Newton's inverse-square law of gravitation, field strength and potential of a point mass, and the physics of circular orbits and geostationary satellites.
11 definitions to learn word for word, 13 equations in their own column, 10 diagrams and graphs. Read it, cover it, test yourself.
a region of space in which a mass experiences a force.
Gravitational field strength (g)
the gravitational force per unit mass acting on a small mass placed at that point.
Field line
a line showing the direction of the force on a small mass placed in the field.
Key points
g is a vector, directed towards the mass producing the field.
Field lines always point towards masses: gravity is only attractive.
Closer field lines mean a stronger field.
Around a point or spherical mass the field is radial; near the Earth's surface it is effectively uniform (parallel, equally spaced lines).
N kg-1 and m s-2 are equivalent units for g.
Radial field around a spherical mass: lines point to the centre and crowd together near the surface, where the field is strongest.
Near the surface the field is uniform: parallel, equally spaced lines, so g is the same everywhere in the region.
Watch out!
Gravitational field lines never point outwards; there is no gravitational repulsion.
g is force per unit mass, not the force itself.
13.2aNewton's law of gravitation
Definitions
Newton's law of gravitation
any two point masses attract each other with a force that is proportional to the product of their masses and inversely proportional to the square of their separation.
Point mass
a mass whose size is negligible compared with the distances involved; a uniform sphere acts as a point mass at its centre.
Key points
For uniform spheres, measure r centre to centre.
The forces on the two masses are equal and opposite (Newton's third law), even if the masses differ.
Inverse square: double r, force falls to one quarter.
G is tiny, so gravity is only noticeable when at least one mass is very large.
Two masses attract with equal and opposite forces; r is measured between the centres.
Inverse-square law: F against r falls steeply; each doubling of r quarters the force.
Watch out!
r is the distance between centres, not the height above the surface.
Forgetting to square r: the force depends on 1/r2.
13.2bCircular orbits and geostationary satellites
Definitions
Geostationary orbit
an equatorial orbit, travelling west to east, with a period of 24 hours, so the satellite stays above the same point on the Earth's surface.
Orbital period (T)
the time taken for one complete orbit.
Key points
For a circular orbit, gravitational force provides the centripetal force: GMm/r2 = mv2/r.
The satellite mass m cancels: orbital speed and period depend only on M and r.
Larger orbit: slower speed, longer period.
T2 ∝ r3 (Kepler's third law): plot T2 against r3, gradient = 4π2/GM, so M can be found.
Geostationary radius ≈ 4.2 × 107 m (about 3.6 × 107 m above the surface).
Geostationary satellites suit communications and weather monitoring: dishes stay fixed; but coverage near the poles is poor.
Circular orbit: the gravitational pull is the centripetal force, always towards the planet's centre, perpendicular to v.
Kepler's third law: T2 against r3 is a straight line through the origin with gradient 4π2/GM.
Geostationary satellite (viewed from above the North Pole): it orbits in the same direction as the Earth spins, with the same 24 h period.
Watch out!
Use r = R + h (orbit radius from the centre) in orbit equations, not the height h.
Geostationary needs all three: equatorial, west to east, 24 h period. A 24 h polar orbit is not geostationary.
13.3Gravitational field of a point mass
Definitions
Gravitational field strength of a point mass
g = GM/r2, directed towards the mass, where r is the distance from its centre.
Key points
Derived from g = F/m with F = GMm/r2: the test mass m cancels.
Outside a uniform sphere, g follows the inverse-square law: double r, g falls to a quarter.
Near the Earth's surface r hardly changes over everyday heights, so g is approximately constant (9.81 N kg-1).
g at altitude compared with the surface: g/g0 = R2/r2.
Useful link: GM = g0R2 (surface field and planet radius).
Field strength outside a planet: g0 at the surface (r = R), falling to g0/4 at r = 2R.
Watch out!
g falls as 1/r2, not 1/r: tripling the distance from the centre cuts g to a ninth.
g = GM/r2 applies only outside (or at the surface of) the sphere.
13.4Gravitational potential and potential energy
Definitions
Gravitational potential (φ)
the work done per unit mass in bringing a small test mass from infinity to that point.
Gravitational potential energy (of a mass m at a point)
the work done in bringing the mass from infinity to that point.
Equipotential
a surface on which every point has the same potential; no work is done moving a mass along it.
Key points
Zero of potential is at infinity, and φ is negative everywhere because gravity is attractive (work is done by the field as mass moves in).
Moving away from a mass, φ increases (becomes less negative).
Change in potential energy: ΔEp = mΔφ = GMm(1/r1 - 1/r2) moving from r1 out to r2.
Near the surface only, ΔEp = mgΔh is a good approximation (g constant).
Field strength = minus the potential gradient: g = -Δφ/Δr (gradient of the φ-r graph).
Equipotentials around a point mass are concentric spheres, perpendicular to the field lines.
The potential well: φ = -GM/r is most negative at the surface and rises towards zero at infinity. The gradient at any point gives g.
Equipotentials (dashed circles) around a mass are perpendicular to the field lines; potential is more negative closer in.
Watch out!
A more negative φ is a lower potential: -6 × 107 J kg-1 is lower than -3 × 107 J kg-1.
Use mgΔh only for small height changes near the surface; use mΔφ for satellites.