Question

Explain the process of finding the dimensional formula of universal gravitational constant (G) and mention its SI unit.

Answer 1

Hint: Apply the Newton’s law of universal gravitation i.e., all objects attract each other with a force of gravitational attraction. This force of gravitational attraction is directly proportional to the product of masses of both objects and inversely proportional to the square of the distance between them. Mathematically it is given by, \[F=\dfrac{G{{m}_{1}}{{m}_{2}}}{{{r}^{2}}}\]

Complete answer:
According to Newton's law of universal gravitation, the force of gravitational attraction is given by,
\[F=\dfrac{G{{m}_{1}}{{m}_{2}}}{{{r}^{2}}}\]
Where,
F = gravitational force of attraction
G = universal gravitational constant
m1 = mass of an object
m2 = mass of another object
r = distance between two objects
Which can be written as,
\[G=\dfrac{F{{r}^{2}}}{{{m}_{1}}{{m}_{2}}}\]
Now for finding the dimension formula of universal gravitational constant G,
we need to write dimensional formula of each quantity in right hand side i.e.,
The dimensional formula of G = \[\dfrac{\left[ \text{ML}{{\text{T}}^{\text{-2}}} \right]\left[ {{\text{L}}^{\text{2}}} \right]}{\left[ \text{M} \right]\left[ \text{M} \right]}\]
\[\Rightarrow \dfrac{\left[ \text{ML}{{\text{T}}^{\text{-2}}} \right]\left[ {{\text{L}}^{\text{2}}} \right]}{\left[ {{\text{M}}^{2}} \right]}\]
\[\Rightarrow \left[ {{\text{M}}^{\text{-1}}}\text{L}{{\text{T}}^{\text{-2}}} \right]\left[ {{\text{L}}^{\text{2}}} \right]\]
\[\Rightarrow \left[ {{\text{M}}^{\text{-1}}}{{\text{L}}^{3}}{{\text{T}}^{\text{-2}}} \right]\]
So, the dimensional formula of universal gravitational constant G is, \[\left[ {{\text{M}}^{\text{-1}}}{{\text{L}}^{3}}{{\text{T}}^{\text{-2}}} \right]\]
And the SI unit of universal gravitational constant (G) is $N{kg^{-2}}{m^2}$

Additional Information:
The gravitational force is directly proportional to the mass of both interacting objects, more massive objects will attract each other with a relatively greater gravitational force. So as the mass of either object increases, the force of gravitational attraction between them also increases. If the mass of one of the objects is doubled, then the force of gravity between them is also doubled.
The gravitational force is inversely proportional to the square of the distance between the two interacting objects, more distance will result in relatively weaker gravitational forces. So as two objects are separated from each other, the force of gravitational attraction between them also decreases.

Note:
Students should understand Newton's law of universal gravitation so that they can write the formula for force of attraction easily. Thereafter students need to write the dimensional formula of each and every quantity in the right hand side, in this way they will get the required dimensional formula. There is one more method to find out the dimensional formula of universal gravitational constant G, by the relation between g and G i.e., \[g=\dfrac{Gm}{{{r}^{2}}}\]. In this method students need to make G as a subject and write dimensional formulas for the quantities in the right hand side.