Gravitation

Concept overview

Newton's law of gravitation states that every point mass attracts every other with force F = Gm₁m₂/r², directed along the line joining them. The universal gravitational constant G = 6.67 × 10⁻¹¹ N·m²/kg². For an extended body with spherical symmetry, you can treat all mass as concentrated at the centre — a result derived from Gauss's law for gravity.

Kepler's three laws describe planetary motion. First law: orbits are ellipses with the Sun at one focus. Second law: the line from the Sun to a planet sweeps equal areas in equal times (this is a direct consequence of angular momentum conservation). Third law: T² ∝ r³, where r is the semi-major axis.

For circular orbits, gravitational force supplies the centripetal force: GMm/r² = mv²/r, giving orbital speed v = √(GM/r). At Earth's surface (r = R_E), setting v² = 2gR_E gives escape velocity v_esc = √(2gR_E) ≈ 11.2 km/s — the minimum speed to leave Earth's gravity entirely.

Gravitational potential energy at distance r from a point mass M (with U = 0 at infinity) is U = −GMm/r. Total energy of a bound orbit is E = KE + U = −GMm/(2r), always negative — an object with positive total energy is unbound and will escape.

Key formulas & concepts

• Newton's law of gravitation: F = G m₁m₂ / r²
• Gravitational PE: U = −G Mm / r (with U = 0 at infinity)
• Orbital velocity: v_orb = √(GM/r)
• Escape velocity: v_esc = √(2GM/R) = √(2gR) at Earth surface
• Kepler's third law: T² = (4π²/GM) r³
• Bound orbit total energy: E = −GMm/(2r)
• Time period of geostationary orbit: 24 h (matches Earth's rotation; r ≈ 42,000 km)

Common mistakes to avoid

• Using U = mgh at large heights — this approximation holds only near Earth's surface where g is constant.
• Forgetting that escape velocity is independent of the escape direction — as long as the speed at the surface exceeds v_esc, the object escapes regardless of the angle.
• Confusing orbital speed with escape speed — v_orb = √(GM/r); v_esc = √(2GM/R) = √2 × v_surface_orbit.
• Ignoring that gravitational potential energy is negative for bound orbits — a positive total energy means the object is unbound.

JEE Main relevance

JEE Main asks 1–2 gravitation problems every paper — usually one Kepler's-law application (period-radius scaling) and one orbital-energy or escape-velocity numerical.

JEE Advanced relevance

Advanced gravitation problems ask about specific transfer orbits (Hohmann), binary star systems, or the gravitational field inside a non-uniform planet. These extend the basic formulas with additional geometry.

Previous-year questions (PYQ)

Filter the PYQ bank by 'Gravitation' — Kepler's-law numerical and escape-velocity problems repeat almost every year.