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Introduction Newton's Universal Law of Gravitation Gravitational Field Gravitational Field Strength Gravitational Field Lines Weight vs Mass Gravitational Potential Energy Gravitational Potential Orbital Motion Orbital Speed Orbital Period Geostationary Orbits Energy in Orbits Escape Velocity Weightlessness Why Gravity MattersDrop your phone. It falls. Toss a ball upward. It comes back down. Jump off a diving board. Gravity pulls you toward the water. These seem like simple observations, almost boring in their predictability.
But zoom out. The Moon orbits Earth. Earth orbits the Sun. The Sun orbits the center of our galaxy along with hundreds of billions of other stars. Galaxies cluster together in enormous groups, held by mutual gravitational attraction.
Gravity is the force that structures the entire universe. It's why planets are spheres. It's why stars shine (gravitational compression heats their cores to fusion temperatures). It's why galaxies exist as coherent objects rather than just random distributions of stars.
And here's the remarkable thing: the same equation that describes your phone falling describes the Moon orbiting Earth and galaxies attracting each other across millions of light-years.
Understanding gravitational fields means understanding how mass creates attraction, how orbits work, why astronauts feel weightless, and how the universe's large-scale structure formed.
F = Gm₁m₂/r²
G = 6.67 × 10⁻¹¹ N·m²/kg²
Key points:
Example: Find the gravitational force between Earth (6.0 × 10²⁴ kg) and a 70 kg person. Earth's radius = 6.4 × 10⁶ m.
F = (6.67 × 10⁻¹¹)(6.0 × 10²⁴)(70)/(6.4 × 10⁶)² = 683 N
This is the person's weight.
Important: Newton's law is universal. The same equation applies to apples falling, the Moon orbiting Earth, planets orbiting the Sun, and galaxies attracting each other.
A gravitational field is a region of space where a mass experiences a force.
Every mass creates a gravitational field around itself. Any other mass placed in that field feels a force. The field exists whether or not there's a second mass there to feel it.
Earth's mass creates a gravitational field filling all of space around it. When you're in that field, you experience a force.
g = F/m (definition)
g = GM/r² (for a point mass or sphere)
SI unit: N/kg (equivalent to m/s²)
On Earth's surface, g ≈ 9.81 N/kg. This means every kilogram experiences 9.81 newtons of gravitational force.
Example: Find g at Earth's surface. (M = 6.0 × 10²⁴ kg, R = 6.4 × 10⁶ m)
g = GM/r² = (6.67 × 10⁻¹¹)(6.0 × 10²⁴)/(6.4 × 10⁶)² = 9.76 N/kg
At higher altitudes, g decreases. At the ISS altitude (400 km), g ≈ 8.7 m/s² — not much less than at the surface!
Near Earth's surface, field is strong (lines close together). Far from Earth, field is weak (lines spread apart).
Amount of matter in an object
Measured in kilograms (kg)
Same everywhere in the universe
Gravitational force on an object
Measured in newtons (N)
Depends on location (W = mg)
Example: An 80 kg astronaut on Earth (g = 9.81 N/kg) vs Mars (g = 3.7 N/kg).
Earth: W = (80)(9.81) = 785 N | Mars: W = (80)(3.7) = 296 N
Her mass is unchanged, but her weight is less than half on Mars.
Near Earth's surface: GPE = mgh
General formula: GPE = -GMm/r
The negative sign is crucial. Gravitational potential energy is always negative (taking zero at infinite distance). The further you are from a mass, the closer GPE gets to zero.
To completely escape a gravitational field, you need to add energy to the system.
Example: GPE of a 1000 kg satellite at Earth's surface vs 400 km altitude.
Surface (r = 6.4 × 10⁶ m): GPE = -6.25 × 10¹⁰ J
Altitude 400 km (r = 6.8 × 10⁶ m): GPE = -5.89 × 10¹⁰ J
The satellite has more (less negative) potential energy at higher altitude.
V = -GM/r
SI unit: J/kg
g = -dV/dr (field strength is the negative gradient of potential)
Gravitational potential at a point tells you how much potential energy each kilogram has at that location.
Where potential changes rapidly with distance, the field is strong.
Why does the Moon orbit Earth instead of crashing into it or flying away?
Answer: Gravitational force provides exactly the centripetal force needed for circular motion.
GMm/r² = mv²/r
The mass m cancels, giving: v² = GM/r
Orbital speed doesn't depend on the orbiting object's mass!
v = √(GM/r)
Example: ISS at 400 km altitude (r = 6.8 × 10⁶ m).
v = √[(6.67 × 10⁻¹¹)(6.0 × 10²⁴)/(6.8 × 10⁶)] = √(5.88 × 10⁷) = 7670 m/s ≈ 7.7 km/s
The ISS travels at 27,600 km/h, completing one orbit in about 90 minutes.
T = 2π√(r³/GM)
Kepler's Third Law: T² ∝ r³
Example: ISS orbital period (r = 6.8 × 10⁶ m).
T = 2π√[(6.8 × 10⁶)³/((6.67 × 10⁻¹¹)(6.0 × 10²⁴))] = 2π√(7.88 × 10⁵) = 5577 s ≈ 93 minutes
A geostationary satellite orbits Earth once every 24 hours, staying above the same point on the equator. It appears stationary in the sky.
r = 42,200 km from Earth's center (≈36,000 km above surface)
All geostationary satellites orbit at this altitude. This is crucial for communications and weather satellites that need to continuously monitor the same region.
KE = GMm/(2r)
PE = -GMm/r
E_total = -GMm/(2r)
Total energy is negative, indicating the satellite is bound to Earth. The more negative, the more tightly bound.
Key insight: Total energy equals negative kinetic energy. Half the potential energy is kinetic, and half stays as potential (virial theorem).
To change orbital altitude, you must change the satellite's energy. Higher orbit → add energy. Lower orbit → remove energy.
v_escape = √(2GM/r)
Escape velocity is the minimum speed needed to completely escape a gravitational field (reach infinite distance with zero final velocity).
It is √2 times the orbital velocity at that radius.
Example: Earth's escape velocity from the surface.
v_escape = √[2(6.67 × 10⁻¹¹)(6.0 × 10²⁴)/(6.4 × 10⁶)] = √(1.25 × 10⁸) = 11,180 m/s ≈ 11.2 km/s
For the Moon, escape velocity is only about 2.4 km/s — much easier to launch from.
Astronauts on the ISS float around, seemingly without gravity. But g at ISS altitude is about 8.7 m/s² — not much less than at Earth's surface!
Why do they feel weightless?
They're in free fall. The ISS and everything in it are continuously falling toward Earth. But they're also moving sideways so fast that as they fall, they miss Earth — that's what an orbit is.
In free fall, you don't feel your weight because there's no normal force pushing up on you. Everything falls together at the same rate.
Understanding gravitational fields is fundamental to modern physics, from Newton's laws to Einstein's general relativity. It's the force that literally holds the universe together.