The complete lecture — every idea comes alive in the live panel on the right as you read. Scroll down; the animation keeps pace, and in the free-fall and Newton's-cannon sections the panel swaps in the real 3D experiments you can run yourself.
1 — The mango and the Moon
A mango drops from a tree in Hyderabad. The Moon circles overhead. Newton's great insight: the same force does both. Every body attracts every other body.
- Law of universal gravitation — F is directly proportional to the product of the masses and inversely proportional to the square of the distance between their centres.
The lawF = G m₁m₂ / r²
G = 6.67 × 10⁻¹¹ N m² kg⁻²
The force is always attractive, mutual (the mango pulls the Earth too — third law!) and acts centre-to-centre.
2 — Inverse square: double the distance, quarter the force
Watch the panel: as the blue mass slides away, the force arrows collapse — at 2r the force is F/4, at 3r only F/9.
| Separation | Force |
|---|
| r | F |
| 2r | F/4 |
| 3r | F/9 |
Exam trick: double both masses AND the distance → F′ = G(2m₁)(2m₂)/(2r)² = F — unchanged.
3 — Cavendish's torsion balance: G vs g
Gravity between lab-sized objects is absurdly weak, so Henry Cavendish (1798) hung two small lead balls on a fibre and brought big spheres near. The fibre's tiny twist measured the force — and gave G. With G known, he had "weighed the Earth".
| G | g |
|---|
| What | universal constant | local acceleration |
| Value | 6.67 × 10⁻¹¹ N m² kg⁻² | 9.8 m/s² (surface) |
| Changes? | never, anywhere | with altitude, depth, latitude, planet |
4 — Finding g & weighing the Earth (live 3D)
Weight is gravitation: set mg equal to the universal law and the body's mass cancels —
g and the Earth's massmg = GMm/R² ⟹ g = GM/R²
M = gR²/G = 9.8 × (6.4 × 10⁶)² / 6.67 × 10⁻¹¹ ≈ 6.0 × 10²⁴ kg
That is why a stone and a coin land together — drop the ball in the 3D panel and check. On the Moon g ≈ 1.6 m/s² (g/6): hammer and feather fall together there too.
worked — force between two students
60 kg and 50 kg, 1 m apart?
F = 6.67 × 10⁻¹¹ × 60 × 50 / 1² = 2.0 × 10⁻⁷ N — about 1% of a mosquito's weight.
5 — How g varies: altitude, depth, latitude
Watch the probe in the panel: g grows linearly from zero at the centre, peaks at 9.8 m/s² at the surface, then dies off as the inverse square above.
The three formulasaltitude: g_h = gR²/(R+h)² ≈ g(1 − 2h/R) for h ≪ R
depth: g_d = g(1 − d/R) → zero at the centre
latitude: max at poles (9.83), min at equator (9.78) — bulge + spin
worked — g on Mount Everest
h = 8848 m.
g_h = 9.80(1 − 2 × 8848/6.4 × 10⁶) = 9.77 m/s² — only 0.3% less.
6 — Free fall & weightlessness
A weighing scale reads the normal reaction, not mg. In a lift accelerating up you read m(g + a); accelerating down, m(g − a); and if the cable snaps — zero.
| Lift | Scale reads |
|---|
| at rest / uniform velocity | mg |
| accelerating up | m(g + a) |
| accelerating down | m(g − a) |
| free fall (a = g) | 0 — weightless |
An astronaut floating in a satellite is in that snapped-cable lift permanently: astronaut and capsule fall together, so nothing presses on anything. Gravity at ISS height is still ≈ 8.7 m/s² — weightlessness is not "no gravity".
7 — Newton's cannon: a satellite is a projectile (live 3D)
Newton's thought experiment: fire a cannonball horizontally from a high peak. Faster → it lands farther. Fast enough (7.9 km/s) → the ground curves away exactly as fast as the ball falls. It never lands.
The idea: a satellite is a projectile that keeps missing the Earth. Launch the 3D projectile on the right and imagine raising its speed until the landing point runs all the way around the globe.
8 — Orbital velocity
For a circular orbit, gravity is the centripetal force. For a satellite skimming the Earth:
Orbital velocitymg = mv²/R ⟹ v = √(gR) = √(9.8 × 6.4 × 10⁶)
v ≈ 7.9 km/s · period T = 2πR/v ≈ 84 min
Note the m cancels: a tiny CubeSat and the 420-tonne ISS orbit at the same speed at the same height. Higher orbit → weaker gravity → slower satellite.
worked — low-orbit speed
v and T just above the surface?
v = √(6.27 × 10⁷) = 7.9 × 10³ m/s · T = 2πR/v ≈ 84 min
9 — Geostationary satellites & PAKSAT-1R
Make the period exactly 24 hours, orbit above the equator, west-to-east — and from the ground the satellite never moves. Your TV dish points at it once, forever.
Geostationary numbersT = 86 400 s → R + h ≈ 4.23 × 10⁷ m
height h ≈ 36 000 km · v ≈ 3.1 km/s
- PAKSAT-1R — Pakistan's communication satellite (launched 2011) parked at 38°E, relaying TV, internet and telephone across the country.
- Three cover the globe — geostationary satellites 120° apart see almost the whole Earth (Arthur C. Clarke, 1945).
10 — Worked numericals & exam recap
numerical — mass of the Earth
g = 9.8, R = 6.4 × 10⁶ m.
M = gR²/G = 9.8 × 4.096 × 10¹³ / 6.67 × 10⁻¹¹ = 6.0 × 10²⁴ kg
numerical — g at height h = R
What is g one Earth-radius up?
g_h = gR²/(2R)² = g/4 = 2.45 m/s² (exact formula — never the 2h/R approximation here)
- F = Gm₁m₂/r² — attractive, mutual, centre-to-centre; G = 6.67 × 10⁻¹¹ N m² kg⁻².
- Cavendish's twist measured G; g = GM/R² — independent of the falling mass.
- M_E = gR²/G ≈ 6.0 × 10²⁴ kg; mean density ≈ 5.5 × 10³ kg/m³.
- g: ↓ with altitude (1/r²), ↓ with depth (linear, 0 at centre), max at poles.
- Weightlessness = body and support in free fall together — gravity is NOT zero.
- v = √(gR) ≈ 7.9 km/s, T ≈ 84 min; geostationary: 24 h, 36 000 km, equator — PAKSAT-1R.