Relation between G and g

An object that falls under the sole influence of gravity is known as a free-falling object. A free-falling object has an acceleration of 9.8 m/s², downward (on Earth). This numerical value is so significant that it is given a special name as the acceleration of gravity. We denote it with the symbol g.

The force of attraction between any two unit masses separated by a unit distance is called the universal gravitational constant. The universal gravitational constant is denoted by the symbol G and is measured in Nm²/kg². The numerical value of G is 6.67 × 10^-11 Nm²/Kg².

The relation between G and g is not proportional. This means that they are independent entities.

Relationship between G and g

In physics, G and g related to each other as follows:

\(\begin{array}{l}g=\frac{GM}{R^{2}}\end{array} \)

Where,

• g is the acceleration due to the gravity measured in m/s².
• G is the universal gravitational constant measured in Nm²/kg².
• R is the radius of the massive body measured in km.
• M is the mass of the massive body measured in Kg

Although there exists a formula to express the relation between g and G in physics, there is no correlation between acceleration due to gravity and universal gravitation constant, as the value of G is constant. The value of G is constant at any point in this universe, and G and g are not dependent on each other.

What Is G and g?

The G and g are distinct entities in physics. Below is the table of the difference between G and g.

	Symbol	Definition	Nature of Value	Unit
Acceleration due to gravity	g	The acceleration experienced by a body under free fall due to the gravitational force of the massive body	Changes from place to place. Acceleration due to gravity of the earth is 9.8 m/s2	m/s2
Universal Gravitational Constant	G	The force of attraction between two objects with unit mass separated by a unit distance at any part of this universe.	Constant at any point in this universe. G = 6.67×10-11 Nm2/kg2	Nm2/kg2

Deriving the relationship between g and G

According to the universal law of gravitation,

\(\begin{array}{l}F=\frac{GMm}{R^{2}} ————(1)\end{array} \)

From Newton’s second law of motion, we know that

\(\begin{array}{l}F={m}{a} ———–(2)\end{array} \)

If the acceleration due to gravity is g at a given point, then the above equation becomes

\(\begin{array}{l}F={m}{g} ———–(3)\end{array} \)

Substituting equation (3) in (1), we get-

\(\begin{array}{l}mg=\frac{GMm}{R^{2}}\end{array} \)

Simplifying the above equation, we get

\(\begin{array}{l}g=\frac{GM}{R^{2}}\end{array} \)

Thus, we arrive at the relationship between g and G as –

\(\begin{array}{l}\Rightarrow g=\frac{GM}{R^{2}}\end{array} \)

Physics Related Topics:

Gravity Waves

Relation between Density and Temperature

Relation between Escape Velocity and Orbital Velocity

Relation between Critical Angle and Refractive Index

Universal Law of Gravitation

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