Newton's Universal Law of Gravitation states that every mass attracts every other mass with a force F = Gm₁m₂/r². Its applications include explaining planetary motion and Kepler's laws, calculating the mass of celestial bodies, predicting tides, determining escape velocity, and explaining the orbits of artificial satellites.
F = Gm₁m₂/r²; G = 6.674×10⁻¹¹ N·m²/kg².
Application 1: Explains planetary motion and Kepler's laws (T² ∝ r³).
Application 2: Enables calculation of mass of Sun, Earth, and Moon.
Also explains: tides, escape velocity, artificial satellite orbits.
Escape velocity from Earth = √(2GM/R) ≈ 11.2 km/s.
Gravitational law is universal — applies to all masses in the universe.
Newton's law of gravitation explains why planets move in elliptical orbits around the Sun. The gravitational force between the Sun and a planet provides the centripetal force for orbital motion.
For circular orbit: GMm/r² = mv²/r v = √(GM/r) Orbital period T = 2πr/v = 2π√(r³/GM)
This gives Kepler's third law: T² ∝ r³
All three of Kepler's empirical laws are derivable from Newton's law of gravitation.
From orbital data (T and r), Newton's law gives: M_sun = 4π²r³/(GT²)
Using Earth's orbit (r = 1.5×10¹¹ m, T = 365.25 days): M_sun ≈ 2 × 10³⁰ kg
Similarly, using Moon's orbit, mass of Earth = 6 × 10²⁴ kg.
Tides: The Moon's gravitational pull on Earth's oceans creates tidal bulges. High tides occur on the side facing the Moon and the opposite side. Newton's law explains both the tidal period (~12.4 hours) and the difference between spring and neap tides.
Escape velocity: Minimum speed to escape a planet's gravity: v_escape = √(2GM/R) For Earth: v_e = √(2 × 6.67×10⁻¹¹ × 6×10²⁴ / 6.4×10⁶) ≈ 11.2 km/s
Artificial satellites: Orbital speed for a satellite at height h: v = √(GM/(R+h)) For low Earth orbit: v ≈ 7.9 km/s, T ≈ 90 min. Geostationary orbit: T = 24 h, h ≈ 36,000 km.
Gravitational acceleration: g = GM/R² Explains why g varies with altitude and latitude.
Statement: Every particle in the universe attracts every other particle with a force that is: • Directly proportional to the product of their masses. • Inversely proportional to the square of the distance between them.
Formula: F = Gm₁m₂/r²
Where: • F = gravitational force (N) • G = 6.674 × 10⁻¹¹ N·m²/kg² (universal gravitational constant) • m₁, m₂ = masses (kg) • r = distance between their centres (m)
G value: 6.674 × 10⁻¹¹ N·m²/kg² Dimensional formula of G: [M⁻¹L³T⁻²]
Key features: • Universal — applies to all masses everywhere in the universe. • Attractive — always pulls, never pushes. • Inverse square law — force ∝ 1/r².
(1) It explains the orbital motion of planets around the Sun and gives Kepler's third law (T² ∝ r³). (2) It allows calculation of the mass of the Sun and Earth using orbital data: M = 4π²r³/(GT²).
Every two masses attract each other with force F = Gm₁m₂/r², where G = 6.674×10⁻¹¹ N·m²/kg², proportional to the product of masses and inversely proportional to the square of distance.
Using Earth's orbital data: M_sun = 4π²r³/(GT²). With r = 1.5×10¹¹ m and T = 365.25 days: M_sun ≈ 2×10³⁰ kg.
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