Gravitational field and orbital mechanics
The gravitational-field block of the PAU (Prueba de Acceso a la Universidad) in Andalusia asks you to move fluently between force, field, potential and energy, and then to apply those ideas to satellites and Kepler's laws. Questions are usually one short calculation plus one conceptual justification, so both the algebra and the sign conventions are marked.
Core rules
- Newton's law: F = GMm/r², always attractive, directed along the line joining the centres.
- Field of a point mass, or of a sphere outside it: g = GM/r², a vector pointing towards the mass.
- Potential: V = −GM/r, a scalar taken as zero at infinity, so it is always negative near a mass.
- Potential energy of a mass m in that field: U = mV = −GMm/r.
- Circular orbit: v = √(GM/r) and T = 2π√(r³/GM), which is Kepler's third law.
- Total energy of a circular orbit: E = −GMm/2r = −½mv², so the binding energy is GMm/2r.
Worked example
A 500 kg satellite orbits 600 km above the Earth's surface. With GM = 3.99×10¹⁴ m³ s⁻² and R = 6.37×10⁶ m, the orbital radius is r = 6.97×10⁶ m, not 6.0×10⁵ m. Then v = √(3.99×10¹⁴/6.97×10⁶) = √(5.72×10⁷) ≈ 7.6×10³ m/s. The period follows from T = 2πr/v ≈ 5.8×10³ s, roughly 96 minutes. The total energy is E = −GMm/2r ≈ −1.4×10¹⁰ J; the negative sign means the satellite is bound, and 1.4×10¹⁰ J would have to be supplied to remove it to infinity.
Marks lost here
Mixing up g and V, or forgetting that V is negative. Writing the field as a scalar and losing its direction. Using the altitude instead of the distance from the Earth's centre. Quoting T² ∝ r³ without noting that the constant depends on the central mass. Claiming a geostationary satellite can hover over any city: it must be equatorial, circular, and have a sidereal period of 23 h 56 min, not exactly 24 h.