What is the Law of Universal Gravitation?

We explain to you what the Law of Universal Gravitation is, how is its formula and its statement. In addition, examples of the use of its formula.

1. What is the Law of Universal Gravitation?

The Law of Universal Gravitation is one of the physical laws formulated by Isaac Newton in his book Philosophiae Naturalis Principia Mathematica of 1687. It describes the gravitational interaction between bodies endowed with mass , and establishes a proportional relationship of the force with which those bodies attract the each other.

To formulate this law, Newton deduced that the force with which two masses attract is proportional to the product of their masses divided by distance that separates them squared. These deductions are the result of empirical verification by observation, as well as the mathematical genius of the English scientist.

Hence, by increasing the distance that separates two bodies, this law acts approximately, as if the entire mass of both bodies is concentrated in its center of gravity . That is, the closer and more massive two bodies are, the more intensely they will attract . Like other Newtonian laws, it represented a leap forward in the scientific calculation of the time.

However, today we know that, from a certain amount of mass, this law loses its validity

 (that is, in the case of supermassive objects), then passing the witness to the General Relativity Law formulated in 1915 by Albert Einstein. However, the Law of Universal Gravitation is still useful to understand most of the gravitational phenomena of the Solar System.

2. Statement of the Law of Universal Gravitation

The formal statement of this Newtonian law says that

” The force with which two objects are attracted is proportional to the product of their masses and inversely proportional to the square of the distance that separates them .”

This means that any two bodies are attracted with a greater or lesser force depending on their mass being greater or lesser, and also depending on the distance between them.

3. Formula of the Law of Universal Gravitation

The fundamental formula of the Law of Universal Gravitation is the following:

Where:

• F is the force of attraction between two masses
• G is the universal gravitation constant (calculated at 6,673484.10 ^-11 Nm ² / kg ² )
• m ₁ is the mass of the first body
• m ₂ is the mass of the second body
• r the distance that separates them.

If the attractive forces of each body (F ₁ and F ₂ ) are calculated , there will be two equal forces in modulus and in the opposite direction, and instead of being canceled, they are combined. This requires a vector formula:

F ₁₂ = -G. (m ₁ .m ₂ ) / ǁr ₂ -r ₁ ǁ ² . ȗ ₁₂

Where ȗ ₁₂ is the unit vector that can be drawn from the center of gravity of one object to the other.

4. Examples of the Law of Universal Gravitation

Let’s solve a couple of exercises as an example of the application of this formula.

• Suppose that a mass of 800 kg and another of 500 kg attract each other in a vacuum, separated by a space of 3 meters. How can we calculate the force of attraction they experience?

Simply by applying the formula:

F = G. (m ₁ .m ₂ ) / r ²

What will it be: F = (6.67 × 10 ^-11 Nm ² / kg ² ). (800 kg. 500 kg) / (3) ²

And then: F = 2,964 x 10 ^-6 N.

• From another point of view, how far should we place two bodies of mass of 1 Kg, so that they attract with a force of 1 N?

Starting from the same formula

F = G. (m ₁ .m ₂ ) / r ²

We will clear the distance, staying that r ² = G. (m ₁ .m ₂ ) / F

Or what is the same: r = √ (G. [m ₁ .m ₂ ]) / F

That is: r = √ (6.67 × 10 ^-11 Nm ² / kg ^2. 1kg x 1kg) / 1N

The result is that r = 8.16 x 10 ^-6 meters.