13.3 Gravitational Field of a Point Mass

2026 Syllabus Objectives

By the end of these notes, you should be able to:

Derive the equation $g = \dfrac{GM}{r^2}$ using Newton's law of gravitation and the definition of gravitational field strength.
Recall and use $g = \dfrac{GM}{r^2}$ to solve problems.
Explain why $g$ is approximately constant for small changes in height near the Earth's surface.

Main Content

1. What Is a Gravitational Field?

A gravitational field is a region of space around a mass where any other mass placed in that region will experience a gravitational force (a pull). Think of it like an invisible zone of influence surrounding every object that has mass.

The strength of a gravitational field at any point tells us how strong the pull is at that specific location.
The further you move away from a mass, the weaker its gravitational field becomes.

2. Gravitational Field Strength — The Definition

Gravitational field strength (given the symbol $g$ ) at a point in a field is defined as:

The gravitational force acting on a mass per unit of that mass placed at that point.

In simple terms: how many newtons of force does each kilogram of mass experience at that point?

The equation that comes directly from this definition is:

$\boxed{g = \frac{F}{m}}$

Symbol	Meaning	Unit
$g$	Gravitational field strength	N kg⁻¹
$F$	Gravitational force on the object	N (newtons)
$m$	Mass of the object placed in the field	kg

This equation can be rearranged to $F = mg$ , which you may already recognise as the equation for weight — the gravitational force acting on an object. This makes sense! Weight is simply the gravitational force, and $g$ tells us how strong that force is per kilogram.

Important: $g$ is a vector quantity — it has both a size and a direction. The direction of $g$ at any point always points towards the centre of the mass producing the field (i.e. it always pulls inward).

3. Newton's Law of Gravitation — A Quick Reminder

Before deriving the formula for $g$ due to a point mass, we need to recall Newton's Law of Gravitation.

This law states that the gravitational force between two point masses is:

Directly proportional to the product of their masses (bigger masses → stronger force)
Inversely proportional to the square of the distance between their centres (further apart → weaker force)

The equation is:

$\boxed{F_G = \frac{GM_1M_2}{r^2}}$

Symbol	Meaning	Unit
$F_G$	Gravitational force of attraction	N
$G$	Universal Gravitational Constant = $6.67 \times 10^{-11}$	N m² kg⁻²
$M_1, M_2$	The two masses	kg
$r$	Distance between the centres of the two masses	m

A point mass is an idealised object — we treat all of its mass as if it is concentrated at a single point at its centre. In practice, when an object is very far away compared to its actual size, treating it as a point mass gives an excellent approximation.

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