Gravitational field equations 

• Describe the attraction between masses and how gravity affects motion in space.

• They explain how gravitational forces govern the movement of objects like planets, satellites, and celestial bodies.

• Essential for solving problems in AP Physics 1 and AP Physics C: Mechanics.

Key applications:

• Newton's Law of Universal Gravitation: F = G(m1m2)/r^2

  • Represents the attractive force between two masses.

  • G is the gravitational constant (6.674 × 10^-11 N(m/kg)^2).

  • The force is inversely proportional to the square of the distance between the centers of the two masses.

• Gravitational Field Strength: g = GM/r^2

  • Represents the gravitational force experienced by a unit mass at a distance (r) from a mass (M).  

  • Describes the intensity of the gravitational pull at a given location in space.  

  • Shows that the gravitational field strength diminishes with the square of the distance from the source mass. 

• Gravitational Potential Energy: U = -GMm/r

  • Represents the work done to bring a mass m from infinity to a distance r from mass M.

  • The negative sign indicates that energy is released when masses come together.

  • Potential energy becomes less negative (increases) as the distance increases.

• Escape Velocity: v_escape = √(2GM/r)

  • This solves for the minimum velocity needed for an object to break free from a gravitational field without further propulsion.

  • Depends on the mass of the celestial body and the distance from its center.

  • Higher mass or smaller radius results in a higher escape velocity.

• Orbital Velocity: v_orbit = √(GM/r)

  • It is the velocity required for an object to maintain a stable orbit around a mass M.

  • Directly related to the mass of the central body and inversely related to the radius of the orbit.

  • Essential for understanding satellite motion and planetary orbits.

• Kepler's Third Law: T^2 ∝ r^3

  • Explains the relation between the square of the orbital period (T) of a planet to the cube of the semi-major axis (r) of its orbit.

  • Shows that more distant planets take longer to orbit the sun.

  • Provides a foundation for understanding planetary motion and distances in the solar system.

• Gravitational Potential: V = -GM/r

  • It fully describes the potential energy per unit mass at a distance r from mass M.

  • Like gravitational potential energy, it is negative, indicating a bound system.

  • Useful for calculating energy changes in gravitational fields.

• Acceleration due to gravity at Earth's surface: g ≈ 9.8 m/s^2

  • Represents the acceleration experienced by an object in free fall near the Earth's surface.

  • Varies slightly with altitude and geographical location.

  • Fundamental for solving problems involving motion under gravity.

• Gravitational Field for a spherical shell: g = 0 (inside), g = GM/r^2 (outside)

  • Inside a uniform spherical shell, the gravitational field is zero.

  • Outside the shell, the gravitational field behaves as if all mass were concentrated at the center.

  • Important for understanding gravitational effects of planets and stars.

• Superposition principle for gravitational fields

  • States that the total gravitational field at a point is the vector sum of the fields due to individual masses.

  • Allows for the analysis of complex systems with multiple masses.

  • Essential for solving problems involving multiple gravitational sources.