NEET Physics — Chapter 8

Gravitation

Newton's law of gravitation, acceleration due to gravity, escape velocity, orbital mechanics, Kepler's laws — complete NEET notes with derivations and exam traps.

1. Newton's Law of Universal Gravitation

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 them:

F = G rac{m_1 m_2}{r^2}

$G = 6.674 imes 10^{-11} ext{N·m}^2 ext{·kg}^{-2}$ (Universal Gravitational Constant)

Dimensional formula of $G$ : $[M^{-1}L^3T^{-2}]$

Key properties of gravitational force:

Always attractive — acts along the line joining the two bodies
Central force — acts along the line joining the masses
Conservative force — work done is path-independent
Weakest fundamental force — but infinite in range
Obeys Newton's third law — equal and opposite forces on both bodies
Independent of the medium between the bodies (unlike electrostatic force)
Superposition principle: Net force = vector sum of individual forces

NEET tip:

G

is a universal constant — same everywhere in the universe. Don't confuse

G

(universal constant,

6.67 imes 10^{-11}

N·m²/kg²) with

g

(acceleration due to gravity,

approx 9.8

m/s² on Earth's surface).

2. Acceleration Due to Gravity — Variation

g on Earth's surface:

g = rac{GM_E}{R_E^2} approx 9.8 ext{m/s}^2

$M_E = 6 imes 10^{24}$ kg, $R_E = 6.4 imes 10^6$ m (Earth's radius)

Variation with altitude (height $h$ above surface):

g_h = rac{GM_E}{(R_E + h)^2} = gleft( rac{R_E}{R_E + h} ight)^2

For $h ll R_E$ : $g_h approx gleft(1 - rac{2h}{R_E} ight)$ (approximate, using binomial expansion)

$g$ decreases as we go up. At height $R_E$ , $g_h = g/4$ .

Variation with depth $d$ below surface:

g_d = gleft(1 - rac{d}{R_E} ight)

$g$ decreases linearly with depth. At Earth's centre ( $d = R_E$ ), $g = 0$ .

Variation with latitude ( $lambda$ ): Earth rotates, so effective $g$ is reduced by centrifugal effect:

g_lambda = g - R_Eomega^2cos^2lambda

Maximum at poles ( $lambda = 90°$ ), minimum at equator ( $lambda = 0°$ ). Also, Earth is oblate — flatter at poles, so $g$ is slightly higher at poles due to smaller $R$ .

Pro tip:

g

at altitude decreases as

(1/r^2)

— faster than at depth (linear decrease). At the same "distance from surface," going up reduces

g

more than going down the same distance (for small distances compared to

R_E

). This is a common NEET comparison question.

3. Gravitational Potential and Potential Energy

Gravitational Potential (V) at a point is the work done per unit mass in bringing a test mass from infinity to that point:

V = - rac{GM}{r}

Always negative (attractive force does positive work as mass approaches; W by external agent is negative). At $r o infty$ , $V o 0$ (maximum — zero is the reference).

Gravitational Potential Energy (U) of mass $m$ at distance $r$ from mass $M$ :

U = - rac{GMm}{r}

Relation: $V = U/m$ . For a system of particles, U = sum of all pairwise potential energies.

Near Earth's surface (reference at surface): $U = mgh$ (only for small $h ll R_E$ ).

Gravitational field strength (g at distance r from centre):

ec{g} = - rac{GM}{r^2}hat{r} quad (r geq R_E, ext{ outside})

ec{g} = - rac{GM}{R_E^3}rhat{r} quad (r < R_E, ext{ inside — linear})

NEET tip: Gravitational PE is always negative (bound system). To remove a mass from Earth's surface to infinity requires energy =

|U_{surface}| = GMm/R_E

. This is the "binding energy" — the energy that keeps the object bound to Earth.

4. Escape Velocity

Escape velocity is the minimum speed needed to escape Earth's gravitational field (reach infinity with zero KE):

v_e = sqrt{ rac{2GM_E}{R_E}} = sqrt{2gR_E} approx 11.2 ext{km/s}

It is independent of the mass of the escaping body and of the direction of projection (ignoring Earth's rotation and air resistance).

Derivation: Set total mechanical energy = 0 at infinity:

rac{1}{2}mv_e^2 - rac{GMm}{R_E} = 0 implies v_e = sqrt{ rac{2GM}{R_E}}

Escape velocity from any planet/moon: $v_e = sqrt{2gR}$ where $g$ and $R$ are for that body.

Relation to orbital speed: $v_e = sqrt{2} cdot v_{orbital}$ (at the surface)

Why Moon has no atmosphere: The rms speed of gas molecules on the Moon exceeds the Moon's escape velocity (

approx 2.4

km/s). So gas molecules escape into space and the Moon retains no appreciable atmosphere.

5. Orbital Motion and Satellites

A satellite in circular orbit at height $h$ above Earth's surface (orbital radius $r = R_E + h$ ):

Orbital speed:

v_o = sqrt{ rac{GM_E}{r}} = sqrt{ rac{GM_E}{R_E + h}}

For $h ll R_E$ : $v_o approx sqrt{gR_E} approx 7.9$ km/s. Orbital speed decreases as $h$ increases.

Time period:

T = rac{2pi r}{v_o} = 2pisqrt{ rac{r^3}{GM_E}}

This is Kepler's Third Law: $T^2 propto r^3$

Total energy of satellite:

KE = rac{GMm}{2r}, quad PE = - rac{GMm}{r}, quad E_{total} = - rac{GMm}{2r}

Total energy is negative (bound orbit). $|E| = KE = rac{1}{2}|PE|$ — the Virial theorem for circular orbits.

Geostationary satellite: Orbital period = 24 hours, altitude ≈ 36,000 km above equator, appears stationary relative to Earth, used for TV/communication.

Energy required to lift satellite to orbit at height $h$ :

Delta E = - rac{GMm}{2(R_E + h)} - left(- rac{GMm}{R_E} ight) = GMmleft( rac{1}{R_E} - rac{1}{2(R_E+h)} ight)

NEET tip: Weightlessness in a satellite is NOT because gravity is zero — gravity provides centripetal force! It is because both the satellite and occupants are in free fall together. The "normal force" from the floor on the person is zero, so they feel weightless.

6. Kepler's Laws of Planetary Motion

Kepler's First Law (Law of Orbits): Every planet moves in an elliptical orbit with the Sun at one focus.

Kepler's Second Law (Law of Areas): The line joining a planet to the Sun sweeps out equal areas in equal times.

rac{dA}{dt} = rac{L}{2m} = ext{constant}

This is a consequence of conservation of angular momentum. Planet moves fastest at perihelion (closest to Sun) and slowest at aphelion (farthest).

Kepler's Third Law (Law of Periods): The square of the orbital period is proportional to the cube of the semi-major axis $a$ :

T^2 propto a^3 quadimpliesquad rac{T^2}{a^3} = rac{4pi^2}{GM_{Sun}} = ext{constant for all planets}

For comparing two planets: $rac{T_1^2}{T_2^2} = rac{r_1^3}{r_2^3}$ (using orbital radii for circular orbits).

Pro tip: Kepler's second law → conservation of angular momentum → no tangential force → gravity is a central (radial) force. This is the logical chain. At perihelion: max speed, max KE, min PE, min r. At aphelion: min speed, min KE, max PE, max r.

7. NEET Traps & Formula Summary

Trap 1 — g ≠ 0 inside Earth:

g

is zero only at Earth's centre. At depth

d

g_d = g(1 - d/R)

. Inside a uniform spherical shell,

g = 0

everywhere (not just at centre).

Trap 2 — Orbital speed decreases with height:

v_o propto 1/sqrt{r}

. Higher orbit → slower speed but longer period. Boosting a satellite to a higher orbit actually slows it down!

Trap 3 — Escape velocity is 11.2 km/s from surface: From height

h

v_e = sqrt{2GM/(R+h)}

— less than the surface value. Escape velocity from orbit ≠ orbital speed.

Trap 4 — Gravitational PE is always negative: Binding energy =

|U| = GMm/r

. Total orbital energy is always negative (bound system). Positive total energy means unbound (hyperbolic path).

Formula Sheet:

Newton's law	$F = Gm_1m_2/r^2$
g at surface	$g = GM/R^2$
g at height h	$g_h = g(1-2h/R)$ (approx)
g at depth d	$g_d = g(1-d/R)$
Gravitational PE	$U = -GMm/r$
Escape velocity	$v_e = sqrt{2gR} approx 11.2$ km/s
Orbital speed	$v_o = sqrt{GM/r}$
Orbital period	$T = 2pisqrt{r^3/GM}$
Kepler's 3rd law	$T^2 propto r^3$
$v_e$ vs $v_o$	$v_e = sqrt{2}, v_o$

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