KAHIBARO
Table of Contents
Meaning of gravitational field strength
Gravitational field strength describes how strongly gravity pulls on mass at a point in space. It tells us the gravitational force per unit mass that a small test object would feel there.
If a mass $m$ placed at some point experiences a gravitational force $\vec{F}_g$, then the gravitational field strength at that point is
$$
\vec{g} = \frac{\vec{F}_g}{m}
$$
This is a vector quantity, so it has both magnitude and direction. Its direction is the same as the direction of the gravitational force on a positive test mass, which means it points toward the attracting body.
Important definition:
$$
\vec{g} = \frac{\vec{F}_g}{m}
$$
Gravitational field strength is force per unit mass, and its direction is toward the source of gravity.
Units and physical interpretation
Since force is measured in newtons, $N$, and mass is measured in kilograms, $kg$, the SI unit of gravitational field strength is
$$
N/kg
$$
This unit is equivalent to
$$
m/s^2
$$
That is why the symbol $\vec{g}$ is also used for the acceleration due to gravity. In fact, gravitational field strength is exactly the acceleration a freely falling object would have at that location, if no other forces act on it.
For example, near Earth's surface,
$$
g \approx 9.8 \, N/kg \approx 9.8 \, m/s^2
$$
This means that every $1 \, kg$ of mass experiences a gravitational force of about $9.8 \, N$ downward near Earth.
Field strength produced by a point mass
Newton's law of gravitation gives the force between two masses. If a source mass $M$ creates the field, and a test mass $m$ is at distance $r$ from its center, then
$$
F = G\frac{Mm}{r^2}
$$
Dividing by $m$, we get the gravitational field strength due to the source mass:
$$
g = G\frac{M}{r^2}
$$
In vector form,
$$
\vec{g} = -G\frac{M}{r^2}\hat{r}
$$
where $\hat{r}$ is the outward radial unit vector. The minus sign shows that the field points inward, toward the mass.
For a point mass or a spherically symmetric mass distribution,
$$
g = G\frac{M}{r^2}
$$
and the field points toward the center of the mass.
Inverse square behavior
The formula
$$
g = G\frac{M}{r^2}
$$
shows that gravitational field strength decreases with the square of distance. If the distance from the source doubles, the field becomes one fourth as large. If the distance triples, the field becomes one ninth as large.
This inverse square pattern is very important in gravitation.
| Distance from mass | Field strength compared with $g_0$ |
|---|---|
| $r$ | $g_0$ |
| $2r$ | $g_0/4$ |
| $3r$ | $g_0/9$ |
| $4r$ | $g_0/16$ |
Relation to weight
The weight of an object is the gravitational force acting on it. If the local gravitational field strength is $\vec{g}$, then the weight is
$$
\vec{W} = m\vec{g}
$$
So gravitational field strength tells us how much weight each kilogram of mass has at that location.
For example, if $g = 9.8 \, N/kg$, then a $2.0 \, kg$ object has weight magnitude
$$
W = mg = (2.0)(9.8) = 19.6 \, N
$$
Direction of the field
Because gravity is always attractive, the gravitational field points toward the mass producing it. Around a spherical body like Earth, the field lines point radially inward from all directions.
At every point, the field is directed toward the center of the sphere.
Near Earth's surface
Close to Earth's surface, the distance from Earth's center does not change very much during ordinary motions, so the gravitational field strength is approximately constant:
$$
g \approx 9.8 \, m/s^2
$$
This approximation is very useful for many mechanics problems. However, farther from Earth, the value of $g$ becomes smaller because $r$ increases.
If Earth has mass $M_E$ and radius $R_E$, then at the surface
$$
g = G\frac{M_E}{R_E^2}
$$
At height $h$ above the surface,
$$
g = G\frac{M_E}{(R_E + h)^2}
$$
So gravity does not suddenly stop above Earth, it just weakens with distance.
Comparison of gravitational field at different locations
The gravitational field strength depends on the mass creating the field and the distance from it. A more massive object creates a stronger field, and a greater distance gives a weaker field.
| Situation | Formula for field strength |
|---|---|
| Near Earth's surface | $g \approx 9.8 \, m/s^2$ |
| Distance $r$ from point mass $M$ | $g = G\frac{M}{r^2}$ |
| Height $h$ above Earth | $g = G\frac{M_E}{(R_E+h)^2}$ |
Superposition of gravitational fields
If more than one mass is present, each mass creates its own gravitational field, and the total gravitational field is the vector sum of the individual fields:
$$
\vec{g}_{\text{total}} = \vec{g}_1 + \vec{g}_2 + \vec{g}_3 + \cdots
$$
This means gravitational fields combine according to vector addition. The total field at a point depends on both the magnitudes and directions of the separate fields.
When several masses act together, add gravitational fields as vectors:
$$
\vec{g}_{\text{total}} = \sum \vec{g}_i
$$
A simple example
Suppose a small test mass of $0.50 \, kg$ is placed where the gravitational field strength is $6.0 \, N/kg$. The gravitational force on it is
$$
F = mg = (0.50)(6.0) = 3.0 \, N
$$
The force points in the same direction as the field.
Now suppose instead that the gravitational force on a $2.0 \, kg$ object is measured to be $14 \, N$. Then the field strength there is
$$
g = \frac{F}{m} = \frac{14}{2.0} = 7.0 \, N/kg
$$
Key ideas
Gravitational field strength tells us how strong gravity is at a point. It is defined as force per unit mass, measured in $N/kg$, and is also equal to the acceleration due to gravity at that point. For a point mass or spherical body, its magnitude is
$$
g = G\frac{M}{r^2}
$$
and its direction is always toward the mass producing the field.
Core facts to remember:
$$
\vec{g} = \frac{\vec{F}_g}{m}
$$
$$
g = G\frac{M}{r^2}
$$
$$
\vec{W} = m\vec{g}
$$
Gravitational field strength is a vector, points toward the attracting mass, and decreases as $1/r^2$.