Understanding the Dimensional Formula of G: From Newton's Law to Gravitational Constant

In the realm of physics, the gravitational constant G is a proportionality constant that appears in the gravitational force equation. Understanding its dimensional formula is essential for grasping the underlying principles of gravitation. This article delves into the dimensional formula of G and its significance within the context of Newton's law of universal gravitation.

The Role of G in Gravitation

When dealing with gravitational forces, the essential equation is the Newton's law of universal gravitation, which states that the force of attraction between two masses is given by:

F - GMM' / d2

This equation describes the attractive force between two masses M and M', where the force is inversely proportional to the square of the distance d between them, and G is the gravitational constant.

Dimensional Formula of G

The dimensional formula of a constant is derived by manipulating the equation to isolate the constant in question. For G, we start with the equation for force and rearrange it to isolate G. The dimensional formula of force F is given by:

[F] MLT-2

The units of force are Newtons (N), and the dimensional formula can be expressed as:

[F] MLT-2

From the gravitational force equation:

F - GMM'/ d2

We can substitute the units of force:

[MLT-2] [G]. M2 L-2

Rearranging to solve for [G], we get:

[G] MLT-2 / M2 L-2

Simplifying, we find:

[G] M-1 L3 T-2

Thus, the dimensional formula for G is:

[G] L3 M-1 T-2

In terms of SI units, the value of G is approximately:

G 6.67 × 10-11 Nm2/kg2

This holds true as both Newton's law of universal gravitation and the dimensional analysis confirm.

The Gravity Field of an Individual Body

The gravity field of an individual body is the result of several factors, including the cosmological acceleration parameter Big G, which has the units of volumetric acceleration divided by mass (cubic meters per second squared per kg, m3/sec2/kg). This complex dimension allows us to break down the gravitational force into more manageable components.

The first step in calculating the total gravitational flux F is to multiply the mass M of the body by the constant 4πG:

F - 4πGM

The negative sign indicates that the flux is inwardly directed. To determine the intensity of the field on the surface of the body, we need to consider the geometry of the surface and the distribution of mass M.

For a uniform density sphere of radius r, we divide the total force by the surface area of the sphere:

4πr2

Thus, the intensity of the field is:

g - 4πGM / 4πr2 - GM / r2

This is equivalent to Newton's gravitational law, often written as F mg, where F is force, m is mass, and g is the acceleration due to gravity.

Interpreting g: Beyond Dimensional Analysis

While g can be expressed as metres/sec2 or as nt/kg, the latter offers more insight into the cause of local g fields. g is the pseudo force created by the action of the global acceleration field Big G acting upon a mass M kg. The intensity of g at any point on the surface of M depends on the geometry, density, and uniformity of M.

Gravity, as seen from the perspective of Newton's second law, is a three-dimensional application where the roles of space and mass are reversed. This insight is crucial for understanding the fundamental nature of gravitation and its profound implications in physics.