Question

How are g and G are related?
A. $g=\dfrac{GM}{{{R}^{3}}}$
B. $g=\dfrac{M}{G{{R}^{2}}}$
C. $g=\dfrac{GM}{{{R}^{2}}}$
D. $g=\dfrac{GM}{R}$

Answer 1

Hint: Study about the universal law of gravitation and obtain an equation which gives us the force between two massive objects. Find the acceleration due to gravity of a free-falling object. Comparing these two, find the relation between them.

Complete Step-by-Step solution:
G is the universal gravitational constant. When we express the gravitational force of attraction between two objects with mass the proportionality constant used is G.
The force of attraction between two objects with mass M and m, which are a distance R apart is directly proportional to the product of the mass and the inversely proportional to the square of the distance,
So, we can write,
$F\propto \dfrac{Mm}{{{R}^{2}}}$
To equate this equation, we will multiply a proportionality constant which is G.
$F=\dfrac{GMm}{{{R}^{2}}}$
Where, g is the universal gravitational constant with value, $G=6.673\times {{10}^{-11}}N{{m}^{2}}k{{g}^{-2}}$
G is the acceleration due to gravity experienced by a massive object falling freely through a gravitational field. Its value changes depending on the place of the object.
Now, according to the universal law of gravitation we have,
$F=\dfrac{GMm}{{{R}^{2}}}$
Again, using Newton's second law of motion, we can define g as,
$g=\dfrac{F}{m}$
Putting the value of F from the universal law of gravitation we get,
$\begin{align}
  & g=\dfrac{\dfrac{GMm}{{{R}^{2}}}}{m} \\
 & g=\dfrac{GM}{{{R}^{2}}} \\
\end{align}$
If we consider an object falling towards earth, in the above equation M will be the mass of earth and R will be the distance to the object from the earth’s centre.
So, the correct option is (C).

Note: Value of g depends on the distance of the object which is falling from the centre of the mass towards which the object is falling. As, the value of g is inversely proportional to the square of this distance, the farther the object is, the less will be the value of g.