The dimensional formula of gravitational force can be derived using the principle of dimensional analysis. The gravitational force between two objects depends on their masses and the distance between them. The formula for gravitational force (F) is given by:
F = G * (m1 * m2) / r^2
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
Let's break down the dimensional analysis for each term:
The gravitational force (F) has the dimensional formula of [M L T^-2], which represents mass times length divided by time squared.
The gravitational constant (G) has the dimensional formula of [M^-1 L^3 T^-2], which represents the inverse of mass times length cubed divided by time squared.
The masses (m1 and m2) have the dimensional formula of [M], representing mass.
The distance (r) has the dimensional formula of [L], representing length.
Substituting the dimensional formulas into the gravitational force equation, we get:
[M L T^-2] = [M^-1 L^3 T^-2] * [M] * [M] / [L^2]
Simplifying the equation, we get:
[M L T^-2] = [M L^3 T^-2] * [M^2] / [L^2]
Canceling out the common terms, we have:
1 = [L^3] * [M] / [L^2]
Simplifying further:
1 = [L] * [M]
Therefore, the dimensional formula of gravitational force is [M L T^-2].