🪐 Gravitational Parameter Solver
(Earth Surface ≈ 6.37e6 m)

Newton's Law of Gravitation

Gravitation is a fundamental force of attraction between any two masses in the universe. Isaac Newton discovered that this force follows an inverse-square law, meaning that doubling the distance between two objects reduces the force between them to one-fourth of its original value.

F = GMm / r²

1. Gravitational Constant (G)

The universal constant G is approximately 6.67 × 10⁻¹¹ N·m²/kg². This constant is universal, applying to everything from the fall of an apple to the orbits of entire galaxies.

2. Gravitational Field Strength (g)

A gravitational field is a region of space where a mass experiences a force. Field strength is defined as the force per unit mass. For a spherical mass (like a planet):

g = F / m = GM / r²

3. Kepler's Third Law

For a circular orbit, gravity provides the necessary centripetal force. By equating the two, we derive the relationship between the orbital period (T) and the orbital radius (r):

T² = (4π² / GM) × r³
Exam Tip: Gravitational potential energy (U) is defined as U = -GMm/r. The negative sign is critical—it signifies that the field is attractive and that energy must be added to "escape" to infinity, where the potential is zero.

Deep Dive: Worked Examples

✅ Example 1: Field Strength at Altitude

Calculate the gravitational field strength experienced by an astronaut in the ISS, orbiting at 400 km above Earth's surface.

Step 1: Calculate total r r = Radius_Earth + altitude = 6370 km + 400 km = 6770 km = 6.77 × 10⁶ m
Step 2: Apply g = GM/r² g = (6.67 × 10⁻¹¹) × (5.97 × 10²⁴) / (6.77 × 10⁶)²
Step 3: Solve g ≈ 8.70 N/kg (This is why astronauts fluctuate between "weightless" and having weight—gravity is still very strong!)

✅ Example 2: Orbital Velocity of the Moon

The Moon orbits Earth at a distance of 3.84 × 10⁸ m. Calculate its orbital speed (v).

Step 1: Link Gravity to Centripetal force GMm/r² = mv²/r → v = √(GM/r)
Step 2: Plug in Earth's data v = √((6.67 × 10⁻¹¹ × 5.97 × 10²⁴) / 3.84 × 10⁸)
Step 3: Solve v ≈ 1018 m/s

✅ Example 3: Geostationary Orbit Height

A geostationary satellite has a period T = 24 hours. Calculate its required orbital radius r.

Step 1: Convert T to seconds T = 24 × 3600 = 86,400 s
Step 2: Use Kepler's 3rd Law r³ = (GM T²) / 4π²
Step 3: Solve for r r = ∛((3.98 × 10¹⁴ × 86400²) / 39.48) ≈ 4.22 × 10⁷ m

✅ Example 4: Escape Velocity from Moon

The Moon has M = 7.35 × 10²² kg and R = 1.74 × 10⁶ m. Find the minimum speed to escape its surface.

Step 1: Conservation of Energy KE + GPE = 0 → ½mv² - GMm/r = 0 → v = √(2GM/r)
Step 2: Plug in values v = √(2 × 6.67 × 10⁻¹¹ × 7.35 × 10²² / 1.74 × 10⁶)
Step 3: Solve v ≈ 2.37 km/s

✅ Example 5: Gravitational Force between People

Two people (70kg and 85kg) stand 2m apart. Calculate the gravitational force they exert on each other.

Step 1: Apply Law of Gravitation F = (6.67 × 10⁻¹¹ × 70 × 85) / 2²
Step 2: Solve F = (3.97 × 10⁻⁷) / 4 ≈ 9.93 × 10⁻⁸ N
Step 3: Reality Check For perspective, this force is roughly 100 million times smaller than a person's weight.