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:

 $$g=\frac{GM}{R^{2}}$$

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,

$$F=\frac{GMm}{R^{2}}$$ ————(1)

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

$$F={m}{a}$$ ———–(2)

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

$$F={m}{g}$$ ———–(3)

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

$$mg=\frac{GMm}{R^{2}}$$

Simplifying the above equation, we get

$$g=\frac{GM}{R^{2}}$$

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

$$\Rightarrow g=\frac{GM}{R^{2}}$$

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1 Comment

1. Abhijeet Das

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