Derivation:
The formula for gravitational force is given by 2F=G⋅r2m1⋅m2.
Alternatively, we can express G using dimensional analysis:
Starting with 2F=G⋅r2m1⋅m2, we rearrange it to solve for G:
2G=m1⋅m2F⋅r2 . . . . . (1)
Here, G represents the Universal Gravitational Constant.
Now, let's analyze the dimensions of the involved quantities:
Mass has dimensions: [M] Radius has dimensions: [L] Force has dimensions: [M⋅L⋅T−2]
Substituting these dimensions into equation (1), we get:
G=[M]⋅[M][M⋅L⋅T−2]⋅[L]2
Simplifying further:
G=[M−1⋅L3⋅T−2]
Hence, the dimension of the Universal Gravitational Constant (G) is [M−1⋅L3⋅T−2].
M-1L3T-2
NaOH is deliquescent
Dimensional Analysis is a process which helps verify any formula by the using the principle of homogeneity. Basically dimensions of each term of a dimensional equation on both sides should be the same.
Limitation of Dimensional Analysis: Dimensional analysis does not check for the correctness of value of constants in an equation.
Let us understand this with an example:
Suppose we don’t know the correct formula relation between speed, distance and time,
We don’t know whether
(i) Speed = Distance/Time is correct or
(ii) Speed =Time/Distance.
Now, we can use dimensional analysis to check whether this equation is correct or not.
By reducing both sides of the equation in its fundamental units form, we get
(i) [L][T]-¹ = [L] / [T] (Right)
(ii) [L][T]-¹ = [T] / [L] (Wrong)
From the above example it is evident that the dimensional formula establishes the correctness of an equation.