# Relation Between G and g

## Small g

An object which falls under the sole influence of our gravity is called a free-falling object.A free-falling object usually has an acceleration of about 9.8 m/s2, downward towards earth. This given numerical value 9.8m/s2 is so significant that it is given a unique and special name which is called the acceleration of gravity. We usually denote it with the symbol g.

Value of g

9.8 m s -2

Dimension of g

M0L1T-2

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## Capital G

The force of attraction which is given between two unit masses which are separated by some unit distance is called the universal gravitational constant. This universal gravitational constant is often denoted by the symbol G (capital G) and is measured in Nm2/kg2.

The numerical value of G (capital G) is usually 6.67 × 10-11 Nm²/Kg².

Both g and G are independent entities,Hence, it is said that the relation between g and G are not proportional

Value of G

C.G.S. value

6.67 x 10⁻⁸ dyne cm² g⁻²

S.I. value

6.67 x 10⁻¹¹ Nm² Kg⁻²

Dimensional Formula for G

G = Fr2 / m1.m2

= {MLT⁻² . L²}

M . M

G = M⁻¹L³T⁻²

## Relationship Between G and g

G and g are usually related to each other as the following formula

g = GM/R2

Here,

g →acceleration due to the gravity measured in m/s2.

G →universal gravitational constant measured in Nm2/kg2.

R →radius of the massive body measured in km.

M →mass of the massive body measured in Kg

In physics there is a formula to express the relation between g and G and there is usually no correlation between the universal gravitation constant and acceleration due to gravity as the value of G (capital G)is constant.

The value of G is always constant at any point in our living universe, and g and G are never dependent on each other.

Derive the relationship between G and g

As per the universal law of gravitation,

F = GMm / R2

———— equation (1)

By Newton’s second law of motion, we get

F = ma

———– equation (2)

At a given point. If the acceleration due to gravity is g, then the above equation (2) becomes.

F=mg

———– equation (3)

Hence, by substituting equation (3) in equation (1), we get….

mg = GMm / R2

Therefore,the relationship between G and g can be recalled as

⇒ g = GM / R2

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