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Calculate the gravitational force between any two masses using Newton's Law of Universal Gravitation (F = Gm₁m₂/r²). Also calculates orbital velocity and escape velocity for planets and satellites.
Newton's Law of Universal Gravitation: every mass attracts every other mass with force F = Gm₁m₂/r², where G = 6.674×10⁻¹¹ N·m²/kg². This inverse-square law governs planetary orbits, tidal forces, satellite motion, and structure of the universe.
Newton's Law of Universal Gravitation states that every particle of matter in the universe 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 = Gm₁m₂/r². The gravitational constant G = 6.674×10⁻¹¹ N·m²/kg² was first measured by Henry Cavendish in 1798 using a torsion balance.
The inverse-square law (1/r²) means that doubling the distance reduces gravitational force to 1/4. Tripling distance: 1/9 the force. This relationship governs planetary orbits, explains why the Moon's orbit is stable, determines satellite orbital speeds, and describes the large-scale structure of the universe. Gravity is the weakest of the four fundamental forces but has infinite range and is always attractive — making it dominant at cosmic scales.
| Body | Mass | Radius | Surface g / Orbital v |
|---|---|---|---|
| Earth | 5.972×10²⁴ kg | 6.371×10⁶ m | 9.81 m/s² |
| Moon | 7.342×10²² kg | 1.737×10⁶ m | 1.62 m/s² |
| Sun | 1.989×10³⁰ kg | 6.96×10⁸ m | 274 m/s² |
| Mars | 6.39×10²³ kg | 3.39×10⁶ m | 3.72 m/s² |
| Jupiter | 1.898×10²⁷ kg | 7.15×10⁷ m | 24.8 m/s² |
| ISS orbit | — | 4.08×10⁵ m alt. | ~7,660 m/s orbital |
For a circular orbit at radius r from planet center: gravitational force = centripetal force. Gm₁m₂/r² = m₂v²/r. Solving: v_orbital = √(GM/r). International Space Station (r = 6,371 + 408 = 6,779 km = 6.779×10⁶ m): v = √(6.674×10⁻¹¹ × 5.972×10²⁴/6.779×10⁶) = √(5.879×10⁷) = 7,668 m/s = 27,600 km/h. Period = 2πr/v = 2π×6.779×10⁶/7,668 = 5,552 s = 92.5 minutes. ISS completes about 15.5 orbits per day — astronauts see 15-16 sunrises/sunsets daily.
Escape velocity is the minimum speed to escape a body's gravity without further propulsion: v_escape = √(2GM/r). Earth surface: v_esc = √(2 × 6.674×10⁻¹¹ × 5.972×10²⁴/6.371×10⁶) = √(1.252×10⁸) = 11,186 m/s = 11.2 km/s = 40,270 km/h. Moon: 2.38 km/s (much easier — explains why lunar missions used small ascent stages). Sun: 617.5 km/s. Black hole: escape velocity ≥ c (speed of light) — even light cannot escape from within the event horizon (Schwarzschild radius = 2GM/c²).
Tidal forces arise from the difference in gravitational attraction across an extended body. The Moon pulls the near side of Earth more strongly than the far side (inverse square law). The differential force stretches Earth's oceans into two bulges — high tides on the Moon-facing side (near) and the opposite side (far). Earth bulges follow the Moon as Earth rotates, creating two high tides daily (roughly). The Sun also creates tides (about 46% as strong as Moon's). Spring tides (new/full Moon): Sun and Moon aligned → stronger tides. Neap tides (quarter Moon): Sun and Moon at 90° → weaker tides. Tidal braking gradually slows Earth's rotation — day was 6 hours long 4 billion years ago.
Always ensure units are consistent before calculating. This calculator uses SI units (International System): meters (m) for distance, kilograms (kg) for mass, seconds (s) for time, Newtons (N) for force, Joules (J) for energy, Watts (W) for power, and Pascals (Pa) for pressure. When converting from other unit systems (imperial, CGS), convert all values before entering them into the calculator.
Gravitational acceleration on Earth: g = 9.81 m/s² (standard: 9.80665 m/s²). Speed of light in vacuum: c = 2.998×10⁸ m/s. Universal gravitational constant: G = 6.674×10⁻¹¹ N·m²/kg². Speed of sound in air at 20°C: 343 m/s. Atmospheric pressure at sea level: 101,325 Pa = 1 atm. These constants appear in virtually every branch of physics and engineering.
The physics concepts calculated here appear in countless real-world engineering and scientific applications. Mechanical engineers design machines, vehicles, and structures using these fundamental equations daily. Civil engineers calculate loads, pressures, and structural forces in bridges and buildings. Aerospace engineers apply kinematics, orbital mechanics, and aerodynamics to aircraft and spacecraft design. Biomedical engineers analyze forces and pressures in the human body for medical device design and injury prevention.
Sports science relies heavily on classical mechanics to analyze and improve athletic performance. Projectile motion governs the trajectory of every thrown, kicked, or hit ball in sport. Torque analysis explains joint loading and muscle force requirements in weight training. Momentum and impulse calculations underpin collision analysis in contact sports and help design protective equipment that extends impact duration to reduce peak force. Power output measurement guides training intensity for cyclists, rowers, and sprinters. Understanding these physics principles gives athletes and coaches a quantitative framework for performance optimization.
Physics calculations directly inform engineering safety standards. Stopping distance calculations (kinetic energy / braking force) set speed limits and vehicle safety requirements. Pressure calculations determine pipe and vessel wall thickness in plumbing and industrial equipment. Torque specifications on fasteners prevent structural failures. Centripetal acceleration limits guide road curve design and roller coaster safety. Every safety standard in transportation, construction, and consumer products has physics calculations at its foundation.