Gravitational Field

A gravitational field is a model used to explain the influence that a massive body extends into the space around itself, producing a force on another massive body. It was first introduced by Isaac Newton as part of his law of universal gravitation.

Gravitational fields are responsible for keeping planets in orbit around stars, moons around planets, and for the formation of galaxies and other cosmic structures.

Newton's law of universal gravitation states that every particle attracts every other particle in the universe with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.

The mathematical form of this law is:

F = G(m₁m₂)/r²

Where:

• F is the gravitational force between the masses
• G is the gravitational constant (6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻²)
• m₁ and m₂ are the masses of the objects
• r is the distance between the centers of the masses

The gravitational field strength (g) at a point is defined as the gravitational force per unit mass experienced by a small test mass placed at that point.

For a point mass or spherically symmetric mass distribution:

g = GM/r²

Where:

• g is the gravitational field strength (N/kg or m/s²)
• G is the gravitational constant
• M is the mass creating the field
• r is the distance from the center of mass

Near the Earth's surface, the gravitational field strength is approximately 9.8 N/kg or 9.8 m/s².

Gravitational potential energy is the energy an object possesses due to its position in a gravitational field.

For an object near the Earth's surface:

E_p = mgh

Where:

• E_p is the gravitational potential energy
• m is the mass of the object
• g is the gravitational field strength
• h is the height above a reference point

For objects in space, the formula becomes:

E_p = -GMm/r

The negative sign indicates that the gravitational potential energy increases (becomes less negative) as objects move further apart.

Objects in orbit are essentially falling around a central body. The centripetal force required for circular motion is provided by the gravitational force.

For a circular orbit:

v = √(GM/r)

Where:

• v is the orbital velocity
• G is the gravitational constant
• M is the mass of the central body
• r is the orbital radius

The period of an orbit (the time taken to complete one orbit) is given by:

T = 2π√(r³/GM)

This relationship is known as Kepler's Third Law of Planetary Motion.