Question

Derive the relation between g and G. $\mathrm{g}=\frac{\mathrm{GM}}{{\mathrm{R}}^2}$

Answer 1

Relationship between 'g' and 'G':

1. The universal law of Gravitation-The force of attraction between any two unit masses separated by the square of the unit distance.
2. G has the same value everywhere in the universe, and G and g are independent of one another.
3. G and g have the following relationship in physics:

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

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 m.

M is the mass of the massive body measured in Kg

Deriving the relationship between 'g' and 'G':

According to the universal law of gravitation:

$\mathrm{F}=\frac{\mathrm{GMm}}{{\mathrm{R}}^2}$ --------------- (1)

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

F = ma ----------------- (2)

Let the acceleration due to gravity be g at a given point.

Then the equations (1) and (2) becomes

F = mg ----------------- (3)

Substitute equations (3) and (1)

Thus, the relationship between ‘g’ and ‘G’ is derived as

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