Question

Which of the following statements is true for the universal gravitational constant?
A.) Its dimensional formula is $\left[ {{M^{ - 1}}{L^3}{T^{ - 2}}} \right]$
B.) Its value does not depend on the size of bodies
C.) It does not depend upon the nature of the medium
D.) All of the above

Answer 1

Hint: Gravitational constant is a universal constant which is encountered in Newton’s law of gravitation. It has the same value at every place in the universe irrespective of size of astronomical bodies or the nature of the medium.

Detailed step by step solution:
Newton’s law of gravitation says that two bodies which have mass exert an attractive force on each other. This force is called gravitational force or gravity. According the Newton, mathematically this force is

Directly proportional to the masses of the two bodies under consideration

$F \propto {m_1}{m_2}{\text{ }}...{\text{(i)}}$

where ${m_1}$and ${m_2}$ are the masses of the two bodies.

Inversely proportional to the square of the distance by which they are separated from each other

$F \propto \dfrac{1}{{{r^2}}}{\text{ }}...{\text{(ii)}}$

where r signifies the separation distance between the two bodies.

This means that greater the mass of the bodies, greater is the gravitational force between the two bodies and if we increase the distance between the two bodies, then the gravitational force between them will also decrease and vice-versa.

Combining equation (i) and (ii), we get

$F \propto \dfrac{{{m_1}{m_2}}}{{{r^2}}}{\text{ }} \\\ \Rightarrow F = G\dfrac{{{m_1}{m_2}}}{{{r^2}}}{\text{ }} \\\$

where G is the constant of proportionality called the universal gravitational constant. Its value is given as

$G = 6.67 \times {10^{ - 11}}{m^3}k{g^{ - 1}}{s^{ - 2}}$

The dimensional formula for G is given as $\left[ {{M^{ - 1}}{L^3}{T^{ - 2}}} \right]$

The value of the gravitational constant remains the same for all astronomical systems and does not depend on the mass of the bodies or nature of the medium.

Therefore, the correct answer is option D.

Note:
1. The gravitational force between two bodies depends on the mass of the objects but the gravitational constant is independent of them.
2. The gravitational force follows the inverse square law which means that its reduces or increases with the square of the distance between two bodies.