Gravitational Fields

A gravitational field is a region of space where any object with mass experiences a non-contact force. This concept is fundamental to understanding the motion of planets, stars, and galaxies. The framework for this topic is built upon Newton's Law of Universal Gravitation.

### Newton's Law of Universal Gravitation

This law states that any two point masses in the universe attract each other with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres. This relationship is an example of an inverse square law.

The mathematical representation is:

F = Gm₁m₂ / r²

Where:

### Gravitational Field Strength (g)

The gravitational field strength (g) at a point is defined as the gravitational force experienced per unit mass placed at that point. It is a vector quantity, with its direction always pointing towards the centre of the mass creating the field.

g = F / m

By substituting Newton's law for F, we can derive the formula for the field strength created by a mass M at a distance r from its centre:

g = (GMm / r²) / m => g = GM / r²

Near the Earth's surface, g is approximately 9.81 N kg⁻¹, which is numerically equal to the acceleration of free fall.

### Gravitational Potential (Φ)

Gravitational potential (Φ) at a point in a gravitational field is defined as the work done per unit mass in bringing a small test mass from infinity to that point. The potential at infinity is defined as zero.

Since the gravitational force is attractive, work is done *by* the field when a mass is brought from infinity. This means the mass loses potential energy, so gravitational potential is always a negative value.

Φ = -GM / r

Where:

Gravitational potential is a scalar quantity. Surfaces connecting points of equal potential are called equipotential surfaces.

The gravitational potential energy (Eₚ) of a mass m at a point is the product of the mass and the gravitational potential at that point:

Eₚ = mΦ = -GMm / r

### Orbital Motion

The motion of satellites and planets is a direct application of these principles. For an object in a stable circular orbit, the gravitational force provides the necessary centripetal force.

Gravitational Force = Centripetal Force

GMm / r² = mv² / r

From this equality, we can derive the speed (v) required for a stable orbit:

v = √(GM / r)

This shows that orbital speed depends only on the mass of the central body and the orbital radius, not the mass of the satellite itself.

The period (T) of the orbit can be found using the relationship v = 2πr / T. Squaring this gives v² = 4π²r² / T². By equating the expressions for v², we get:

GM / r = 4π²r² / T²

Rearranging for T² gives Kepler's Third Law for circular orbits:

T² = (4π² / GM) * r³

This shows that the square of the orbital period is directly proportional to the cube of the orbital radius (T² ∝ r³).

A geostationary orbit is a special type of orbit where a satellite remains fixed above a single point on the Earth's equator. This requires three conditions: