Gravitation
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Direct answer
Newton's law of gravitation gives the attraction between point masses as F = G m1 m2/r^2, and the shell theorem lets spheres be treated as point masses at their centres for external points. From g = G M/R^2 follow the two JEE-critical speeds — orbital v_o = sqrt(GM/r) and escape v_e = sqrt(2GM/R) = sqrt(2 g R), larger than the near-surface orbital speed by a factor of sqrt(2) — and the satellite energies: KE = GMm/(2r), PE = −GMm/r, total = −GMm/(2r).
What you must remember
- Variation with height: g' = g/(1 + h/R)^2, about g(1 − 2h/R) when h is small; with depth d: g' = g(1 − d/R); g is maximum at the surface and zero at the centre.
- g is smaller at the equator than at the poles, from the Earth's rotation and its equatorial bulge.
- Kepler's laws: elliptical orbits with the Sun at a focus; equal areas in equal times (angular momentum conservation); T^2 proportional to a^3.
- Near-surface orbital speed is about 7.9 km/s and escape speed about 11.2 km/s for the Earth; escape speed is independent of the direction of projection (ignoring air and rotation).
- Satellite energies in an orbit of radius r: KE = GMm/(2r), PE = −GMm/r, total = −GMm/(2r); the binding energy equals the magnitude of the total energy.
- Inside a uniform spherical shell the field is zero and the potential is constant at −G M/R — constant, not zero.
- Geostationary satellite: 24-hour period, circular orbit over the equator moving west to east, altitude about 36,000 km.
Common confusion
The stubborn myth is that astronauts float because gravity vanishes in orbit. At orbital altitude g is only slightly weaker than at the surface; spacecraft and astronaut accelerate toward the Earth identically, so the astronaut presses on nothing — apparent weight is zero while gravity fully acts. The second confusion mixes depth with height: with depth the attracting mass shrinks linearly, with height the distance grows, giving different formulas.
Exam-focused takeaway
JEE Main tests percentage variation of g with height or depth, the escape-to-orbital speed ratio, satellite energies and orbit changes as numerical-value questions. JEE Advanced adds binding-energy bookkeeping, the energy cost of orbit raising, binary-star systems analysed from the centre of mass, and motion through a tunnel along a diameter (simple harmonic with acceleration g r/R). Anchor everything in U = −G m1 m2/r and V = −G M/r, and most algebra falls into place.
Frequently asked questions
Why do astronauts float in an orbiting spacecraft?
Craft and crew are in continuous free fall under nearly full-strength gravity; with both accelerating identically, no normal force acts between them and apparent weight is zero.
Why does g decrease with depth but not for the same reason as with height?
With depth, only the mass inside your radius attracts net (the outer shell contributes nothing inside), so effective mass falls linearly; with height, the distance from the whole mass simply grows.
How do escape and orbital speeds differ?
Orbital speed sqrt(GM/r) maintains a circular orbit; escape speed sqrt(2GM/R) makes total energy zero so the body just reaches infinity at rest — larger by a factor of sqrt(2).
Why is an orbiting satellite's total energy negative?
Negative total energy marks a bound system; its magnitude GMm/(2r) is exactly the energy that must be supplied to free the satellite, and it equals the kinetic energy.
Why do satellites obey Kepler's laws like planets?
The laws follow from an inverse-square force directed along the joining line, so any small body orbiting a much larger central mass inherits all three.