Unit 6: Gravitation

NEET sets 1–2 MCQs from Gravitation each year. The questions are tightly bound to a small set of formulas (Kepler, escape, orbital, PE, $g$-variation), so this is one of the easiest scoring units once the formulas are memorised.

The trickiest sub-topic is the variation of $g$ with altitude, depth and latitude — students often mix up the formulas. Take time to derive each from scratch.

Concept Map

• Kepler's three laws
• Universal law of gravitation
• Gravitational field $g(r)$
• Variation of $g$: altitude, depth, rotation/latitude, shape of Earth
• Gravitational PE (general)
• Escape velocity and orbital velocity
• Satellite energies and time period
• Types of satellites (geostationary, polar)
• Weightlessness

Topic 1: Kepler's Laws

Sub-topic A: First Law (Law of Orbits)

Every planet revolves around the Sun in an ellipse, with the Sun at one focus.

Sub-topic B: Second Law (Law of Areas)

The line joining the planet to the Sun sweeps equal areas in equal times. This is a statement of conservation of angular momentum since gravity is central.

A consequence: planets move faster at perihelion (close to Sun) and slower at aphelion.

Sub-topic C: Third Law (Law of Periods)

where $a$ is the semi-major axis. For two planets,

Topic 2: Newton's Universal Law

It acts along the line joining the two masses, always attractive.

Sub-topic A: Acceleration Due to Gravity at Earth's Surface

Equivalently $G M_E = g R_E^2 \approx 4 \times 10^{14}\ \text{m}^3/\text{s}^2$.

Sub-topic B: Gravitational Field

Outside a uniform sphere, the field equals that of a point mass at the centre (shell theorem).

Inside a uniform solid sphere of mass $M$ and radius $R$, at distance $r < R$ from centre:

So $g$ grows linearly from 0 (at centre) to $g_\text{surface}$ (at $r = R$), then falls as $1/r^2$.

Topic 3: Variation of g

Sub-topic A: With Altitude h Above Surface

So $g$ decreases with altitude.

Sub-topic B: With Depth d Below Surface

So $g$ also decreases with depth (linearly), reaching zero at the centre.

For the same fractional decrease in $g$, altitude $h = d/2$ (approximately).

Sub-topic C: With Latitude (rotation effect)

Earth's rotation introduces an apparent reduction in $g$ at latitude $\phi$:

• At equator ($\phi = 0$): $g'_\text{eq} = g - \omega^2 R$. The reduction is $\omega^2 R \approx 0.034\ \text{m/s}^2$.
• At poles ($\phi = 90°$): no reduction.

If Earth rotated 17× faster, $g$ at equator would vanish (objects would fly off).

Sub-topic D: Shape of Earth

Earth is an oblate spheroid: $R_\text{equator} > R_\text{pole}$ by about 21 km. Hence $g_\text{pole} > g_\text{equator}$ (combined with rotation effect, polar $g$ is about 0.5% larger).

Topic 4: Gravitational PE

Sub-topic A: General Formula

For two point masses $m$ and $M$ separated by $r$:

The choice $U(\infty) = 0$ makes $U$ always negative (bound states).

Sub-topic B: PE Near Earth's Surface

For small height $h$ above surface, $\Delta U = mgh$ (taken from surface as reference).

Sub-topic C: PE for a Body of Mass $m$ at Earth's Surface

Topic 5: Escape and Orbital Velocity

Sub-topic A: Escape Velocity

Minimum speed at the surface needed to escape Earth's gravity (reach infinity with zero KE):

Notes:

• Independent of mass of the escaping body and direction of launch.
• $v_e$ at the Moon is about 2.4 km/s.
• For a planet of density $\rho$: $v_e = R\sqrt{8 \pi G \rho / 3}$, so denser/larger planets have higher $v_e$.

Sub-topic B: Orbital Velocity

For circular orbit at radius $r$ from Earth's centre:

For low Earth orbit ($r \approx R_E$): $v_o = \sqrt{g R_E} \approx 7.9\ \text{km/s}$.

Relation: $v_e = \sqrt{2}\,v_o$. Hence at any altitude $v_e/v_o = \sqrt{2}$.

Sub-topic C: Time Period

For low Earth orbit: $T \approx 84\ \text{min}$.

Topic 6: Satellite Energies

For a satellite of mass $m$ orbiting at radius $r$:

• KE: $K = \tfrac{1}{2} m v_o^2 = G M m/(2 r)$.
• PE: $U = -G M m/r$.
• Total energy: $E = K + U = -G M m/(2r) = -K$.

So total energy is negative (bound) and equals half the PE in magnitude. To move a satellite from radius $r_1$ to $r_2$ ($r_2 > r_1$), the energy required is

Topic 7: Geostationary and Polar Satellites

Sub-topic A: Geostationary Satellite (GEO)

• Period $T = 24\ \text{h}$ (synced to Earth's rotation).
• Orbits in the equatorial plane, west to east.
• Altitude: solving $T^2 = 4\pi^2 r^3/(GM)$ gives $r \approx 42\,200\ \text{km}$, i.e. altitude $\approx 36\,000\ \text{km}$ above surface.
• Used for telecommunications, weather monitoring (Insat, etc.).

Sub-topic B: Polar Satellite

• Lower altitude (~ 800 km), short period (~ 100 min).
• Orbits in polar plane.
• Used for remote sensing, mapping; covers entire Earth surface as Earth rotates beneath.

Sub-topic C: Geosynchronous vs Geostationary

A geosynchronous satellite has 24-h period but need not be equatorial; a geostationary is geosynchronous and equatorial.

Topic 8: Weightlessness

A body in free-fall (in a falling lift, orbiting satellite, parabolic-flight plane) experiences apparent weight zero — the only forces on it are inertial and gravitational, which combine to produce $g$ acceleration, so the normal force from any contact surface is zero. This is apparent weightlessness, not absence of gravity.

In an orbiting satellite the astronaut is in continuous free fall around the Earth.

NEET Pattern MCQ Tips

• Kepler's third law numerical: given $T_1$ and $a$'s, find $T_2$.
• Escape velocity recall: $v_e = \sqrt{2gR}$ or $\sqrt{2GM/R}$.
• Orbital vs escape: $v_e = \sqrt{2}\,v_o$.
• Variation of g: pick the right formula for altitude (quadratic decrease) vs depth (linear decrease).
• Geostationary altitude: 36,000 km.
• Energy of orbit: $E = -GMm/(2r)$.
• Assertion-reason: weightlessness in satellites, value of $g$ at centre of Earth (zero).

Common Confusions and Traps

• $v_e$ is independent of the mass of the escaping body and the angle of projection — many students think a steeper angle requires more speed.
• $g$ decreases with both altitude and depth — but with different formulas (quadratic vs linear).
• Earth's rotation reduces apparent $g$ only at non-polar latitudes; no effect at the poles.
• Astronauts in orbit are not free of gravity — they are in continuous free fall.
• $T^2 \propto r^3$ uses the semi-major axis for elliptical orbits, not the radius (only equals radius for circular orbit).
• The gravitational field inside a uniform sphere is linear in $r$, not $1/r^2$.
• Geostationary satellites must orbit in the equatorial plane.

Quick Revision Card

• $F = Gm_1m_2/r^2$; $g = GM/R^2$.
• Inside uniform sphere: $g(r) = GMr/R^3$.
• $g_h \approx g(1 - 2h/R)$ (altitude).
• $g_d = g(1 - d/R)$ (depth).
• Latitude effect: $g' = g - \omega^2 R \cos^2\phi$.
• $v_e = \sqrt{2gR} \approx 11.2$ km/s.
• $v_o = \sqrt{gR}$ at surface, $\approx 7.9$ km/s.
• $v_e = \sqrt{2}\,v_o$.
• $T^2 = 4\pi^2 r^3/(GM)$.
• Total satellite energy: $E = -GMm/(2r)$, half of PE in magnitude.
• Geostationary: $T = 24$ h, altitude $\approx 36,000$ km, equatorial.

Worked NEET Examples

Example 1: Find $g$ at Height $h = R/2$

$g_h = g(R/(R + h))^2 = g(R/(3R/2))^2 = g(2/3)^2 = 4g/9$.

Example 2: Time Period of Pendulum at Depth $d$

$g_d = g(1 - d/R)$, so $T_d = 2\pi\sqrt{L/g_d} = T_0/\sqrt{1 - d/R}$. At $d = R$ (centre): $T \to \infty$ (no gravity, no oscillation).

Example 3: Two Stars Orbiting Each Other

Two stars of equal mass $M$ orbiting their common center, each at radius $r$ (separation $2r$). Each provides gravitational force on the other:

Centripetal: $F = M\omega^2 r$. So $\omega^2 = GM/(4r^3)$ and $T = 2\pi\sqrt{4r^3/(GM)} = 4\pi\sqrt{r^3/(GM)}$.

Example 4: Satellite Reaches Half the Earth's Radius Above Surface

Orbit radius $r = R + R/2 = 3R/2$. Orbital velocity:

Period: $T = 2\pi\sqrt{(3R/2)^3/(GM)} = 2\pi(3R/2)^{3/2}/(gR^2)^{1/2}$.

Example 5: Escape Velocity from a Planet

A planet has half Earth's mass and a quarter of Earth's radius. Then $v_e \propto \sqrt{M/R} = \sqrt{(1/2)/(1/4)} = \sqrt{2}$ times Earth's $v_e$. So $v_e \approx 1.41 \times 11.2 = 15.9$ km/s.

Derivations

Escape Velocity from Energy Conservation

At surface, total energy $E_\text{surf} = \tfrac{1}{2}mv_e^2 - GMm/R$. To just escape, $E_\infty = 0$, so

Orbital Velocity

Gravitational force provides centripetal: $GMm/r^2 = mv_o^2/r$, so $v_o = \sqrt{GM/r}$.

Kepler's Third Law from Newton's Gravity

For circular orbit: $GM/r^2 = \omega^2 r$, so $\omega^2 = GM/r^3$. Period:

Generalises to elliptical orbits with $r \to a$ (semi-major axis).

g Variation with Depth

Treat Earth as uniform sphere of density $\rho$. At depth $d$ from surface (so distance $r = R - d$ from center), only mass within radius $r$ contributes (shell theorem):

At surface: $g_R = (4/3)\pi G \rho R$. So $g(r)/g_R = r/R$, and $g_d = g(1 - d/R)$.

Total Energy of a Satellite

KE: $K = \tfrac{1}{2}m v_o^2 = GMm/(2r)$.

PE: $U = -GMm/r$.

Total: $E = K + U = -GMm/(2r) = -K$.

The negative total energy signifies a bound orbit. To escape, energy must be supplied to make $E = 0$.

Satellites and Practical Applications

Geostationary Orbit Calculation

Requirements: $T = 24$ h = 86400 s. From $T^2 = 4\pi^2 r^3/(GM)$:

So $r \approx 4.22 \times 10^7$ m. Altitude above surface: $\approx 4.22 \times 10^7 - 6.4 \times 10^6 \approx 3.58 \times 10^7$ m or about 36,000 km.

Polar Satellites

Polar satellites orbit in north-south planes. As Earth rotates beneath, they cover the entire surface over time. Used for mapping (IRS), weather (NOAA), spy satellites. Altitude: 700–800 km, period: ~100 min.

Sun-Synchronous Orbits

A type of polar orbit where the satellite passes over the same point on Earth at the same local solar time daily. Useful for consistent lighting in imaging.

Inertial Mass vs Gravitational Mass

• Inertial mass appears in $F = ma$ — resistance to acceleration.
• Gravitational mass appears in $F = GMm/r^2$ — gravitational charge.

The equivalence principle (Einstein) states these are equal — confirmed to high precision by Eötvös experiment. In free fall all bodies accelerate identically because gravity force ($m_g g$) divided by inertial mass ($m_i$) is the same for all — hence the universality of free fall, and the local indistinguishability between gravity and acceleration.

Black Holes

A black hole is an object so dense that escape velocity exceeds the speed of light. Schwarzschild radius:

For Earth mass: $R_s \approx 9$ mm. For Sun: $R_s \approx 3$ km. For supermassive black hole at galactic center: $\sim$ AU-scale.

Formula Sheet