Gravitational force is one of the four fundamental forces in nature that governs the interactions between all objects with mass or energy. It is the force that keeps the planets in orbit around the sun, the moon around the earth, and the stars in the galaxy. But how does this force depend on the distance between two objects? In this article, we will explore the answer to this question using Newton’s law of gravitation.
Newton’s Law of Gravitation
Newton’s law of gravitation states that every object in the universe attracts every other object with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. The mathematical expression for this law is:
$$F_g = \frac{Gm_1m_2}{r^2}$$
where $F_g$ is the gravitational force, $m_1$ and $m_2$ are the masses of the two objects, $r$ is the distance between their centers, and $G$ is the universal gravitational constant, which has a value of approximately $6.67 \times 10^{-11} Nm^2/kg^2$.
This formula can be used to calculate the gravitational force between any two objects, such as the earth and the sun, or an apple and the earth. For example, if we want to find the gravitational force between an apple of mass 0.1 kg and the earth of mass $5.97 \times 10^{24} kg$, and we assume that the distance between their centers is equal to the radius of the earth, which is about $6.37 \times 10^6 m$, we can plug these values into the formula and get:
$$F_g = \frac{(6.67 \times 10^{-11})(0.1)(5.97 \times 10^{24})}{(6.37 \times 10^6)^2}$$
$$F_g = 0.98 N$$
This means that the earth pulls the apple with a force of 0.98 newtons, which is equal to its weight.
The Inverse-Square Law
One important feature of Newton’s law of gravitation is that it follows an inverse-square law, which means that the gravitational force decreases as the square of the distance increases. This means that if we double the distance between two objects, the gravitational force becomes four times weaker; if we triple the distance, it becomes nine times weaker; and so on.
To see why this makes sense, imagine a point source of light that emits light rays in all directions. The light rays form a spherical surface around the source, and as they move away from it, they spread out over a larger area. The intensity of light at any point on this surface is proportional to how much light passes through a unit area perpendicular to the rays. Since the area of a sphere is proportional to its radius squared, we can see that as the radius increases, the intensity decreases as its inverse square.
Similarly, we can think of gravity as a field that emanates from a point mass in all directions. The gravitational field strength at any point in space is proportional to how much gravitational force acts on a unit mass placed at that point. Since the gravitational field lines form a spherical surface around the mass, and as they move away from it, they spread out over a larger area, we can see that as the distance increases, the field strength decreases as its inverse square.
The inverse-square law implies that gravity becomes weaker as we move farther away from a mass, but it never becomes zero. This means that even very distant objects can still exert some gravitational influence on each other, although it may be negligible compared to other forces.
Conclusion
In this article, we have learned how to use Newton’s law of gravitation to calculate the gravitational force between two objects, and how this force depends on their masses and their distance. We have also seen that gravity follows an inverse-square law, which means that it decreases rapidly as we increase the distance between two objects.
