Gravitation is one of the most reliably tested topics in NEET Physics, consistently contributing 2–3 questions per year. Mastery requires understanding Newton's law, variation of g, Kepler's laws, satellite mechanics, and energy relationships — all of which appear in direct numerical and conceptual forms.

Newton's Universal Law of Gravitation establishes that every particle in the universe attracts every other particle with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres:

$F = \frac{G m_1 m_2}{r^2}$

where $G = 6.674 \times 10^{-11}\ \text{N m}^2\ \text{kg}^{-2}$ is the universal gravitational constant. Its dimensional formula is $[M^{-1} L^3 T^{-2}]$. The acceleration due to gravity at Earth's surface is $g = GM/R^2 \approx 9.8\ \text{m/s}^2$, dimensionally $[M^0 L^1 T^{-2}]$.

Variation of g is the most frequently tested sub-topic. Three independent effects modify g:

1. Altitude (exact): $g' = \dfrac{gR^2}{(R+h)^2}$ — inverse-square decrease. At $h = R$, $g' = g/4$.
2. Altitude (approximate, $h \ll R$): $g' \approx g\!\left(1 - \dfrac{2h}{R}\right)$ — valid only when $h$ is small.
3. Depth: $g' = g\!\left(1 - \dfrac{d}{R}\right)$ — linear decrease. At Earth's centre ($d = R$), $g' = 0$.
4. Latitude $\lambda$: $g_\text{eff} = g - R\omega^2\cos^2\lambda$ — maximum at poles ($\lambda = 90°$), minimum at equator ($\lambda = 0°$).

The critical distinction: g decreases linearly with depth but by an inverse-square law with altitude. At equal fractions (e.g., $h = R/2$ vs $d = R/2$), the depth gives a higher g than the altitude.

Kepler's Three Laws govern planetary and satellite orbits:

• First Law (Orbits): Planets move in ellipses with the Sun at one focus.
• Second Law (Equal Areas): The radius vector sweeps equal areas in equal times. Mathematically, areal velocity $dA/dt = L/(2m) = \text{constant}$. This means the planet moves fastest at perihelion and slowest at aphelion.
• Third Law (Periods): $T^2 \propto r^3$, or $T^2 = \dfrac{4\pi^2}{GM} r^3$. This is used to compare the periods of satellites at different orbital radii.

Gravitational Potential Energy is $U = -GMm/r$ (in joules, $[M^1 L^2 T^{-2}]$). The negative sign is fundamental: it indicates a bound system. $U = 0$ at $r = \infty$. The gravitational potential (PE per unit mass) is $V = -GM/r$, dimensional formula $[M^0 L^2 T^{-2}]$, unit J/kg.

Escape Velocity is the minimum speed needed to escape the gravitational field of a planet:

$v_e = \sqrt{2gR} = \sqrt{\frac{2GM}{R}} \approx 11.2\ \text{km/s for Earth}$

Key fact: escape velocity is completely independent of the mass and direction of the projected body. It depends only on the planet's mass $M$ and radius $R$.

Orbital Velocity for a satellite at radius $r$ from Earth's centre:

$v_0 = \sqrt{\frac{GM}{r}} = \sqrt{\frac{gR^2}{r}}$

Near Earth's surface ($r \approx R$): $v_0 \approx 7.9$ km/s. The fundamental relation: $v_e = \sqrt{2}\, v_0$.

Satellite Energy: For a satellite of mass $m$ at orbital radius $r$:

$KE = \frac{GMm}{2r}, \quad PE = -\frac{GMm}{r}, \quad E_{total} = -\frac{GMm}{2r}$

The energy ratio is $KE : PE : E = 1 : {-2} : {-1}$. The relation $|PE| = 2KE$ always holds. Total energy is negative for a bound orbit. If total energy $\geq 0$, the satellite escapes.

Geostationary Satellite: Period $T = 24$ hours, orbital radius $r \approx 42{,}164$ km from Earth's centre (altitude $\approx 35{,}786$ km), orbits in the equatorial plane from west to east, and appears stationary relative to any ground observer.

Numerical Example: A satellite at $h = R$ (so $r = 2R$, $m = 200$ kg, $R = 6.4 \times 10^6$ m, $g = 10$ m/$s^{2}$):

• Orbital velocity: $v_0 = \sqrt{gR/2} = \sqrt{3.2 \times 10^7} \approx 5{,}657$ m/s
• Period: $T = 2\pi r/v_0 \approx 14{,}225$ s $\approx 3.95$ hours
• Total energy: $E = -gRm/4 = -3.2 \times 10^9$ J $= -3.2$ GJ