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Class XI · Physics · Gravitation · Lecture

Gravitation

The full lecture, side by side with a live panel. Read the text on the left; on the right every idea is animated through a real-life picture — an apple and the Moon, a person weighing differently on three worlds, an astronaut floating, Newton's cannon, PAKSAT over Pakistan. Press ▶ and the narration plays straight through, scrolling and animating each section for you.

Newton asked a simple question: if an apple (or a mango in Hyderabad) falls because the Earth pulls it, does that same pull reach all the way to the Moon? His answer became one of the great laws of physics: every body attracts every other body.

  • Law of universal gravitation — the force is directly proportional to the product of the two 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 (universal gravitational constant)

The force is always attractive, never repulsive; it is mutual — the apple pulls the Earth with the same force the Earth pulls the apple (Newton's third law) — and it acts centre-to-centre along the line joining the bodies. For uniform spheres the whole mass behaves as if it sits at the centre, so r is measured between centres, in metres.

The 1/r² factor is the heart of the law. Watch the panel: as the masses move apart the force arrow collapses. Double the separation and F drops to a quarter; triple it and F is a ninth.

SeparationForce
rF
2rF/4
3rF/9
r/24F

Newton checked this against the Moon, about 60 Earth-radii away: its falling acceleration should be g/60² ≈ 0.0027 m/s² — exactly what the orbit needs. The apple and the Moon obey the same law.

Exam point: if both masses are doubled and the distance is doubled, F′ = G(2m₁)(2m₂)/(2r)² = F — no change.
worked — force between two students
60 kg and 50 kg, 1.0 m apart.
F = 6.67 × 10⁻¹¹ × 60 × 50 / 1² = 2.0 × 10⁻⁷ N — about 1% of a mosquito's weight, which is why measuring G needed Cavendish's delicacy.

Your weight on a planet is the gravitational pull of that planet on you. Setting weight equal to the universal force makes the body's own mass cancel:

Deriving gmg = G M m / R² ⟹ g = G M / R²
g depends only on the planet — not on the falling body's mass

That is why a stone and a coin land together (with air removed): the m cancels. The panel shows one person on three worlds — the scale changes because g changes.

Worldg (m/s²)Scale shows
Moon1.6about g/6 — astronauts bounce
Earth9.8full weight
Jupiter24.8about 2.5× — you can barely stand

The newton (N) is the SI unit of weight; mass stays the same everywhere in kilograms, but weight = mg changes with g.

Rearranging g = GM/R² gives the mass of the Earth from three measurable numbers — g, the Earth's radius R, and the constant G that Cavendish's 1798 torsion balance measured. This is why finding G was called "weighing the Earth".

Mass of the EarthM = g R² / G
M = (9.8)(6.4 × 10⁶)² / (6.67 × 10⁻¹¹) ≈ 6.0 × 10²⁴ kg
worked — mean density of the Earth
Using M = 6.0 × 10²⁴ kg and R = 6.4 × 10⁶ m.
V = (4/3)πR³ ≈ 1.1 × 10²¹ m³
ρ = M/V ≈ 5.5 × 10³ kg/m³ — about 5.5× water, so the iron core must be far denser than the rocky crust (ρ ≈ 2700 kg/m³).

Watch the probe in the panel and the g-meter beside it. Above the surface the distance grows to (R + h), so g falls off by the inverse square; below the surface only the inner sphere pulls, so g shrinks linearly to zero at the centre.

Altitudeg_h = g R² / (R + h)²
for h ≪ R: g_h ≈ g (1 − 2h/R)
Depthg_d = g (1 − d/R) → zero at the centre (d = R)

With latitude: the Earth bulges at the equator and spins, so g is smallest at the equator (9.78 m/s²) and largest at the poles (9.83 m/s²) at sea level.

worked — g on Mount Everest
h = 8848 m.
g_h ≈ 9.80(1 − 2 × 8848/6.4 × 10⁶) = 9.77 m/s² — even the highest peak loses only 0.3%.

A weighing scale reads the normal reaction it pushes back with — your apparent weight — not the true weight mg. In a lift accelerating up you read m(g + a); accelerating down, m(g − a); and if the cable snaps and a = g, the scale reads zero.

Lift motionApparent weight
at rest / uniform velocitymg (true weight)
accelerating up with am(g + a) — feel heavier
accelerating down with am(g − a) — feel lighter
free fall (a = g)0 — weightless

An astronaut floating in a satellite is in that snapped-cable lift permanently: capsule, astronaut and water droplets all fall together with the same acceleration, so nothing presses on anything. Gravity at the ISS is still ≈ 8.7 m/s² — weightlessness is not "no gravity", it is apparent weight zero from shared free fall.

Newton's thought experiment, alive in the panel: fire a cannonball horizontally from a high mountain. Faster → it lands farther. Fast enough (7.9 km/s) → the ground curves away exactly as fast as the ball falls. It never lands. A satellite is a projectile that keeps missing the Earth.

For a satellite skimming the surface, gravity supplies the centripetal force, so the orbital speed follows from mg = mv²/R:

Orbital velocity (low orbit)mg = mv²/R ⟹ v = √(gR)
v = √(9.8 × 6.4 × 10⁶) ≈ 7.9 × 10³ m/s ≈ 7.9 km/s
period T = 2πR/v ≈ 5060 s ≈ 84 min
Exam point: orbital velocity is independent of the satellite's mass — a 50 kg CubeSat and the 420-tonne ISS at the same height orbit at the same speed. Higher orbit → weaker gravity → slower satellite.

Make the period exactly 24 hours, orbit above the equator moving west-to-east, and from the ground the satellite never moves — your TV dish points at it once, forever.

Geostationary orbitT² = 4π²(R + h)³ / (GM)
T = 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° East, 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).
  • The tides — the same gravitation: the Moon pulls the near ocean harder than the far one, raising two tidal bulges that sweep the coast as the Earth turns.
🌍 Live panelGravitation
Press ▶ or scroll the lecture — this panel animates each concept through a real-life picture.