The dimensional formula for the gravitational constant (G) is [M⁻¹L³T⁻²].
Understanding the Dimensional Formula
The dimensional formula expresses a physical quantity in terms of its fundamental dimensions: Mass (M), Length (L), and Time (T). Let's break down how we arrive at the dimensional formula for G.
Derivation from Newton's Law of Universal Gravitation
Newton's Law of Universal Gravitation states:
F = G (m₁ m₂) / r²
Where:
- F is the gravitational force
- G is the gravitational constant
- m₁ and m₂ are the masses of the two objects
- r is the distance between the centers of the two objects
To find the dimensional formula for G, we can rearrange the formula:
G = F r² / (m₁ m₂)
Now, let's substitute the dimensional formulas for each quantity:
- Force (F) = [MLT⁻²] (Mass x Acceleration)
- Distance (r) = [L]
- Mass (m₁ and m₂) = [M]
Therefore:
G = ([MLT⁻²] [L]²) / ([M] [M])
G = [ML³T⁻²] / [M²]
G = [M⁻¹L³T⁻²]
Significance
The dimensional formula helps in:
- Checking the consistency of equations: Ensuring both sides of an equation have the same dimensions.
- Converting units: Converting physical quantities from one system of units to another.
- Understanding the nature of physical quantities: Providing insight into how a quantity relates to fundamental dimensions.
