Introduction
Newton's Law of Universal Gravitation describes the attractive force between any two masses. It explains both terrestrial gravity (falling apples) and celestial motion (planetary orbits) with a single elegant law. This unification was a landmark achievement in physics, establishing that the same laws govern heaven and earth.
Statement of the Law
Every particle 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: F = G(m1m2)/r2, where G is the universal gravitational constant (6.674×10¹ N·m2/kg2), m1 and m2 are the masses, and r is the distance between their centers. The force is always attractive and acts along the line joining the centers.
\nIntroduction
Newton's Law of Universal Gravitation describes the attractive force between any two masses. It explains both terrestrial gravity (falling apples) and celestial motion (planetary orbits) with a single elegant law. This unification was a landmark achievement in physics, establishing that the same laws govern heaven and earth.
Statement of the Law
Every particle 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: F = G(m1m2)/r2, where G is the universal gravitational constant (6.674×10¹ N·m2/kg2), m1 and m2 are the masses, and r is the distance between their centers. The force is always attractive and acts along the line joining the centers.
\nProperties of Gravitational Force
Key properties: (1) Always attractive, never repulsive; (2) Action-at-a-distance through gravitational field; (3) Follows inverse-square law (F ∝ 1/r2); (4) Proportional to each mass (F ∝ m1 and F ∝ m2); (5) Obeys superposition principle - total force is vector sum of forces from all masses; (6) Weakest of fundamental forces but has infinite range and is always attractive, so dominates at large scales.
Universal Gravitational Constant G
G = 6.67430(15)×10¹ N·m2/kg2 is one of nature's fundamental constants. First measured by Cavendish in 1798 using torsion balance. Remarkably small value means gravity is negligible for everyday objects (two 1 kg masses 1 m apart attract with only 6.67×10¹ N). G is difficult to measure precisely because gravity is so weak compared to electromagnetic forces. Modern experiments achieve about 10^-5 relative precision.
\nShell Theorems
Newton proved two important theorems for spherical mass distributions: (1) A uniform spherical shell attracts external objects as if all its mass were concentrated at the center; (2) A uniform spherical shell exerts zero gravitational force on objects inside it. These theorems explain why we can treat planets and stars as point masses for orbital calculations, and why gravitational force decreases linearly with r inside a uniform sphere.
Applications and Verification
Applications include: explaining planetary orbits, predicting comet returns (Halley), calculating satellite trajectories, understanding tides, weighing Earth (Cavendish experiment), measuring planetary masses. Verifications: precise prediction of planetary positions, spacecraft navigation, galaxy rotation curves. Limitations: modified by General Relativity in strong fields, but Newtonian gravity is adequate for most practical purposes.
\nSolved Example: Gravitational Force Between Two People
Calculate the gravitational force between two 70 kg people standing 1 m apart. Compare with the weight of one person. Solution: F = G(m1m2)/r2 = (6.67×10¹ × 70 × 70) / (1)2 = 3.27×10^-7 N. This is about 0.3 micronewtons. Weight of one person: W = mg = 70 × 9.8 = 686 N. Ratio: F_gravity / W = 3.27×10^-7 / 686 ≈ 4.8×100. The gravitational attraction between two people is about 2 billion times smaller than Earth's gravitational pull on one person. This shows why we don't notice gravitational forces between everyday objects but do notice Earth's gravity (Earth's mass is 6×10^24 kg, enormous compared to a person).
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