Lesson 11 of 11 · 15 min

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Must-know facts

16 facts

1F = G m₁ m₂ / r², with r measured between centres; G = 6.67 × 10⁻¹¹ N m² kg⁻².
2g = G M_E / R_E² ≈ 9.8 m/s² and is independent of the mass of the falling body.
3g(h) = g R_E²/(R_E + h)² exactly; ≈ g(1 − 2h/R_E) only for h ≪ R_E.
4g(d) = g(1 − d/R_E) for a uniform earth; zero at the centre.
5g is largest at the surface and decreases both above and below it.
6Shell outside → acts as a point at its centre; point inside a shell → zero net pull.
7Kepler: ellipse with the sun at a focus; equal areas in equal times; T² ∝ a³.
8Law of areas = conservation of angular momentum; holds for any central force.
9v_P r_P = v_A r_A: fastest at perihelion.
10U = −G M m / r, zero at infinity; mgh is its near-surface approximation.
11Escape speed vₑ = √(2gR_E) ≈ 11.2 km/s; independent of the body's mass and launch direction.
12Orbital speed near the surface v₀ = √(gR_E) ≈ 7.9 km/s; vₑ = √2 v₀.
13Orbital speed v = √(G M_E / r) falls with r; period T ∝ r^(3/2).
14Near-surface orbit period ≈ 85 min.
15Satellite: K = GMm/2r, U = −GMm/r, E = −GMm/2r = −K.
16Negative total energy = bound orbit; zero or positive = escape.

Common traps

Where marks are lost

Putting the height above the surface into F = G M m / r².

r is always measured from the centre of the earth: r = R_E + h.

Using g(1 − 2h/R_E) for a height like R_E/2.

The shortcut only works for h ≪ R_E. At h = R_E/2 use g/(1.5)² = 0.44 g, not zero.

Thinking g keeps increasing as you go down a mine, because you are closer to the centre.

The shell above you pulls with zero net force and the mass below you shrinks; g(d) = g(1 − d/R_E) falls to zero at the centre.

Believing a heavier body needs a bigger escape speed.

The mass m cancels: vₑ = √(2GM/R) depends only on the planet (and launch height).

Thinking astronauts float because there is no gravity in orbit.

At a few hundred km g is still close to its surface value; they float because they and the craft fall freely together.

Giving a satellite a positive total energy because it is moving.

E = −GMm/2r is negative for every bound orbit; K is positive but U is negative and twice as large.

Saying the law of areas proves the inverse-square law.

Equal areas follows from any central force (angular momentum is conserved); it is T² ∝ a³ that points to 1/r².

Treating G and g as the same kind of constant.

G is universal (6.67 × 10⁻¹¹ N m² kg⁻²); g is a local acceleration that changes with height, depth and planet.

Formulas

13 to know

Law of gravitation

F = G m₁ m₂ / r²

r between centres; attractive.

Kepler's third law

T² = (4π² / G M) a³

M is the central body; a is the semi-major axis (radius for a circle).

Law of areas

ΔA/Δt = L / 2m = constant

At the ends of the orbit: v_P r_P = v_A r_A.

Cavendish balance

G M m L / d² = τθ

τ is the restoring torque per unit twist.

g at the surface

g = G M_E / R_E²

Gives M_E = g R_E² / G.

g at height h

g(h) = g R_E² / (R_E + h)² ≈ g (1 − 2h/R_E)

Approximation only for h ≪ R_E.

g at depth d

g(d) = g (1 − d/R_E)

Uniform earth; zero at the centre.

Potential energy

U = −G m₁ m₂ / r

Zero at infinity; add over all pairs for a system.

Gravitational potential

V = −G M / r

Potential energy per unit mass.

Escape speed

vₑ = √(2 G M / R) = √(2 g R)

≈ 11.2 km/s for the earth.

Orbital speed

v = √(G M_E / (R_E + h)); v₀ = √(g R_E)

v₀ ≈ 7.9 km/s just above the surface.

Orbital period

T = 2π (R_E + h)^(3/2) / √(G M_E); T₀ = 2π √(R_E / g)

T₀ ≈ 85 min.

Satellite energies

K = G M m / 2r, U = −G M m / r, E = −G M m / 2r

r = R_E + h; E = −K = U/2.

Key terms

13 terms

Geocentric model: A picture of the heavens with the earth at the centre.
Heliocentric model: A picture with the sun at the centre and the planets going round it.
Ellipse: A closed curve on which the distances to two fixed foci always add to the same total.
Perihelion: The point of a planet's orbit nearest the sun.
Aphelion: The point of a planet's orbit farthest from the sun.
Semi-major axis: Half of the longest diameter of an ellipse.
Central force: A force always directed along the line joining the body to a fixed centre.
Gravitational constant (G): The universal constant in Newton's law, 6.67 × 10⁻¹¹ N m² kg⁻².
Acceleration due to gravity (g): The acceleration of a freely falling body near a planet; about 9.8 m/s² at the earth's surface.
Gravitational potential: Gravitational potential energy per unit mass at a point.
Escape speed: The least launch speed with which a body can reach infinity.
Satellite: A body that revolves around a planet.
Bound system: A system with negative total energy, whose orbit stays closed.

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