Gravity
Gravity
Definition: The fundamental force of attraction between any two masses, responsible for weight on Earth and the large-scale structure of the universe.
How It Works
- Newton’s law of universal gravitation states that the attractive force between two masses grows with the product of their masses and shrinks with the square of the distance between them.
- Gravity is always attractive, never repulsive, in classical physics; because mass never cancels the way electric charge does, gravity dominates at cosmic scales even though it’s the weakest fundamental force.
- Weight is the force of gravity acting on an object, W = mg, and depends on local gravitational acceleration. Mass is the amount of matter an object contains and stays the same everywhere.
- Local gravitational acceleration g varies slightly with altitude, latitude, and nearby geology, decreasing with the square of distance from a planet’s center.
- Einstein’s general relativity reframes gravity not as a force but as the curvature of spacetime caused by mass and energy; objects in free fall simply follow the straightest possible path (a geodesic) through that curved spacetime.
- General relativity is required to accurately describe extreme cases: black holes, neutron stars, GPS satellite timing, gravitational lensing of light, and gravitational waves from merging massive objects.
- Gravitational potential energy near a planet’s surface is approximated as PE = mgh, but the exact form, PE = −GMm/r, is needed at large distances where the field’s strength changes noticeably.
- Escape velocity is the minimum speed an object needs to break free of a gravitational body’s pull permanently, without further propulsion.
- Tides arise because the Moon’s (and Sun’s) gravitational pull is slightly stronger on the near side of Earth than the far side, stretching the oceans into two bulges.
Illustration
Under the Hood
Key equations:
F = Gm₁m₂ / r² (universal gravitation, G = 6.674 × 10⁻¹¹ N·m²/kg²)g = GM / r² (surface gravitational acceleration)PE = −GMm / r (general gravitational potential energy, zero at infinity)v_escape = √(2GM/r)T² = (4π²/GM) r³ (Kepler's third law, circular orbit)Worked Example 1: Weight on Earth vs. the Moon Given: person with mass m = 70 kg, g_Earth = 9.8 m/s², g_Moon = 1.62 m/s² Step 1: Weight on Earth: W = mg = 70 × 9.8 = 686 N Step 2: Weight on Moon: W = mg = 70 × 1.62 = 113.4 N Answer: The person weighs about 6 times less on the Moon, even though their mass of 70 kg is unchanged in both places.
Worked Example 2: Gravitational force between two everyday objects Given: two spheres, each m = 1000 kg, separated by r = 1 m Step 1: F = Gm₁m₂/r² = (6.674 × 10⁻¹¹)(1000)(1000)/1² Answer: F ≈ 6.67 × 10⁻⁵ N, a force smaller than the weight of a single grain of rice, showing how weak gravity is compared to electromagnetic forces at everyday scales.
Worked Example 3: Escape velocity from Earth Given: Earth’s mass M = 5.97 × 10²⁴ kg, radius r = 6.37 × 10⁶ m Step 1: v = √(2GM/r) = √[2(6.674 × 10⁻¹¹)(5.97 × 10²⁴)/(6.37 × 10⁶)] Step 2: Numerator: 2 × 6.674 × 10⁻¹¹ × 5.97 × 10²⁴ ≈ 7.97 × 10¹⁴; divide by 6.37 × 10⁶ ≈ 1.251 × 10⁸ Answer: v ≈ √(1.251 × 10⁸) ≈ 11,185 m/s ≈ 11.2 km/s, the real escape velocity used in spacecraft launch planning.
Worked Example 4: Radius of a geostationary orbit Given: orbital period T = 24 h = 86,400 s, Earth’s GM = 3.9846 × 10¹⁴ m³/s² Step 1: Rearranged Kepler’s third law: r³ = T²GM/(4π²) = (86,400)²(3.9846 × 10¹⁴)/(4π²) Step 2: r³ ≈ (7.465 × 10⁹)(3.9846 × 10¹⁴)/39.48 ≈ 7.53 × 10²² m³ Step 3: r ≈ (7.53 × 10²²)^(1/3) ≈ 4.23 × 10⁷ m Answer: r ≈ 42,300 km from Earth’s center, or about 35,900 km altitude, matching the real geostationary belt used by communication satellites.
Worked Example 5: Why PE = mgh breaks down at large heights Given: 10 kg object lifted from Earth’s surface (r₁ = 6.371 × 10⁶ m) to 1000 km altitude (r₂ = 7.371 × 10⁶ m) Step 1: Exact PE change: ΔPE = −GMm/r₂ − (−GMm/r₁) = GMm(1/r₁ − 1/r₂) Step 2: ΔPE ≈ (3.9846 × 10¹⁴)(10)(1/6.371 × 10⁶ − 1/7.371 × 10⁶) ≈ 8.48 × 10⁷ J Step 3: Compare to the flat-Earth approximation: mgh = 10 × 9.8 × 1,000,000 = 9.8 × 10⁷ J Answer: The two estimates differ by about 15%, showing PE = mgh is only accurate near the surface, where g doesn’t change much over the height involved.
Worked Example 6: Tidal acceleration from the Moon Given: Moon’s mass 7.35 × 10²² kg, Earth-Moon distance 3.844 × 10⁸ m, Earth’s radius 6.371 × 10⁶ m Step 1: Near-side distance: r_near = 3.844 × 10⁸ − 6.371 × 10⁶ ≈ 3.780 × 10⁸ m; g_near = GM_moon/r_near² ≈ 3.43 × 10⁻⁵ m/s² Step 2: Far-side distance: r_far = 3.844 × 10⁸ + 6.371 × 10⁶ ≈ 3.908 × 10⁸ m; g_far = GM_moon/r_far² ≈ 3.21 × 10⁻⁵ m/s² Answer: The difference, about 2.2 × 10⁻⁶ m/s², is the tidal acceleration that stretches Earth’s oceans into the bulges responsible for tides.
Surface Gravity Around the Solar System
| Body | Surface gravity (m/s²) | Relative to Earth |
|---|---|---|
| Mercury | 3.7 | 0.38× |
| Earth | 9.8 | 1.0× |
| Moon | 1.62 | 0.17× |
| Mars | 3.71 | 0.38× |
| Jupiter | 24.8 | 2.53× |
| Sun (photosphere) | 274 | 27.9× |
Why It Matters
- GPS satellites must correct for both special and general relativistic effects on their onboard clocks, or position errors would accumulate at roughly 10 km per day.
- Spacecraft trajectory planning, from Moon landings to interplanetary probes, depends entirely on precise gravitational calculations for orbits and gravity-assist maneuvers.
- Structural and civil engineers calculate gravitational loads on every building, bridge, and dam as a baseline design requirement.
- Coastal engineers use tidal gravity predictions for harbor design, flood planning, and tidal power generation.
- Astronomers use gravitational lensing, the bending of light by massive objects predicted by general relativity, to detect dark matter and distant galaxies otherwise too faint to see directly.
- Rocket launch planning depends on precise escape velocity and gravity-loss calculations to determine fuel requirements and optimal trajectories.
Common Pitfalls
- Confusing mass (invariant amount of matter, kg) with weight (a force that depends on local gravity, newtons).
- Thinking of gravity in general relativity as a literal pulling force; it’s spacetime curvature, and objects in free fall actually feel weightless because they’re following the natural, force-free path through that curved geometry.
- Applying PE = mgh over large altitude changes, where the uniform-field approximation breaks down and the exact −GMm/r formula is needed.
- Believing heavier objects fall faster than lighter ones in a vacuum; Galileo’s insight, confirmed by general relativity’s equivalence principle, is that all objects fall at the same rate regardless of mass, air resistance aside.
- Forgetting the inverse-square law means gravity weakens rapidly with distance, not linearly; doubling the distance cuts the force to one quarter, not one half.
- Equating “weightlessness” in orbit with “no gravity”; astronauts on the ISS still experience about 90% of Earth’s surface gravity, they’re simply in continuous free fall alongside their spacecraft.
Comparison
| Aspect | Newtonian gravity | General relativity |
|---|---|---|
| Nature of gravity | A force acting at a distance | Curvature of spacetime |
| Math | Algebraic (F = Gm₁m₂/r²) | Tensor field equations |
| Accurate for | Everyday scales, most orbital mechanics | Strong fields, high speeds, precision timing |
| Predicts | Orbits, tides, free fall | All of Newtonian gravity, plus light bending, time dilation, gravitational waves |
| Breaks down | Near black holes, at relativistic speeds | Not yet found to break down, but incompatible with quantum mechanics |
FAQ
Why do all objects fall at the same rate in a vacuum? Because gravitational mass (which determines the force) and inertial mass (which resists acceleration) are experimentally identical, so mass cancels out of the acceleration entirely, a fact called the equivalence principle.
Do astronauts in orbit really feel no gravity? No, they feel none of gravity’s effects relative to their spacecraft because both are in free fall together, but gravity itself is still acting on them almost as strongly as on the ground.
Does gravity travel instantly or at a finite speed? At a finite speed, exactly the speed of light, confirmed directly by the 2015 detection of gravitational waves from merging black holes, which arrived exactly when general relativity predicted.
Is gravity really the weakest fundamental force? Yes, by an enormous margin, roughly 10³⁸ times weaker than the electromagnetic force between two protons; it only seems dominant at large scales because it always adds up instead of canceling.
Why is Jupiter’s surface gravity only about 2.5 times Earth’s despite being 318 times more massive? Jupiter’s radius is about 11 times Earth’s, and surface gravity depends on mass divided by radius squared, so the much larger radius largely offsets the huge mass increase.
Example
GPS satellites orbit at about 20,200 km altitude, where Earth’s gravity is weaker than at the surface, causing their onboard atomic clocks to tick faster by about 45 microseconds per day (a general relativistic effect). Their high orbital speed causes those same clocks to tick slower by about 7 microseconds per day (a special relativistic effect). Engineers apply a net correction of about 38 microseconds per day, without it, GPS position errors would accumulate by roughly 10 km every day.
Related Terms
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