Universal Gravitation Formula: Superposition & Derivation

Universal law of gravitation gives the relationship between the force of attraction between any point objects in space states with respect to the product of their masses and the shortest distance between these two objects. 

• The universal law of gravitation simply is a physical law responsible to demonstrate the attraction between any two objects with mass in the universe.
• The law was discovered by Sir Isaac Newton in the 17th century and is considered one of the most important and fundamental laws of physics.
• The mathematical form of the universal law of gravitation was deduced by Johannes Kepler. This law explains the universality of gravity. The law was experimentally proven by Henry Cavendish in 1798. 

The Gravitational force between the two objects is calculated by the formula:

The law states that “every object in the universe can be seen to attract every other object with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.”

Key Terms: Gravitation, Universal law of gravitation, Gravitational formula, Gravitational Constant, Acceleration due to gravity, Mass, Weight, Newton's Laws, Kepler’s Laws

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Universal Law of Gravitation

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Sir Issac Newton, known as the father of modern science revolutionized our understanding of the concept of Gravity, by introducing the Universal law of gravitation in 1665. He came up with the concept of Gravity wherein, he explains that “all objects were attracted to each other when separated by a distance”.

The Universal Law of Gravitation states that: 

• This law is stated as the “Universal Law of Gravity” since it not only holds true for the attraction between the Earth and other objects but also holds true for all objects in the universe. 
• This law also accurately explains the elliptical motion of the planets around the sun pertaining to the fact that the objects exert gravitational force on each other.
• This discovery helped in further understanding how the sun keeps the planets in their orbits and further advanced our understanding of the structure of the universe and how gravity plays an extremely important role in it. 
• It is important to note that gravity is the weakest of all basic forces.

Gravitation Detailed Video Explanation

Read More: Difference between Gravitation and Gravity

Characteristics of Gravitational Force

Some Gravitational force characteristics include:

• Gravitational force is a central force of attraction between two objects which acts along the line joining the centres of the interacting bodies.
• It is independent of the nature of the medium.
• Magnitude of Gravitational force is extremely small hence it is the weakest of all basic forces
• The gravitational force between the two particles forms an action-reaction pair.

Graviational Force

Gravitational force is a force of attraction between any two objects in the universe that have mass.

• It is one of the four fundamental forces of nature. For example, it is also responsible for holding the planets in orbit around the sun, and the moon in orbit around the Earth.
• The strength of the gravitational force between two objects depends on the mass of the objects and the distance between them.
• The greater the mass of the objects, the greater the gravitational force between them.
• Similarly, the closer the objects are to each other, the stronger the gravitational force between them.

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Universal law of Gravitation Formula

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Consider two bodies A and B,

Representation of Universal Law of Gravitational 

Where,

• m₁ and m₂ = the mass of the two objects A and B
• r = the shortest distance between the centers of the two objects
• F = Gravitational force (measured in Newtons(N)).

By the law stated by Newton, we understand that;

→ F ∝ m₁×m₂ ;

→ F ∝ \(\frac{1}{r^2}\);

→ F ∝ \(\frac{m_1 * m_2}{r^2}\) ;

Therefore, the universal graviation equation is given by:

→ F = \(\frac{G \times \ m_1 \times\ m_2}{r^2}\)

Where, G is the Gravitational Constant

Universal Law of Gravitation Derivation

As per the Force = G × m₁ × m₂ × [r²]-¹

Or, G = Force × r² × [m₁ × m₂]-¹ . . . . . (1)

Herein, G = Universal Gravitational Constant

Now, as per the dimensions,

Mass = [M¹ L⁰ T⁰] . . . . (2)

Radius = [M⁰ L¹ T⁰] . . . . (3)

Force = [M¹ L¹ T-²] . . . . . (4)

Upon replacing the equations (2), (3), and (4) in equation (1), we get,

Universal Gravitational Constant = Force × r² × [m₁ × m₂]^–2

It can also be expressed as, G = [M¹ L¹ T^–2] × [M⁰ L¹ T⁰]² × [M¹ L⁰ T⁰]-1 × [M¹ L⁰ T⁰]^–1 = M^–1 L³ T^–2.

Hence, the Universal Gravitational Constant is dimensionally expressed as [M^–1 L³ T^-2].

Gravitational constant (G)

The proportionality constant G termed the Gravitational constant, in the universal law of gravitation has a value equivalent to 6.67259 x 10^–11N m²/kg². This precise value was experimentally determined by Henry Cavendish. G is a scalar quantity and this value remains constant in the universe and is not dependent on the nature and the size of the body.

→ (G = 6.67259 x 10^–11N m²/kg²) 

The dimensional formula of G is [M^-1 L³ T^-2]

Read More: Gravitational potential energy

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Universal Gravitation Formula Vectorial Representation

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Vectorially the Gravitational force is given as:

where,

• \(\widehat{r}\) is the unit vector from m₁ to m₂
• ​and r = r₂ - r₁

Vector diagram

Principle of Superposition of Gravitation

This principle states that the resultant gravitational force F acting on a particle due to a number of point masses is equal to the vector sum of the individual forces exerted by the individual masses on the given particle.

\(\overrightarrow{F}\) = \(\overrightarrow{F_{01}}\) + \(\overrightarrow{F_{02}}\) + \(\overrightarrow{F_{03}}\)+ ……. +\(\overrightarrow{F_{0n}}\)

Where  \(\overrightarrow{F_{01}}\) , \(\overrightarrow{F_{02}}\) , \(\overrightarrow{F_{03}}\), ……. ,\(\overrightarrow{F_{0n}}\) are all gravitational forces exerted. 

Weight and Mass

We often interchange the terms weight and mass, but there is a major difference between the two.

• Mass is purely the amount of matter that the object consists of whereas weight is considered to be the gravitational force applied on an object of a certain mass. 
• Weight is the product of the mass and the acceleration due to gravity. (w = mg)
• Hence, Mass and weight are directly proportional to each other. 
• Mass always remains the same whereas Weight varies. This is the exact same reason why we weigh differently on different planets. 

Acceleration due to gravity (g)

Acceleration due to gravity is defined as the acceleration that is gained by an object due to the gravitational force applied on it. 

• It is always in the direction towards the centre of the Earth. 
• Acceleration due to gravity has both magnitude and direction. Hence, it is a vector quantity.
• Acceleration due to gravity (g) is different in different parts of the earth. It is highest at the poles and at the center of the earth, it becomes zero.

Read More: The value of g on the moon

The Formula for acceleration due to gravity is

g = \(\frac{GM}{r^2}\)

where,

• M = Mass of the Earth
• G = Gravitational Constant
• r = radius of the Earth

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Universality of Gravity

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The universality of gravity is generally refers that every particle in the universe is affected by the force of gravity. This includes all matter and energy, regardless of its composition or state.

• The force of gravity between two objects is directly proportional to their masses and inversely proportional to the square of the distance between them.
• This means that every object with mass exerts a gravitational force on every other object with mass in the universe, no matter how small or far away.

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Advantages and Disadvantages of Universal law of Gravitation

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The many advantages and disadvantages of Universal law of Gravitation include:

Advantages of Universal law of Gravitation:

• The law explains as to why all objects fall towards the Earth.
• Explains the tides in the ocean due to the gravitational pull between the moon and the ocean.
• This law also helps in determining the Gravitational force between any planets in the Universe.
• It also helps us to calculate the trajectory of astronomical bodies which further, helps in determining their motion.
• The law helps in predicting the orbits and the time period of artificial satellites.
• It also helps in calculating the occurrences of solar and lunar eclipses.

Disadvantages of Universal Law of Gravitation

• The law cannot be applied when the objects are travelling at the speed of light.
• The law also cannot be applied when very small objects such as atoms or sub-atomic particles are taken into consideration

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Things to Remember

• Gravitational force between two objects is independent of the nature and size of the interacting objects.
• ​The force of Gravitation is directly proportional to the Product of their masses and inversely proportional to the square of the distance between them.
• The Force of Gravitation between two objects is calculated using. F = \(\frac{G*m_1 * m_2}{r^2}\)
• The value of the gravitational constant is equal to 6.67259 x 10^–11N m²/kg². 
• Acceleration due to gravity is dependent on the radius of the Earth’s surface and hence varies in different parts of the earth.

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