:
- 70
- r:
- 6371000
Find: F
Solution
F = GMm/r² = 6.674e-11 × 5.97e24 × 70 / (6.371e6)² = 686.6 N ≈ 687 N
Mars surface gravity
Given:
- G:
- 6.674e-11
- M:
- 6.39e+23
- r:
- 3390000
Find: g_mars
Solution
g = GM/r² = 6.674e-11 × 6.39e23 / (3.39e6)² = 3.71 m/s²
Scenarios
What if…
- scenario:
- What if Earth were twice as massive?
- answer:
- Surface gravity doubles to 19.6 m/s². You'd weigh twice as much. Jumping height halves.
- scenario:
- What if you're at the ISS altitude (408 km)?
- answer:
- r = 6779 km. g = GM/r² = 8.67 m/s² — still 88% of surface gravity. Astronauts float because they're in freefall, not because gravity vanishes.
- scenario:
- What if G were 10× larger?
- answer:
- Stars would burn faster, planets would be denser, and life as we know it couldn't exist. G is finely tuned.
Limiting cases
- condition:
- r → ∞
- result:
- F → 0
- explanation:
- Gravity vanishes at infinite separation.
- condition:
- r → 0
- result:
- F → ∞
- explanation:
- The formula diverges — in reality, objects have finite size.
- condition:
- M or m → 0
- result:
- F → 0
- explanation:
- No mass means no gravitational attraction.
Context
Isaac Newton · 1687
Inspired by the falling apple and the Moon's orbit, Newton showed the same force governs both in Principia.
Hook
How much weaker is gravity on Mars compared to Earth? Calculate it from first principles.
Compare g on Mars (M=6.39e23 kg, r=3.39e6 m) vs Earth (M=5.97e24 kg, r=6.37e6 m) using F = GMm/r² to find surface gravity.
Dimensions:
- lhs:
- F → [MLT⁻²]
- rhs:
- G·M·m/r² → [L³M⁻¹T⁻²]·[M]·[M]·[L⁻²] = [MLT⁻²]
- check:
- Both sides are [MLT⁻²] = Newton. ✓
Validity: Valid for non-relativistic, weak-field gravity. Fails near black holes or at speeds approaching 299792458 m/s — use General Relativity instead.