Dimensional Formula of Gravitational Constant

The dimensional formula of Gravitational Constant is 

[ M-1  L3  T-2 ]

where [M], [L], and [T] are the fundamental quantities: Mass, Length, and Time.

From the above formula, we can derive the dimensions of the gravitational constant as (1, 2, -1).


Derivation of Dimensional Formula of Gravitational Constant

We can derive this formula using the definition of the gravitational constant.

In physics, the gravitational constant is the constant of proportionality that has the same value everywhere.

Newton’s law of gravitation states that the gravitational force of attraction between the earth and a body is given by,


  F = GMm / R2

Here,

  • G is the Gravitational Constant
  • M represents the mass of the earth
  • m is the mass of the body
  • R represents the radius of the earth

Hence, the formula of gravitational constant becomes


  G = F * R2 / M m

We know that Force = mass (m) * acceleration (a), substituting the value of Force in the above equation.


  G = (m * a * R2 ) / (M * m)

Dividing m, we get


  G = (a * R2 ) / M

We know,

  • acceleration = speed / time
  • speed = distance / time
  • acceleration = distance (d) / time2 (t2)

Substituting the value of acceleration, we get


  G = ( d / t2) * R2 / M
    
    = ( d * R2 ) / ( M * t2)
    
    = d1 * R2 * M-1 * t-2

Dimensionally, we use

  • [L] to represent both the distance (d) and radius (r
  • [M] for mass (M)
  • [T] to represent the time (t)

So, the equation becomes


  Gravitational Constant = [L1] * [L2] * [M-1] * [T-2]

                         = [M-1] [L3] [T-2]

Hence, the dimensional equation of the gravitational constant is G = [ M-1 L3 T-2 ].


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