Gravitation formulas for JEE Main & Advanced

23 formula entries for Gravitation, in the order the chapter builds them. Each shows the formula, when to use it and the first trap to avoid.

Newton's Law of Gravitation

F = G \frac{m_{1} m_{2}}{r^{2}}

When to use. Use to compute gravitational force between two point masses or spherically symmetric objects.

Trap. Forgetting that the force is always attractive and acts along the line joining the centers.

Gravitational Field

{\vec{E}}_{g} = \frac{\vec{F}}{m} = - \frac{G M}{r^{2}} \hat{r}

When to use. Use to find the gravitational force on any mass at a point, or to find field due to a mass distribution.

Trap. Confusing field with force; field is a property of space, force depends on test mass.

Gravitational Potential

V = - \frac{G M}{r}

When to use. Use to compute work done by gravity or to find potential energy of a system.

Trap. Forgetting that potential is always negative (zero at infinity).

Gravitational Potential Energy

U = - \frac{G m_{1} m_{2}}{r}

When to use. Use to compute energy of a system or to apply conservation of mechanical energy in gravitational fields.

Trap. Forgetting the negative sign; it indicates bound system.

Superposition Principle

{\vec{F}}_{n e t} = \sum {\vec{F}}_{i}

{\vec{E}}_{n e t} = \sum {\vec{E}}_{i}

V_{n e t} = \sum V_{i}

When to use. Use whenever multiple masses are present, e.g., finding net force on a mass in a system.

Trap. Forgetting that force and field are vectors; adding them without considering direction.

Acceleration due to Gravity (g)

g = \frac{G M}{R^{2}}

(at surface)

When to use. Use to compute weight, free-fall motion, or pendulum period.

Trap. Confusing $g$ with $G$ ; $g$ varies, $G$ is constant.

Variation of g with Altitude and Depth

g_{h} = g (1 - \frac{2 h}{R})

(for

h ≪ R

);

g_{d} = g (1 - \frac{d}{R})

(inside Earth)

When to use. Use to compute g at a given height or depth, e.g., in satellite problems or mining.

Trap. Using the altitude formula for large $h$ ; use $g_{h} = \frac{G M}{(R + h)^{2}}$ instead.

Gravitational Field Inside a Uniform Sphere

E_{g} = \frac{G M}{R^{3}} r

(for

r < R

);

E_{g} = \frac{G M}{r^{2}}

(for

r \geq R

)

When to use. Use to find field inside a planet or star (assuming uniform density).

Trap. Forgetting that field inside a spherical shell is zero.

Gravitational Potential Inside a Uniform Sphere

V = - \frac{G M}{2 R^{3}} (3 R^{2} - r^{2})

(for

r < R

);

V = - \frac{G M}{r}

(for

r \geq R

)

When to use. Use to compute potential energy of a mass inside a planet or to find escape speed from inside.

Trap. Forgetting that potential is continuous at $r = R$ .

Kepler's First Law (Law of Orbits)

Ellipse:

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1

, with eccentricity

e = \sqrt{1 - \frac{b^{2}}{a^{2}}}

When to use. Use to describe planetary motion or satellite orbits.

Trap. Thinking orbits are circular; they are elliptical (circular is a special case).

Kepler's Second Law (Law of Areas)

\frac{d A}{d t} = \frac{L}{2 m} = constant

When to use. Use to relate speed at different points in an orbit (e.g., faster at perihelion).

Trap. Forgetting that this law implies angular momentum is conserved.

Kepler's Third Law (Law of Periods)

T^{2} = \frac{4 π^{2}}{G M} a^{3}

When to use. Use to find period of a satellite or planet, or to compare orbits.

Trap. Forgetting that $M$ is the mass of the central body.

Orbital Velocity

v_{o} = \sqrt{\frac{G M}{r}}

When to use. Use to find speed of a satellite in circular orbit.

Trap. Forgetting that $r$ is distance from center, not altitude above surface.

Escape Velocity

v_{e} = \sqrt{\frac{2 G M}{R}} = \sqrt{2 g R}

When to use. Use to determine if an object can leave a planet's gravity.

Trap. Forgetting that escape velocity is independent of mass of the object.

Energy of an Orbiting Satellite

E = - \frac{G M m}{2 r}

(kinetic:

\frac{G M m}{2 r}

, potential:

- \frac{G M m}{r}

)

When to use. Use to compute energy required to change orbit or to find binding energy.

Trap. Forgetting that total energy is half the potential energy (in magnitude).

Geostationary and Polar Satellites

Geostationary:

T = 24 h

r \approx 42, 000 km

from Earth's center

When to use. Use to describe communication satellites (geostationary) or remote sensing (polar).

Trap. Thinking geostationary satellites are at rest; they move with Earth's rotation.

Weightlessness

Apparent weight =

m (g - a)

; in free fall

a = g

, so weight = 0

When to use. Use to explain phenomena in spacecraft or elevators.

Trap. Thinking weightlessness means no gravity; gravity still acts.

Gravitational Binding Energy

U_{b i n d} = \frac{3 G M^{2}}{5 R}

(for uniform sphere)

When to use. Use to compute energy released in formation of a star or planet.

Trap. Using this formula for non-uniform bodies.

Gravitational Potential Energy of a System of Particles

U_{t o t a l} = - \sum_{i < j} \frac{G m_{i} m_{j}}{r_{i j}}

When to use. Use to find total energy of a system of point masses (e.g., stars in a cluster).

Trap. Double counting pairs; sum over $i < j$ only.

Gravitational Lensing

Deflection angle

θ = \frac{4 G M}{c^{2} b}

(where

b

is impact parameter)

When to use. Use to explain astronomical observations like Einstein rings.

Trap. Thinking this is a classical Newtonian effect; it's relativistic.

Gravitational Time Dilation

Δ t^{'} = Δ t \sqrt{1 - \frac{2 G M}{r c^{2}}}

When to use. Use to explain GPS corrections or time differences near massive objects.

Trap. Confusing with special relativistic time dilation.

Black Holes and Schwarzschild Radius

R_{s} = \frac{2 G M}{c^{2}}

When to use. Use to determine if an object will become a black hole (if $R < R_{s}$ ).

Trap. Thinking black holes are 'holes'; they are extremely dense objects.

Tidal Forces

Tidal acceleration difference

Δ a \approx \frac{2 G M}{r^{3}} Δ r

When to use. Use to explain ocean tides, Roche limit, or spaghettification near black holes.

Trap. Thinking tides are only due to Moon; Sun also contributes.

Values to remember

$G = 6.67 \times 10^{- 11} {N m}^{2} / {kg}^{2}$ Newton's Law of Gravitation
$g = 9.8 {m/s}^{2}$ Acceleration due to Gravity (g)
$R_{E} = 6.4 \times 10^{6} m$ Acceleration due to Gravity (g)
$v_{e} \approx 11.2 km/s$ for Earth Escape Velocity
$r_{g e o} \approx 4.2 \times 10^{7} m$ Geostationary and Polar Satellites
$h_{g e o} \approx 3.6 \times 10^{7} m$ Geostationary and Polar Satellites
$c = 3 \times 10^{8} m/s$ Gravitational Lensing

Read the full chapter: Gravitation for JEE, with what JEE asks and the common traps.