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Gravitation

Source lecture(s): SC133 Lec 16

Intuition

Newton's great unification: the force that drops an apple is the force that holds the Moon. The Moon is falling — continuously — it just moves sideways fast enough to keep missing the Earth. One law, inverse-square in distance, universal in mass, governs apples, moons, planets, and galaxies.

The law

\[\boxed{\,F = G\,\frac{m_1 m_2}{r^2}\,} \qquad G = 6.67\times10^{-11}\ \text{N·m}^2/\text{kg}^2\]

— always attractive, along the line joining the (centers of the) masses. Full reference: equation page.

Two theorems make spheres tractable (Newton's shell theorems): a uniform spherical shell attracts outside bodies as if all its mass sat at the center, and exerts zero net force on a body inside it. Hence planets act like points — and gravity decreases as you tunnel into the Earth.

Superposition

Gravity adds as vectors: the field of several masses is the sum of the individual fields,

\[\vec g(\vec r) = \sum_i \left(-\frac{Gm_i}{r_i^2}\hat r_i\right)\]

Surface gravity: \(g = GM_E/R_E^2 = 9.8\,\text{m/s}^2\) — this equation weighs the Earth once \(G\) is measured (Cavendish, 1798: "the experiment that weighed the world").

Gravitational potential energy

Taking \(U(\infty) = 0\) (why we may choose this):

\[U = -\frac{G M m}{r}\]

Negative everywhere: bound systems sit in an energy hole. Escape requires total energy \(E = K + U \geq 0\):

\[v_\text{esc} = \sqrt{\frac{2GM}{R}}\]

Earth: \(11.2\,\text{km/s}\). Moon: \(2.4\) (why it kept no atmosphere). Push \(R\) small enough at fixed \(M\) that \(v_\text{esc} \to c\) and you have sketched a black hole — dimensional analysis gets the same radius.

The apple–Moon check (Newton's own)

Moon distance \(\approx 60 R_E\), so lunar gravity should be \(g/60^2 = 2.7\times10^{-3}\,\text{m/s}^2\). Required centripetal acceleration: \(\omega^2 r = \left(\frac{2\pi}{27.3\,\text{d}}\right)^2 (3.84\times10^8\,\text{m}) \approx 2.7\times10^{-3}\,\text{m/s}^2\). ✓ The inverse-square law, confirmed with 17th-century data.

Weight vs weightlessness

Astronauts float not because gravity is absent (at ISS altitude it's still ~90% of \(g\)!) but because they and the station fall together — orbit is perpetual free fall. "Weightlessness" = no contact force, not no gravity.

Common mistakes

  • "No gravity in space." Gravity extends to infinity; orbiting is falling.
  • Using \(U = mgh\) far from the surface. That's the near-surface linearization; use \(-GMm/r\) for anything orbital.
  • Doubling \(r\) halves the force? Inverse square: quarter.
  • Forgetting both masses attract each other equally (third law) — the Earth accelerates toward the apple too, just imperceptibly.

Knowledge graph position

Prerequisites: Newton's laws, Circular motion, Potential energy. Leads to: Kepler's laws, astrophysics, N-body simulation (PHY653).

Quiz

Q1 (computational). How fast must a satellite move in a circular orbit just above Earth's surface (\(R_E = 6.4\times10^6\) m)?

Answer

\(\frac{GM}{R^2} = \frac{v^2}{R} \Rightarrow v = \sqrt{gR_E} = \sqrt{9.8\times6.4\times10^6} \approx 7.9\,\text{km/s}\) — Newton's cannonball; note \(v_\text{esc} = \sqrt2\, v_\text{orbit}\).

Q2 (conceptual). You tunnel halfway to Earth's center (uniform density). How does \(g\) there compare with the surface?

Answer

Half. Only the mass inside your radius pulls (shell theorem): \(g(r) = \frac{G}{r^2}\cdot M\frac{r^3}{R^3} = g_\text{surf}\,\frac{r}{R}\) — gravity is linear in \(r\) inside a uniform sphere.

Q3 (multiple choice). Two satellites orbit Earth at radii \(r\) and \(4r\). The outer one's orbital speed is: (a) half (b) quarter (c) \(1/\sqrt2\)

Answer

(a). \(v = \sqrt{GM/r} \propto r^{-1/2}\): four times the radius, half the speed — outer planets amble.