 # Relation Between G and g

### What is Gravitation?

Gravitation is the name given to a phenomenon that is acting as a force of attraction between any two bodies in the universe. It was discovered by Issac Newton in the year 1665.

The saying goes that Newton was sitting under an apple tree when an apple fell down on the earth. This observation gave birth to a new phenomenon that the earth attracts all the bodies towards its center by some force. Hence the apple fell freely from the tree under the effect of a gravitational pull.

This led to the introduction of concepts like G and g.

• What are G and g?

• Difference between G and g

• G and g

• Relation between G and g

• Value of capital g

• Difference between small g and capital g

### What is G?

Consider two bodies of mass $m_{1}$ and $m_{2}$. Let r be the distance between the center of these bodies and is inversely proportional to the square of the distance between them.

F ∝ $\frac{m_{1}m_{2}}{r^{2}}$ (Newton’s law of gravitation)

Or F = $\frac{Gm_{1}m_{2}}{r^{2}}$

Here, G is the constant of proportionality called the universal gravitational constant.

Let $m_{1}$ = $m_{2}$ = 1 and r = 1

Then from equation F = $\frac{Gm_{1}m_{2}}{r^{2}}$

F = $\frac{G\times 1\times 1}{1}$ = G

We get, F = G

Here we can conclude that Universal gravitational constant is equal to the force of attraction acting between two bodies each unit mass and their centers are placed a unit distance apart.

The G is a scalar quantity whose value is given by,

## Value of G

 S.I. value =  6.67 x 10⁻¹¹ N m² Kg⁻²CGS value = 6.67 x 10⁻⁸  dyne cm ² g ⁻²

### Dimensional Formula for G

G = $(\frac{Fr^{2}}{m_{1}.m_{2}})$

=  {[M L T⁻²] . [L²]} / [M] . [M]

 G =  [M ⁻¹ L ³ T ⁻²]

### What is g?

A gravitational pull attracts the body towards the earth and that body which freely falls under the effect of gravity by a constant acceleration. Such an acceleration is called the acceleration due to gravity, denoted by g.

Where g is a vector quantity and it is directed towards the center of the earth.

Its value is taken as 9.8 m s⁻² for all practical purposes.

## Difference Between G and g

 S.No. Parameters Gravitational constant G Gravitational acceleration g 1. Definition It is defined as the gravitational force between two bodies of each unit mass separated by a unit distance. It states that a body freely falls under the effect of gravity by a constant acceleration. 2. Value 6.67 x 10⁻¹¹ N m² kg⁻² 9.8 m s⁻² 3. Attribute (Quantity) Scalar quantity Vector quantity 4. Variations The value of G is constant in the universe or simply a dimensional constant. Value varies with altitude, depth, and rotation of the earth. 5. Nature Independent of the nature and size of the bodies and the medium in which they are kept. Independent of the shape, size, and mass, but depends upon the mass and radius of the earth or planet due to which there is a gravity pull. 6. Unit Nm² /kg² ms² 7. Dimensional formula [M ⁻¹ L ³ T ⁻²] [M⁰ L¹ T⁻²]

### The Relation Between G and g

Consider earth to be a spherical body of mass M, radius R with center O, as shown in the figure below:

The acceleration due to gravity is g. If the density of the earth is uniform,  its mass can be supposed to be concentrated at its center O.

Let F be the force of attraction between the body and the earth.

According to Newton’s law of gravitation

F = $\frac{G mM}{r^{2}}$ ….(1)

From the gravity pull, F = mg…(2)

Equating (1) and (2) we get,

 g  = $\frac{GM}{ r^{2}}$

The equation so obtained states that g depends upon the mass and radius of the earth.

### SAQs

Q1: What is the acceleration due to gravity at the Moon’s surface if the moon’s mass is 7.35 x 10²² kg, and the radius is $1.74\times 10^{6}$ m.

Ans: Given M = 7.35 x 10²² kg and r = 1.74 x 10⁶ m

Considering equation g = GM / r² …(3)

Putting the values in above eq(3)

g = $\frac{(6.67\times 10^{-11})\times (7.35\times 10^{22})}{(1.74\times 10^{6})^{2}}$

On solving we get,

 g = 16.1925 x 10-1 or 1.61925

Q1: How does Acceleration due to Gravity vary with the Altitude of the Earth? Explain Giving Complete Mathematical Expression.

Effect of Altitude

Let us consider the earth to be of mass M, radius R with a center O. Let g be the acceleration due to gravity g at a point P on the surface of the earth.

Since g = GM/ R² …(1)

If g’ is the acceleration due to gravity at point h above the surface of the earth then g’ is given by,

g’ = GM /(R+h)² …..(2)

eq (2) / eq(1) we get,

g’ / g =  R² / (R+h)²  =  R² / (1+h/R)²

= (1 + h / R)⁻² = (1 - 2 h /R)

(By binomial expansion, (1+x)⁻ⁿ = 1 - n x n(n+1)/2! x² +......)

When h << R, then h/R is very less (less than unity)

Hence g’ = g (1 -2h/R).

Q2: Which Law Explains the Rotation of Planets Around the Sun and the Rotation of Satellites Around the Planets?

The universal law of gravitation (stated by Issac Newton) forms the basis of the explanation for the rotation of planets around the sun and the rotation of satellites around the planets.

Q3: What is Gravity Made of?

The gravity is made of quantum particles called gravitons. These gravitons are massless, but they do carry energy.