Newton's Law of Universal Gravitation revolutionized our understanding of the cosmos. It unified celestial and terrestrial mechanics, showing that the same laws govern both. This groundbreaking theory explained planetary motion and the behavior of objects on Earth.

The law states that every particle in the universe attracts every other particle with a force proportional to their masses and inversely proportional to the square of the distance between them. This simple yet powerful equation allows us to calculate gravitational forces between objects.

Historical Development and Fundamentals of Newton's Law of Universal Gravitation

Evolution of gravitational theory

• Ancient times
  • Aristotle proposed idea of "natural place" for elements (earth, water, air, fire) with each seeking its natural place
  • Heavenly bodies believed to follow different laws than objects on Earth, with perfect circular motion
• Middle Ages
  • Ibn Sina (Avicenna) introduced concept of momentum and inertia, laying groundwork for future developments
  • Jean Buridan and Albert of Saxony developed concept of impetus, a precursor to momentum, to explain motion
• Renaissance
  • Nicolaus Copernicus proposed heliocentric model of solar system, challenging Earth-centered view
  • Galileo Galilei made key observations and experiments that laid foundation for Newton's work
    • Discovered moons of Jupiter (Io, Europa, Ganymede, Callisto), supporting Copernican model
    • Studied motion of falling objects and projectiles, leading to concept of acceleration
• Isaac Newton's formulation
  • Unified celestial and terrestrial mechanics under single law of gravitation, showing same laws govern both
  • Developed inverse-square law of gravitation based on Kepler's laws of planetary motion (elliptical orbits, equal areas, period-distance relation)
  • Published groundbreaking work Principia Mathematica in 1687, which presented three laws of motion and Law of Universal Gravitation

Calculations with Newton's law

• Newton's Law of Universal Gravitation expressed as $F = G \frac{m_1 m_2}{r^2}$
  • $F$ represents gravitational force between two objects, always attractive force
  • $G$ is gravitational constant with value $6.67 \times 10^{-11} \text{ N} \cdot \text{m}^2/\text{kg}^2$
  • $m_1$ and $m_2$ are masses of the two interacting objects (kg)
  • $r$ is distance between centers of the two objects (m)
• Force acts along line connecting centers of objects
• Magnitude of force directly proportional to product of masses and inversely proportional to square of distance (inverse square law)
• To calculate force, identify masses and distance, then substitute values into equation
  • Example: Calculate force between Earth (mass $5.97 \times 10^{24}$ kg) and Moon (mass $7.34 \times 10^{22}$ kg) with distance of $3.84 \times 10^8$ m between centers

Complex gravitational interactions

• Superposition principle applies to gravitational forces
  • Net gravitational force on object is vector sum of individual forces exerted by all other objects
  • To find net force, calculate each individual force and add as vectors
    • Example: Net force on Earth due to Sun and Moon is vector sum of individual forces
• Symmetry considerations simplify calculations for symmetric mass distributions (spherical, cylindrical)
  • Net gravitational force can be calculated as if all mass concentrated at center
  • Applies to objects like planets or stars, treating them as point masses
• Continuous mass distributions require integration over mass distribution
  • Uses calculus to sum contributions from infinitesimal mass elements
  • Example: Calculating gravitational force between two galaxies or within a star
• Gravitational potential energy $U$ given by $U = -G \frac{m_1 m_2}{r}$
  • Work done by gravitational force in moving object equals change in gravitational potential energy
  • Symbols have same meaning as in force equation
  • Example: Lifting object from Earth's surface to higher altitude increases gravitational potential energy

Mass and Gravitation

• Inertial mass: measure of object's resistance to acceleration when force is applied
• Gravitational mass: measure of object's response to gravitational force
• Equivalence principle: states that inertial mass and gravitational mass are equivalent
• This equivalence leads to gravitational acceleration being independent of an object's mass
• Newton's law implies action at a distance, where objects can influence each other without physical contact