D.1.1 Laws and properties of gravitational fields

Newton’s Law of Gravitation

The Universal Law of Gravitation

Newton's law of gravitation describes the attractive force between any two objects with mass.
The force is given by:

$F = G \frac{m_{1} m_{2}}{r^{2}}$

Where:

$F$ is the gravitational force between the masses (in newtons, N).
$G$ is the universal gravitational constant, approximately $6.67 \times 10^{- 11} {N m}^{2} / {kg}^{2}$ .
$m_{1}$ and $m_{2}$ are the masses of the two objects (in kilograms, kg).
$r$ the distance between the centers of the two masses (in meters, m).

Note

The gravitational force is always attractive and acts along the line joining the two masses.

Illustration of the universal law of gravitation.

Application to Spherical Masses

Newton proved that for spherical bodies with uniform density, the gravitational force can be calculated as if all the mass were concentrated at the center of the sphere.

This simplifies calculations for planets, stars, and other spherical objects.

Example question

Gravitational Force Between Earth and the Moon

Calculate the gravitational force between Earth and the Moon, given:

Mass of Earth: $m_{1} = 5.97 \times 10^{24} kg$
Mass of Moon: $m_{2} = 7.35 \times 10^{22} kg$
Distance between centers: $r = 3.84 \times 10^{8} m$

Solution

Substitute the given values into the formula:

$F = G \frac{m_{1} m_{2}}{r^{2}} = (6.67 \times 10^{- 11}) \frac{(5.97 \times 10^{24}) (7.35 \times 10^{22})}{(3.84 \times 10^{8})^{2}}$

$F \approx 1.98 \times 10^{20} N$

This is the gravitational force between Earth and the Moon.

Conditions for Point Mass Approximation

Newton’s law of gravitation is derived for point masses, but it can be applied to extended bodies under certain conditions:

Spherical Symmetry: The body must be spherical and have a uniform mass distribution.
Distance: The distance between the bodies must be much larger than their sizes.

Tip

For non-spherical bodies, the gravitational force can be complex.
However, if the body is far enough away, it can often be approximated as a point mass.

Gravitational Field Strength

Definition of Gravitational Field Strength

Definition

Gravitational field

A gravitational field is a region of space where a mass experiences a gravitational force.

Definition

Gravitation field strength

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

Mathematically:

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

where:

$g$ is the gravitational field strength (in newtons per kilogram, N/kg).
$F$ is the gravitational force (in newtons, N).
$m$ is the mass experiencing the force (in kilograms, kg).

Gravitational Field Strength of a Spherical Mass

For a spherical mass $M$ , the gravitational field strength at a distance $r$ from its center is given by:

$g = \frac{G M}{r^{2}}$

This formula shows that the field strength depends only on the mass $M$ and the distance $r$ , not on the mass of the object experiencing the field.

Note

The gravitational field strength decreases with distance from the mass. For example, at a height equal to Earth’s radius above the surface, the field strength is one-quarter of its surface value.

Example question

Gravitational Field Strength on Earth

Calculate the gravitational field strength on Earth given:

Mass of Earth: $M = 5.97 \times 10^{24} kg$
Radius of Earth: $R = 6.37 \times 10^{6} m$

Solution

Substitute these values into the formula:

$g = \frac{G M}{R^{2}} = \frac{(6.67 \times 10^{- 11}) (5.97 \times 10^{24})}{(6.37 \times 10^{6})^{2}}$

$g \approx 9.81 N/kg$

This is the gravitational field strength at Earth’s surface, which is also the acceleration due to gravity.

Field Lines

Representation of Gravitational Fields

Gravitational field lines are a visual tool to represent the strength and direction of a gravitational field.

Direction:
- Field lines point towards the mass creating the field.
- Indicate the direction of the force on a test mass.
Density:
- The closer the lines, the stronger the field.
- Field lines spread out as the distance from the mass increases, reflecting the inverse-square relationship.

Hint

In the diagram, the field lines are radial and point towards the center of the mass.
The lines are denser near the mass, indicating a stronger field.

Tip

Gravitational field lines never cross each other. If they did, it would imply two different directions for the force at a single point, which is impossible.

Why This Matters

Theory of Knowledge

How does the concept of a gravitational field challenge the idea of "action at a distance"?
How might this relate to other fields, such as electric or magnetic fields?

Self review

What is the formula for the gravitational force between two point masses?
How does the gravitational field strength vary with distance from a spherical mass?
Why can a spherical mass be treated as a point mass in gravitational calculations?
What do the density and direction of gravitational field lines represent?

Newton’s law of gravitation and the concept of gravitational fields are foundational to understanding the motion of planets, satellites, and other celestial bodies.

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