Essential concepts and memory tricks for mastering Gravitation

Newton's Universal Law of 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 = Gm₁m₂/r². This force acts along the line joining the centers of the masses and is always attractive.

Gravitational Constant G and Its Significance

G = 6.67 × 10⁻¹¹ N⋅m²/kg² is the universal gravitational constant. It represents the gravitational force between two unit masses separated by unit distance. Measured by Cavendish experiment, it's universal - same everywhere in the universe.

Acceleration Due to Gravity and Variations

g = GM/R² ≈ 9.8 m/s² at Earth's surface. Varies with height: g_h = g(R/(R+h))², with depth: g_d = g(1-d/R), and with latitude due to Earth's rotation. Maximum at poles, minimum at equator.

Kepler's Laws of Planetary Motion

First Law: Planets move in elliptical orbits with Sun at one focus. Second Law: Line joining planet and Sun sweeps equal areas in equal times (areal velocity constant). Third Law: Square of period proportional to cube of semi-major axis (T² ∝ a³).

Gravitational Potential and Potential Energy

Gravitational potential V = -GM/r is work done per unit mass to bring from infinity. Potential energy U = -GMm/r is always negative (attractive force). Zero at infinity, becomes more negative as objects approach.

Escape Velocity and Orbital Velocity

Escape velocity v_e = √(2GM/R) = √(2gR) ≈ 11.2 km/s for Earth. Minimum speed to escape gravitational field. Orbital velocity v_o = √(GM/r) for circular orbit. Relationship: v_e = √2 × v_o.

Satellite Motion and Energy

Satellites in circular orbits have KE = GMm/2r, PE = -GMm/r, Total Energy = -GMm/2r. Binding energy equals negative of total energy. Higher orbit = less speed but more energy needed to reach there.

Tides and Gravitational Effects

Tides caused by differential gravitational force of Moon (and Sun) on Earth. Tidal force ∝ 1/r³ (more sensitive to distance than gravitational force ∝ 1/r²). Spring tides when Sun and Moon align, neap tides when perpendicular.

All essential gravitation formulas with LaTeX equations and simple explanations

Concept Formula Meaning in Simple Words When to Use
Newton's Law of Gravitation \(F = G\frac{m_1 m_2}{r^2}\) Gravitational force between two masses is proportional to masses and inversely proportional to distance squared To calculate gravitational force between any two objects
Gravitational Field \(g = \frac{F}{m} = \frac{GM}{r^2}\) Gravitational field is force per unit mass at any point To find gravitational acceleration at distance r from mass M
Acceleration Due to Gravity (Surface) \(g = \frac{GM}{R^2}\) Surface gravity depends on planet's mass and radius To calculate surface gravity of any planet or celestial body
Variation of g with Height \(g_h = g\left(\frac{R}{R+h}\right)^2 \approx g\left(1 - \frac{2h}{R}\right)\) Gravity decreases with height above Earth's surface For objects at altitude h above Earth's surface (h << R)
Variation of g with Depth \(g_d = g\left(1 - \frac{d}{R}\right)\) Gravity decreases linearly with depth inside Earth For objects at depth d below Earth's surface
Gravitational Potential \(V = -\frac{GM}{r}\) Gravitational potential is work done per unit mass to bring from infinity To calculate potential at distance r from mass M
Gravitational Potential Energy \(U = -\frac{GMm}{r}\) Potential energy between two masses (always negative for attractive force) For energy calculations in gravitational systems
Escape Velocity \(v_e = \sqrt{\frac{2GM}{R}} = \sqrt{2gR}\) Minimum velocity needed to escape gravitational field completely To find speed needed for object to escape from planet's surface
Orbital Velocity (Circular) \(v_o = \sqrt{\frac{GM}{r}}\) Velocity needed for circular orbit at distance r from center For satellites in circular orbits around planets
Orbital Period \(T = 2\pi\sqrt{\frac{r^3}{GM}}\) Time for one complete orbit around mass M at distance r To calculate time period of satellites or planetary motion
Kepler's Third Law \(\frac{T^2}{a^3} = \frac{4\pi^2}{GM}\) or \(T^2 \propto a^3\) Square of orbital period proportional to cube of semi-major axis For relating periods and distances of different orbits
Energy of Satellite (Circular Orbit) \(E = -\frac{GMm}{2r}\), \(KE = \frac{GMm}{2r}\), \(PE = -\frac{GMm}{r}\) Total energy is negative, kinetic energy is positive, potential energy is negative For energy analysis of satellites in circular orbits
Binding Energy \(BE = \frac{GMm}{2r}\) Energy required to remove satellite from orbit to infinity To calculate energy needed to escape from orbit
Geostationary Satellite Height \(h = \left(\frac{GMT^2}{4\pi^2}\right)^{1/3} - R\) Height above Earth's surface for 24-hour orbital period For geostationary satellite calculations (T = 24 hours)
Relationship Between Escape and Orbital Velocity \(v_e = \sqrt{2} \times v_o\) Escape velocity is √2 times the orbital velocity at same distance To relate escape and orbital speeds for any celestial body

Systematic approach to solve Gravitation problems efficiently

1

Identify the Gravitational System

Determine what objects are involved (planets, satellites, masses), their masses, distances, and what needs to be found. Identify if it's about gravitational force, motion, energy, or orbital mechanics.

2

Draw Clear Diagrams

Sketch the system showing all masses, distances, orbital paths, and directions. Label known quantities and mark what needs to be found. Include coordinate system if needed for complex problems.

3

Choose Appropriate Formula

Select correct formula based on problem type: F = GMm/r² for forces, energy formulas for orbital problems, Kepler's laws for planetary motion, or variations of g for height/depth problems.

4

Apply Conservation Laws When Needed

Use conservation of energy for escape velocity and orbital problems. For satellite problems: Total Energy = KE + PE = constant. Consider mechanical energy conservation in gravitational fields.

5

Handle Height and Depth Variations

For problems involving altitude, use g_h = g(R/(R+h))² or approximation g_h ≈ g(1-2h/R) for small heights. For depth problems, use g_d = g(1-d/R). Remember g varies!

6

Use Proper Sign Conventions

Gravitational potential energy is always negative: U = -GMm/r. Potential V = -GM/r. Total energy is negative for bound systems. Work done against gravity is positive.

7

Check Units and Convert If Necessary

Ensure consistent units: masses in kg, distances in meters, G = 6.67×10⁻¹¹ N⋅m²/kg². Convert km to m, hours to seconds, etc. Verify final answer units make sense.

8

Verify Physical Reasonableness

Check if answers make physical sense: escape velocity > orbital velocity, gravity decreases with height/depth, satellite period increases with orbital radius, binding energy is positive.

Avoid these common errors to improve your exam performance

Mistake How to Avoid
Confusing mass and weight in gravitational problems Remember mass (kg) is invariant, weight = mg varies with g. Use mass in gravitational formulas, not weight. Weight depends on local gravitational field strength.
Using constant g = 9.8 m/s² for all heights and depths Use g = 9.8 m/s² only at Earth's surface. For height h: g_h = g(R/(R+h))². For depth d: g_d = g(1-d/R). Gravity varies with position!
Incorrect application of escape velocity formula Escape velocity v_e = √(2GM/R) is from surface. For height h, use v_e = √(2GM/(R+h)). Remember escape velocity is independent of object's mass.
Mixing up orbital velocity and escape velocity Orbital velocity keeps object in orbit: v_o = √(GM/r). Escape velocity allows escape: v_e = √(2GM/r). Relationship: v_e = √2 × v_o. They're different!
Wrong sign for gravitational potential energy Gravitational PE is always negative: U = -GMm/r. Zero at infinity, becomes more negative as objects approach. This reflects attractive nature of gravity.
Misunderstanding Kepler's laws applications Kepler's laws apply to elliptical orbits. T² ∝ a³ where a is semi-major axis, not radius. For circular orbits, a = r. Use GM_total in the constant of proportionality.
Incorrect use of conservation of energy in gravity Total energy E = KE + PE = ½mv² - GMm/r. For escape: set E = 0. For circular orbits: E = -GMm/2r. Remember PE is negative, KE is positive.
Forgetting to convert units in calculations Always convert to SI units: km→m, g→kg, hours→seconds. G = 6.67×10⁻¹¹ N⋅m²/kg². Check that final units match expected answer type (m/s, J, N, etc.).
Wrong interpretation of negative total energy Negative total energy means bound system (satellite in orbit). Positive total energy means unbound (escape trajectory). Zero total energy is limiting case for escape.
Confusion between gravitational field and acceleration Gravitational field g = GM/r² is force per unit mass. Acceleration due to gravity has same value but conceptually different. Field exists even without test mass.

Quick memory aids and essential information for last-minute revision

Quick Gravitation Formulas

  • Newton's law: F = GMm/r²
  • Surface gravity: g = GM/R²
  • Escape velocity: v_e = √(2GM/R) = √(2gR)
  • Orbital velocity: v_o = √(GM/r)

Kepler's Laws Memory Aids

  • Law 1: Elliptical orbits, Sun at focus
  • Law 2: Equal areas in equal times (areal velocity constant)
  • Law 3: T² ∝ a³ (period² ∝ distance³)
  • All laws explained by Newton's gravitation + circular motion

Escape & Orbital Velocity Relations

  • v_e = √2 × v_o (escape = √2 × orbital)
  • v_e independent of object mass
  • Earth's v_e ≈ 11.2 km/s, v_o ≈ 7.9 km/s (at surface)
  • Higher orbit → slower orbital speed but more energy

Important Constants & Values

  • G = 6.67 × 10⁻¹¹ N⋅m²/kg²
  • Earth: g = 9.8 m/s², R = 6.4 × 10⁶ m
  • Earth escape velocity = 11.2 km/s
  • Geostationary orbit height ≈ 36,000 km

Variation of g Formulas

  • Height: g_h = g(R/(R+h))² ≈ g(1-2h/R)
  • Depth: g_d = g(1-d/R)
  • At center of Earth: g = 0
  • Maximum g at surface, zero at infinity and center

Energy in Gravitational Fields

  • PE = -GMm/r (always negative)
  • For circular orbit: KE = GMm/2r, E = -GMm/2r
  • Binding energy = GMm/2r (positive)
  • Conservation: E_initial = E_final

Common Problem Types

  • Force problems → use F = GMm/r²
  • Orbital motion → use v = √(GM/r), T = 2π√(r³/GM)
  • Escape problems → use energy conservation
  • Height/depth variations → use modified g formulas

Exam Strategy Tips

  • Always draw diagrams showing masses and distances
  • Check if g varies or remains constant in problem
  • Use energy methods for escape/orbital problems
  • Verify units: force (N), energy (J), velocity (m/s)