Gravitation Formula Sheet

Every formula, key relation, and must-know result from the Gravitation chapter, grouped by sub-topic for last-minute revision. Use it to scan before the exam, not to learn for the first time.

Universal Law of Gravitation

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

Vector form (attractive, along line of centres):

$$\vec{F}_{12} = -G\frac{m_1 m_2}{r^2}\hat{r}_{12}, \qquad \vec{F}_{12} = -\vec{F}_{21}$$

Superposition for several masses:

$$\vec{F}_{net} = \vec{F}_1 + \vec{F}_2 + \vec{F}_3 + \dots$$

Quantity	Formula / Value	Notes
Force between two masses	$F = G\dfrac{m_1 m_2}{r^2}$	$r$ = centre-to-centre distance
Dimensions of $G$	$[G] = M^{-1}L^3T^{-2}$	i.e. $\text{m}^3\,\text{kg}^{-1}\,\text{s}^{-2}$
Acceleration of a mass	$a = \dfrac{F}{m} = G\dfrac{M}{r^2}$	independent of $m$
Distance doubled	$F \to F/4$	inverse-square law
Both masses doubled	$F \to 4F$	—

Acceleration Due to Gravity (g) and Its Variation

Surface value (from $mg = GMm/R^2$):

$$\boxed{g = \frac{GM}{R^2}} \qquad (GM = gR^2)$$

Situation	Formula	Notes
At height $h$ (general)	$g_h = g\left(\dfrac{R}{R+h}\right)^2$	inverse-square
At height $h$ ($h \ll R$)	$g_h \approx g\left(1 - \dfrac{2h}{R}\right)$	note factor of 2; valid $h < 0.1R$
At depth $d$	$g_d = g\left(1 - \dfrac{d}{R}\right)$	linear; assumes uniform density
At Earth’s centre	$g = 0$	all forces cancel
Due to rotation, latitude $\lambda$	$g' = g - \omega^2 R\cos^2\lambda$	$\omega = 2\pi/T$
At poles ($\lambda = 90^\circ$)	$g' = g$	maximum
At equator ($\lambda = 0^\circ$)	$g' = g - \omega^2 R$	minimum

Headline forms:

$$\boxed{g_h = g\left(\frac{R}{R+h}\right)^2 \approx g\left(1 - \frac{2h}{R}\right)} \qquad \boxed{g_d = g\left(1 - \frac{d}{R}\right)}$$

Combined height + latitude (small $h$):

$$g' \approx g\left(1 - \frac{2h}{R}\right) - \omega^2 R\cos^2\lambda$$

Gravitational Field, Potential, and Potential Energy

$$\boxed{\vec{E} = \frac{\vec{F}}{m} = -\frac{GM}{r^2}\hat{r}} \qquad \boxed{V = -\frac{GM}{r}} \qquad \boxed{U = -\frac{GMm}{r}}$$

Quantity	Formula	Units	Scalar / Vector
Force	$F = \dfrac{GMm}{r^2}$	N	Vector (attractive)
Field intensity	$E = \dfrac{GM}{r^2}$	N/kg = m/s²	Vector (toward $M$)
Potential	$V = -\dfrac{GM}{r}$	J/kg	Scalar (negative)
Potential energy	$U = -\dfrac{GMm}{r}$	J	Scalar (negative)

Key relations:

$$E = \frac{F}{m}, \qquad V = \frac{U}{m}, \qquad E = -\frac{dV}{dr}, \qquad F = -\frac{dU}{dr}$$

Superposition (potential is scalar — add algebraically):

$$V_{total} = -G\left(\frac{M_1}{r_1} + \frac{M_2}{r_2} + \frac{M_3}{r_3}\right)$$

Values for Earth

Quantity	At surface ($r = R$)	At height $h$ ($r = R+h$)
Potential	$V = -\dfrac{GM}{R} = -gR$	$V_h = -\dfrac{GM}{R+h}$
Potential energy	$U = -\dfrac{GMm}{R} = -mgR$	$U_h = -\dfrac{GMm}{R+h}$

Change in PE moving from surface to height $h$:

$$\boxed{\Delta U = \frac{GMmh}{R(R+h)}} \xrightarrow{h \ll R} \Delta U \approx mgh$$

Work done against gravity (move $m$ from $r_1$ to $r_2$):

$$\boxed{W = m(V_2 - V_1) = GMm\left(\frac{1}{r_1} - \frac{1}{r_2}\right)}$$

Surface to infinity (binding energy of mass on surface):

$$W = \frac{GMm}{R}$$

Self-energy of a uniform solid sphere:

$$\boxed{U = -\frac{3GM^2}{5R}}$$

Escape Velocity

$$\boxed{v_e = \sqrt{\frac{2GM}{R}} = \sqrt{2gR}}$$

Derived from total mechanical energy $= 0$ at the surface (just reaches infinity with $v = 0$).

Quantity	Formula	Value for Earth
Escape velocity	$v_e = \sqrt{2gR}$	$\approx 11.2$ km/s
From height $h$	$v_e(h) = \sqrt{\dfrac{2GM}{R+h}} = v_e\sqrt{\dfrac{R}{R+h}}$	decreases with height
Relation to orbital	$v_e = \sqrt{2}\,v_o \approx 1.414\,v_o$	—
Binding energy (surface)	$E = \dfrac{GMm}{R} = mgR = \tfrac{1}{2}mv_e^2$	$= mgR$
Proportionality	$v_e \propto \sqrt{\dfrac{M}{R}} \propto \sqrt{gR}$	mass of object independent

Energy classification of trajectories:

• $E < 0$ → bound orbit
• $E = 0$ → parabolic escape (minimum escape, $v_\infty = 0$)
• $E > 0$ → hyperbolic escape ($v_\infty > 0$)

Speed at infinity when projected at $v = k\,v_e$ from the surface:

$$\boxed{v_\infty = v_e\sqrt{k^2 - 1}} \qquad (\text{generally } v_\infty = \sqrt{v^2 - v_e^2})$$

Atmospheric retention condition (gas RMS speed $v_{rms} = \sqrt{3kT/m}$):

$$v_{rms} < \frac{v_e}{6}$$

Body	$v_e$ (km/s)
Earth	11.2
Moon	2.4
Mars	5.0
Jupiter	60
Sun	618

Satellite Motion

Gravitational force provides centripetal force: $\dfrac{GMm}{r^2} = \dfrac{mv^2}{r}$.

$$\boxed{v_o = \sqrt{\frac{GM}{r}} = \sqrt{gR}\cdot\sqrt{\frac{R}{r}}} \qquad \boxed{T = 2\pi\sqrt{\frac{r^3}{GM}} = 2\pi\sqrt{\frac{r^3}{gR^2}}}$$

Parameter	Formula	Variation with $r$	Value (LEO, $h \approx 0$)
Orbital velocity	$v_o = \sqrt{\dfrac{GM}{r}}$	$v \propto r^{-1/2}$	$\approx 7.9$ km/s
Time period	$T = 2\pi\sqrt{\dfrac{r^3}{GM}}$	$T \propto r^{3/2}$	$\approx 84$ min
Kinetic energy	$KE = \dfrac{GMm}{2r}$	$\propto r^{-1}$	$\tfrac{1}{2}mgR$
Potential energy	$PE = -\dfrac{GMm}{r}$	$\propto r^{-1}$	$-mgR$
Total energy	$E = -\dfrac{GMm}{2r}$	$\propto r^{-1}$	$-\tfrac{1}{2}mgR$

Energy relations (circular orbit):

$$\boxed{E = -\frac{GMm}{2r} = \frac{PE}{2} = -KE} \qquad |PE| = 2\,KE, \quad |E| = KE$$$$\boxed{KE : |PE| : |E| = 1 : 2 : 1}$$

Period at surface ($r = R$): $T_0 = 2\pi\sqrt{\dfrac{R}{g}} \approx 84$ minutes (minimum possible period; same as a body dropped through a tunnel along Earth’s diameter).

Energy to launch from surface to circular orbit at height $h$:

$$\boxed{\Delta E = GMm\left(\frac{1}{R} - \frac{1}{2(R+h)}\right) = mgR\left(1 - \frac{R}{2(R+h)}\right)}$$

For low orbit ($h \ll R$): $\Delta E \approx \dfrac{mgR}{2}$ (half the escape energy).

Types of Satellites

Type	Height	Period	Notes
Low Earth Orbit (LEO)	200–2000 km	~90–120 min	ISS, Hubble; atmospheric drag
Medium Earth Orbit (MEO)	2,000–35,786 km	2–12 h	GPS at ~20,200 km, 12 h
Geostationary (GEO)	$\approx 35{,}786$ km	exactly 24 h	equatorial, west→east, appears fixed
Polar	~500–800 km	~100 min	covers whole Earth, mapping/weather

Kepler’s Laws of Planetary Motion

graph LR
    K[Kepler's Laws] --> L1["1st: Ellipse — Sun at a focus"]
    K --> L2["2nd: Equal areas in equal times → L conserved"]
    K --> L3["3rd: T² ∝ a³"]

First Law (Orbits): planets move in ellipses with the Sun at one focus.

$$r = \frac{a(1-e^2)}{1 + e\cos\theta}, \qquad e = \sqrt{1 - \frac{b^2}{a^2}}, \qquad r_1 + r_2 = 2a$$

Point	Distance from Sun
Perihelion (closest)	$a(1-e)$
Aphelion (farthest)	$a(1+e)$

Eccentricity cases: circle $e = 0$; ellipse $0 < e < 1$; parabola $e = 1$; hyperbola $e > 1$ (Earth’s orbit $e = 0.017$).

Second Law (Areas): equal areas swept in equal times.

$$\boxed{\frac{dA}{dt} = \frac{1}{2}r^2\omega = \frac{L}{2m} = \text{constant}}$$

This is conservation of angular momentum (gravity is central → zero torque), giving:

$$\boxed{r_p v_p = r_a v_a} \quad\Rightarrow\quad v_p > v_a \ (\text{faster at perihelion})$$

Third Law (Periods):

$$\boxed{T^2 \propto a^3} \qquad \boxed{\frac{T^2}{a^3} = \frac{4\pi^2}{GM} = \text{constant}}$$

For circular orbits $a = r$, so $T^2 = \dfrac{4\pi^2}{GM}r^3$. Mass of central body from an orbiting object:

$$\boxed{M = \frac{4\pi^2 a^3}{GT^2}}$$

Quantity	Proportionality
Period	$T \propto r^{3/2}$
Orbital speed	$v \propto r^{-1/2}$
Angular velocity	$\omega \propto r^{-3/2}$

Standard Constants and Reference Values

Quantity	Value
Universal gravitational constant $G$	$6.67 \times 10^{-11}$ N·m²/kg²
$g$ at Earth’s surface	$9.8$ m/s² (use 10 for quick work)
Earth’s mass $M_E$	$\approx 6 \times 10^{24}$ kg
Earth’s radius $R_E$	$\approx 6.4 \times 10^6$ m
$\omega^2 R$ (rotation term)	$\approx 0.034$ m/s²
Escape velocity (Earth)	$11.2$ km/s
Orbital velocity (surface)	$7.9$ km/s
Surface-orbit / tunnel period	$\approx 84$ minutes
Geostationary height	$\approx 35{,}786$ km

One-Line Connectors

• $GM = gR^2$ — the bridge between $G$-form and $g$-form of every formula.
• $v_e = \sqrt{2}\,v_o$ — escape vs orbital velocity at the same point.
• $E = -KE = PE/2$ — satellite energy shortcut ($1:2:1$).
• $r_p v_p = r_a v_a$ — angular-momentum bridge for elliptical orbits.
• $T^2 = \dfrac{4\pi^2}{GM}a^3$ — Kepler’s third law, same form for planets and satellites.

Related Topics

• Newton’s Law of Universal Gravitation
• Acceleration due to Gravity and Its Variation
• Gravitational Field and Potential
• Kepler’s Laws of Planetary Motion
• Escape Velocity
• Satellite Motion