Fundamentals of Gravitational Force and Field Calculation

Gravitational Acceleration and Field Strength

The strength of the gravitational field depends on the amount of mass $M$ causing the field. Let us define a characteristic of the field that depends only on the mass $M$ and the distance $r$ to the point we consider.

Defining Gravitational Field Strength ($g$)

The gravitational field strength, $g$, at a point in space is the force that would act on a unit mass located at that point. Its unit is Newtons per kilogram ($N/kg$). This term is often used interchangeably with gravitational field intensity.

Calculating the Gravitational Field

To determine the gravitational field created by a point mass $M$, we place a test mass $m$ at a point $P$ in space at a distance $r$ from mass $M$. We calculate the force $F$ per unit mass:

$$g = \frac{F}{m} = \frac{(-GMm/r^2) \cdot \mathbf{u}_r}{m} = \left(-\frac{GM}{r^2}\right) \cdot \mathbf{u}_r$$

The gravitational field has the following properties:

• It is a central field and decreases with the square of the distance ($1/r^2$).
• The negative sign indicates that the unit vector $\mathbf{u}_r$ and the gravitational field vector $g$ have opposite directions.
• Gravitational forces are always attractive.

Newton's Law of Universal Gravitation

All physical bodies possess mass, and consequently, they exert attractive forces on one another. In our daily life, this force is not significant because, relatively speaking, no object in our immediate environment has enough mass to noticeably attract other objects.

However, when we consider massive bodies like the Earth, we encounter a force which causes, among other things, the existence of gravity.

The Statement of Universal Gravitation

Isaac Newton was the first to quantify this force, establishing the following statement:

The force that two masses exert on each other is proportional to the product of the masses and inversely proportional to the square of the distance that separates them.

We must not forget that this force is always attractive between two bodies, never repulsive.

Mathematical Expression of the Law

The force ($F$) can be expressed mathematically as follows:

$$F = G \frac{m M}{r^2}$$

Where:

• $G$: The Universal Gravitational Constant, determined by Henry Cavendish experimenting with a torsion balance. $G \approx 6.67 \times 10^{-11} \, N \cdot m^2 / kg^2$.
• $m$: The mass of the smaller body (measured in kilograms, $kg$).
• $M$: The mass of the larger body (measured in kilograms, $kg$), for example, the Sun or Earth.
• $r$: The distance between the centers of the two masses (measured in meters, $m$).
• $F$: The gravitational force (measured in Newtons, $N$).