Gravitational force is directly proportional to the product of the masses of two bodies, and inversely proportional to the square of the distance between their centres. (p.79)
F = Gm1m2 / r2, where G = 6.67 × 10−11 N m2 kg−2 (gravitational constant). (p.80)
Relating g to G: equating F = mg and F = GMm/r2 gives g = GM/r2. (p.83)
Beyond the Earth's surface, gravitational acceleration decreases — it is inversely proportional to r2. (p.84)
On the Earth's surface: g = GM/R2. At height h above the surface: g = GM/(R + h)2. (p.85)
Centripetal Force in Satellite and Planet Motion
Centripetal force, F = mv2/r (p.90)
Centripetal acceleration, a = v2/r (p.91)
For an orbiting satellite, gravitational force = centripetal force, giving the mass of the central body: M = 4π2r3 / (GT2). (p.94)
3.2 Kepler's Laws
First Law (Law of Orbits): All planets move in elliptical orbits with the Sun at one focus. (p.96)
Second Law (Law of Areas): A line connecting a planet to the Sun sweeps out equal areas in equal times. (p.97)
Third Law (Law of Periods): T2 ∝ r3 (T = orbital period, r = radius of orbit). (p.98)
T2 = (4π2/GM) r3(p.99)
Comparing two orbits: T12/T22 = r13/r23(p.100)
3.3 Man-made Satellites
Linear speed of a satellite in a stable orbit: v = √(GM/r), where r = R + h. (p.103)
Geostationary satellite: orbital period T = 24 hours, moves in the same direction as Earth's rotation, always above the same geographical location (e.g. MEASAT). (p.105)
Non-geostationary satellite: orbital period differs from 24 hours; positioned above different geographical locations at different times (e.g. TiungSAT, ISS, GPS satellites). (p.105)
Escape velocity, v = √(2GM/r); from Earth's surface, v = √(2GM/R) ≈ 1.12 × 104 m s−1 (11.2 km s−1). (p.107)
Escape velocity does not depend on the mass of the escaping object. (p.107)
Earth's high escape velocity keeps the atmosphere from escaping into space, since the average air molecule speed (~500 m s−¹) is far lower than the escape velocity. (p.108)