Gravitational Force Calculator

Gravity and Gravitation Guide

What's the key thing to understand about Newton's Law of Universal Gravitation?

F = Gm₁m₂/r², where G = 6.674×10⁻¹¹ N⋅m²/kg², m₁ and m₂ are the two masses in kg, and r is the distance between their centres in metres. Weight on Earth = mass × g, where g = 9.81 m/s² at sea level. Your 'weight' actually varies slightly across Earth's surface — it is about 0.5% less at the equator than at the poles due to Earth's rotation and slightly flattened shape (centrifugal effect and slightly larger radius at the equator).

What do I need to know about Weight on Other Planets?

Gravitational acceleration varies with planetary mass and radius. On the Moon (g = 1.62 m/s²): a 70kg person weighs 113N — about 16% of Earth weight. On Mars (g = 3.72 m/s²): about 38% of Earth weight. On Jupiter (g = 24.79 m/s²): 252% — a 70kg person would weigh about 1,735N and have extreme difficulty even standing. On the Moon, the lack of gravity and atmosphere makes daily activities drastically different — astronauts must be careful not to develop muscle and bone loss (which begins within days of exposure to microgravity), requiring dedicated daily exercise routines to slow the loss during long space missions.

What's the key thing to understand about Escape Velocity?

Escape velocity is the minimum speed needed to escape a body's gravitational pull without further propulsion: v = √(2GM/r). Earth: 11.2 km/s (40,320 km/h). Moon: 2.38 km/s. Mars: 5.03 km/s. Jupiter: 59.5 km/s. This is why it is so difficult (energetically) to leave Earth's gravity well — a rocket must accelerate to 11.2 km/s, requiring enormous fuel. The Saturn V moon rocket used more fuel in its first stage than all previous rocket launches combined.

What do I need to know about Tidal Forces and Roche Limit?

Gravitational force varies with distance — the near side of a body (like the Moon relative to Earth) experiences stronger gravitational pull than the far side, creating a tidal force that stretches the body. The Roche limit is the minimum orbital radius at which a satellite can survive without being torn apart by tidal forces — bodies orbiting within it (like Saturn's rings) are shredded into ring particles. Earth's tides are caused by the Moon's gravitational tidal force — the ocean bulges both on the side facing the Moon and the opposite side, producing two high tides roughly every 24 hours.

What should I know about Newton's Law of Gravitation Formula?

F = G×m₁×m₂/r², where G = 6.674×10⁻¹¹ N·m²/kg² (the gravitational constant), m₁ and m₂ are masses in kg, and r is separation in metres, centre to centre. For two 1000kg masses 1m apart: F = 6.674×10⁻¹¹ × 1000 × 1000 / 1² ≈ 6.674×10⁻⁵ N — an almost immeasurably tiny force, which is exactly why gravity between everyday objects is never noticeable; it only becomes significant when at least one mass is planet-sized. For Earth (5.97×10²⁴ kg) and a 70kg person at Earth's surface (r≈6.371×10⁶ m): F = 6.674×10⁻¹¹×5.97×10²⁴×70/(6.371×10⁶)² ≈ 686 N, matching the familiar weight = mg calculation (70×9.8=686N) — confirming that g=9.8 m/s² is itself just GM_Earth/r² evaluated at Earth's radius.

What do I need to know about Gravity's Inverse-Square Law in Practice?

Because gravitational force falls off as 1/r², doubling the distance between two masses reduces the force to a quarter, and tripling the distance reduces it to a ninth. This is why astronauts on the International Space Station (orbiting roughly 400km above Earth's surface, barely 6% further from Earth's centre than someone standing at sea level) still experience about 90% of surface gravity — their apparent weightlessness is due to orbital free-fall, not distance from Earth. It also explains why gravitational force changes so dramatically near a black hole or neutron star: as r shrinks toward zero, 1/r² grows without bound, producing the extreme tidal forces that would stretch an object apart (different parts of the object are at meaningfully different r values, experiencing meaningfully different force) long before it reached the object itself — the effect known as spaghettification.

What should I know about Gravitational Field Strength and Altitude?

Gravitational field strength g = GM/r² falls off with the square of distance from a planet's centre, which is why 'r' must be measured from the centre of mass, not from the surface. At the top of Mount Everest (about 8.8 km up), Earth's g drops only slightly, from 9.81 to about 9.78 m/s², because Earth's radius (≈6,371 km) is so much larger than the altitude gained. At the orbital altitude of the ISS (≈400 km), g is still about 8.7 m/s² — astronauts experience weightlessness not because gravity has vanished, but because they and their spacecraft are in continuous free fall together.

What should I know about Gravitational Potential Energy and Orbits?

Gravitational potential energy between two masses is given by U = −GMm/r, where the negative sign reflects the convention that potential energy is zero at infinite separation and becomes increasingly negative as the masses move closer together — meaning gravity does positive work and the system becomes more tightly bound. This formula differs from the simpler U = mgh used near Earth's surface, which is only a local, flat-Earth approximation valid for small height changes where g is essentially constant; the full −GMm/r form is needed for satellites, planetary motion, and anything where distance from the central mass varies significantly. For a circular orbit, the total mechanical energy (kinetic plus potential) works out to E = −GMm/(2r), exactly half the magnitude of the potential energy, a result called the virial theorem for gravitational orbits. This negative total energy signifies that the orbiting body is gravitationally bound: it does not have enough energy to escape to infinity. An object given exactly enough kinetic energy to make its total energy zero reaches escape velocity, the threshold between a bound orbit and an unbound trajectory that eventually leaves the gravitational system entirely, connecting orbital energy directly to the escape velocity formula.

Gravitational Acceleration on Different Worlds

Each value in this table comes from the same formula, g = GM/r², using that body's own mass and radius — Mars has roughly a tenth of Earth's mass but also a much smaller radius, and the two effects nearly cancel to give surface gravity close to Mercury's despite the two planets being very different in size. Jupiter's enormous mass dominates despite its much larger radius, giving it the highest surface gravity of any planet in the solar system — about 2.5 times Earth's.

Worked Example: Comparing Gravitational Force at Two Distances

Find the gravitational force between the Earth (M = 5.97×10²⁴ kg) and a 70 kg person at the surface (r = 6.37×10⁶ m), then compare it to the force at twice that distance (as if standing on a tower one Earth-radius high). Using F = GMm/r² with G = 6.674×10⁻¹¹ N·m²/kg²: F₁ = (6.674×10⁻¹¹ × 5.97×10²⁴ × 70) / (6.37×10⁶)² ≈ 686 N (matching the expected weight, mg ≈ 70 × 9.81 ≈ 687 N) At double the distance (r₂ = 2r₁), the inverse-square law means the force drops to a quarter: F₂ = F₁/4 ≈ 171.5 N. This is why astronauts on the International Space Station (orbiting at roughly 1.06 Earth radii from the centre, not far above the surface at all) still experience close to 90% of surface gravity — their famous 'weightlessness' comes from continuous free-fall during orbit, not from being far enough away for gravity to have meaningfully weakened.

Worked Example: Escape Velocity From Earth

Escape velocity is the minimum speed needed to leave a body's gravitational field without further propulsion, given by v_esc = √(2GM/r). For Earth (M = 5.97×10²⁴ kg, r = 6.37×10⁶ m): v_esc = √(2 × 6.674×10⁻¹¹ × 5.97×10²⁴ / 6.37×10⁶) = √(1.253×10⁸) ≈ 11,190 m/s ≈ 11.2 km/s This is why rockets need such enormous amounts of fuel just to leave Earth's gravity well entirely (as opposed to merely reaching orbit, which requires roughly 7.9 km/s — the lower 'orbital velocity' rather than escape velocity). The Moon, with far less mass and a smaller radius, has an escape velocity of only about 2.4 km/s, which is part of why landing on and leaving the Moon requires dramatically less fuel than doing the same on Earth.

Gravitational Potential Energy Near a Planet's Surface

Near a planet's surface, gravitational potential energy is often approximated as PE = mgh, treating g as constant. This approximation is only valid when h is small compared to the planet's radius — lifting an object 2 m near Earth's surface changes the distance from Earth's centre so little (6,370,002 m vs 6,370,000 m) that g barely changes at all. For large changes in altitude, such as a satellite's orbit, the full formula PE = -GMm/r must be used instead, since g itself weakens noticeably with distance at those scales and the simple mgh approximation would give a significantly wrong answer.