The Universal Gravitational Constant G in SI Units

The concept of the universal gravitational constant, G, was a pivotal element in Isaac Newton's development of his law of universal gravitation. This fundamental constant represents the strength with which two masses attract each other through gravity, and understanding its SI units is crucial for precise calculations and scientific endeavors.

Deriving the SI Units of G

The equation for Newton's law of universal gravitation is given by:

F G * m1m2 / r2

Where:

F is the gravitational force between the two masses G is the gravitational constant m1 and m2 are the masses of the two bodies r is the distance between the centers of the two bodies

To find the SI units of G, we rearrange the formula to make G the subject:

G F * r2 / m1m2

The SI units of the quantities involved are:

Force (F): Newtons (N) Mass (m1, m2): Kilograms (kg) Distance (r): Metres (m)

Substituting these units into the equation, we get:

N * m2 / (kg * kg) N m2 / kg2

Therefore, the SI units of G are:

N m2 / kg2 or equivalently, N kg-2 m2

Dimensional Analysis of G

Using dimensional analysis, we can further break down the units of G:

kg * m / s2 * m2 / (kg * kg) kg * m / (kg * s2) * m2 / kg m3 / (kg * s2)

This simplifies to:

kg-1 m3 s-2

Or, another equivalent form:

m3 / kg s2

Historical Context and Informed Guess

As Mr. Hoath rightly points out, the formula for Newton's law of universal gravitation was not derived from any existing theory but was rather an intelligent guess based on existing knowledge and geometric reasoning. Newton's three laws of motion, which define mass and force, were developed before his law of gravitation.

Newton himself proposed that the force of gravity follows an inverse square law, which is a concept that can be derived from the idea that a field of flux spreading out from a source will decrease with distance in an inverse square fashion. However, this is not how gravity works in general relativity, which provides a more accurate description of gravity.

Despite its limitations, Newton's law of gravitation was found to successfully describe planetary motion, ordinary falling, and tides. This historical context highlights the importance of both empirical observation and theoretical speculation in the development of scientific theories.