The gravitational constant, denoted by "G," is a fundamental physical constant in the equation for gravitational force. The equation, as you mentioned, is:
F = G * (m₁ * m₂) / r²
In this equation:
- F represents the gravitational force between two objects.
- m₁ and m₂ are the masses of the two objects.
- r is the distance between the centers of mass of the two objects.
The dimensions of the gravitational constant "G" can be determined by analyzing the units on both sides of the equation.
On the left-hand side, the unit of force is typically Newton (N), which can be expressed as kg·m/s² (kilogram times meter per second squared).
On the right-hand side, the mass "m₁" and "m₂" have units of kilogram (kg), and the distance "r" is measured in meters (m). Therefore, the product (m₁ * m₂) has units of kg².
Putting these together, the dimensions of the gravitational constant "G" can be determined as:
[G] = [F] * [r²] / [(m₁ * m₂)]
[G] = (kg·m/s²) * (m²) / (kg²)
Simplifying further:
[G] = kg⁻¹·m³·s⁻²
Thus, the dimensions of the gravitational constant "G" are kilograms to the power of negative one (kg⁻¹), meters cubed (m³), and seconds to the power of negative two (s⁻²).