Understanding Gravity — The Physics Behind the Game
Gravity is the weakest of the four fundamental forces of nature — yet it dominates the universe at large scales because it acts over infinite range and only attracts, never repels. It holds planets in orbit around stars, governs the motion of galaxies, shaped the formation of the solar system from a rotating disk of gas and dust, and keeps your feet on the ground right now. The Gravity Lab game lets you explore how gravitational force, mass, and distance interact in real-time — building the physical intuition that equations alone cannot provide.
Newton's Law of Universal Gravitation
Isaac Newton formulated the law of universal gravitation in 1687: every particle of matter attracts every other particle with a force proportional to the product of their masses and inversely proportional to the square of the distance between them. In equation form: F = GMm/r², where G = 6.674 × 10⁻¹¹ N·m²/kg² is the gravitational constant, M and m are the masses, and r is the separation.
The inverse-square law — force proportional to 1/r² — is one of the most important mathematical relationships in physics. It arises because gravity spreads in three dimensions: the gravitational "influence" from a point mass spreads outward over the surface of a sphere, and sphere surface area increases as r². Doubling the distance reduces the force to one-quarter. Tripling the distance reduces it to one-ninth.
The law applies between all masses, not just planets. The gravitational force between you and a friend standing 1 m away is approximately F = 6.674×10⁻¹¹ × 70 × 70 / 1² ≈ 3.3 × 10⁻⁷ N — detectable in principle but dwarfed by the electromagnetic forces between your atoms. Gravity becomes dominant only at planetary scales because it accumulates over enormous masses of matter.
Gravitational Field Strength
The gravitational field strength g at a point is the gravitational force per unit mass experienced there: g = F/m = GM/r². For Earth's surface: g = GM_Earth/R_Earth² = (6.674×10⁻¹¹ × 5.972×10²⁴)/(6.371×10⁶)² = 9.81 N/kg = 9.81 m/s². This is the acceleration all objects experience in free fall near Earth's surface — independent of mass.
Field strength decreases with altitude: at 400 km (ISS altitude), g = 9.81 × (6371/6771)² = 8.67 m/s² — about 11% less than at the surface. At the Moon's distance (384,400 km), Earth's gravitational field has fallen to about 0.0027 m/s² — still enough to maintain the Moon in orbit.
The gravitational field is a vector field — at every point in space, it has both magnitude (the field strength) and direction (toward the source mass). For a planet, the field lines point radially inward toward the centre. For multiple masses, the total field at any point is the vector sum of contributions from each mass. This superposition principle is what the Gravity Lab game demonstrates when you add multiple masses.
Orbital Motion and Circular Orbits
For a satellite in a circular orbit of radius r, gravity provides the centripetal force: GMm/r² = mv²/r → v = √(GM/r). This is the orbital velocity. Faster and the orbit rises; slower and it falls. For a given orbit radius, there is exactly one correct orbital speed — too fast and the satellite spirals outward, too slow and it falls inward.
The period T = 2πr/v = 2π√(r³/GM) — Kepler's Third Law. Squaring: T² = (4π²/GM)r³. This means T² ∝ r³: doubling the orbital radius increases the period by 2^(3/2) = 2.83×. The Moon's period (27.3 days) and orbital radius (384,400 km) and Earth's mass can be used to verify this relationship precisely.
Geostationary orbit — at radius ~42,164 km where the orbital period equals exactly 24 hours — keeps satellites stationary above a fixed point on Earth. This is where communication satellites, weather satellites, and GPS infrastructure live. The specific radius comes from solving T = 24 hours in T = 2π√(r³/GM_Earth).
Gravitational Potential Energy
Near Earth's surface, GPE = mgh (choosing ground as zero). At larger distances, the formula GPE = −GMm/r applies, with zero at infinity. The negative sign means objects are bound to the planet — they have less energy than at infinity. The minimum energy needed to escape is zero total energy: ½mv_esc² − GMm/R = 0 → v_esc = √(2GM/R) = 11.2 km/s for Earth.
Conservation of energy in orbits: total mechanical energy E = ½mv² − GMm/r. For a circular orbit: E = −GMm/(2r) — exactly half the potential energy. Lower orbits have more negative total energy — they are more tightly bound. This is why lowering a satellite's orbit requires removing energy (firing retro-rockets), while raising it requires adding energy — a counterintuitive but fundamental result of orbital mechanics.
General Relativity — Gravity as Curved Spacetime
Einstein's General Theory of Relativity (1915) provided a deeper understanding of gravity: it is not a force but a curvature of spacetime caused by mass and energy. Objects follow the straightest possible paths (geodesics) through curved spacetime — what we observe as gravitational acceleration is simply the effect of following a geodesic in curved spacetime near a massive object.
For most practical purposes (engineering, spacecraft navigation, everyday physics), Newtonian gravity gives correct answers. General relativistic corrections become important for: GPS satellite timing (clocks run faster at altitude — requires a 45 μs/day correction), gravitational waves (ripples in spacetime from accelerating masses, detected by LIGO in 2015), black holes (where gravity is so strong even light cannot escape), and the large-scale structure of the universe.
The equivalence principle — that gravitational acceleration is locally indistinguishable from acceleration due to any other force — was Einstein's key insight. A person in a closed box cannot tell whether they are on Earth's surface (gravitational field g) or accelerating at g in empty space. This equivalence connects gravity to inertia in a profound way and is the foundation of general relativity.
Tidal Forces and the Roche Limit
Gravity is not uniform — it decreases with distance. When a large body orbits near another, the gravitational pull on the near side is stronger than on the far side. This difference — the tidal force — stretches the body along the line connecting the two objects and compresses it perpendicular to that line. Earth's tides are caused by tidal forces from the Moon (and to a lesser extent the Sun).
The Roche limit is the minimum orbital radius at which tidal forces exceed the self-gravity holding a smaller body together. Inside the Roche limit, tidal forces would disrupt a fluid satellite — breaking it into a ring. Saturn's rings exist because the material was unable to coalesce into a moon inside Saturn's Roche limit (~140,000 km). This is pure gravitational physics playing out at planetary scale.