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Chapter 09 · Part I · Mechanics
Revised on ____________________

Chapter 09 · Gravitation

One inverse square, from a dropped stone to Neptune

sheets
diagrams
results

Formulas and conditions only — no derivations, no solved numbers. The diagrams are the reader's own.

The law and the field

SUPERPOSE, NEVER ADD MAGNITUDES

I.Newton's law F = G m₁ m₂r² attractive, along the join

II.Field g = Fm = G Mr² an acceleration, N kg⁻¹

III.Superposition g = Σ g_i vector sum, always

IV.Shell theorem, outside a shell acts as if all its mass sat at the centre

V.Shell theorem, inside g = 0 everywhere inside a uniform shell

VI.Surface value g = G MR² = 43 π G ρ R

G is the same everywhere in the universe; g is a local number that changes with height, depth, latitude and the planet you are standing on. They are not the same constant.

How g varies

HEIGHT, DEPTH, SPIN, SHAPE

I.At height h g_h = g(1 + h/R)² ≈ g(1 − 2h/R) for h ≪ R

II.At depth d g_d = g ( 1 − dR ) zero at the centre

III.Inside a uniform sphere g ∝ r outside, g ∝ 1/r²

IV.Latitude λ g_λ = g − ω² R cos²λ largest at the poles

V.Loss at the equator ω² R ≈ 0.034 m s⁻²

VI.Weightless spin ω = √( gR ) a day of about 84 minutes

III

Potential and energy

ZERO AT INFINITY, NEGATIVE EVERYWHERE ELSE

I.Potential V = − G Mr a scalar; add them arithmetically

II.Potential energy U = m V = − G M mr

III.Escape speed v_e = √( 2 G MR ) = √(2 g R) 11.2 km s⁻¹ on Earth

IV.Bound or free E < 0 ellipse, E = 0 parabola, E > 0 hyperbola

Orbits

HALF THE ESCAPE SPEED, TWICE THE ENERGY

I.Orbital speed v_o = √( G Mr ) = √(gR) in a low orbit

II.Period T = 2π √( r³G M )

III.Total energy E = − G M m2 r = U2 = − K

IV.Speeds compared v_e = √2 · v_o

V.Geostationary r = 4.22×10⁴ km equatorial, T = 24 h

VI.Weightlessness in orbit free fall, not the absence of gravity

Kepler's laws

TRUE FOR EVERY CENTRAL 1/r² FORCE

I.First an ellipse with the Sun at one focus

II.Second dAdt = L2m = constant conservation of angular momentum

III.Its consequence v_p r_p = v_a r_a fastest at perihelion

IV.Third T² ∝ a³

V.In full T² = 4 π²G M a³

Where marks are lost in this chapter

1Carrying g = 9.8 onto another planet. G is universal; g = GM/R² is local — recompute it.

2Using mgh at orbital altitude. Past about 100 km the error already exceeds 3 % — use −GMm/r.

3“Astronauts float because there is no gravity.” g ≈ 8.7 m s⁻² at the ISS; they are in free fall.

4Applying T² ∝ a³ across different central bodies. The constant carries the mass M you orbit.