Dimensional Formula of Gravitational constant

Gravity is an enormous force in nature which is present in macro objects like planets, stars and galaxies to micro-objects like atoms, molecules etc. It is the attraction force present between the objects. Whenever there exist two objects with mass, the gravitational constant can be used to determine the force between them. The Gravitational constant between two bodies is directly proportional to the square of the distance and inversely proportional to their masses.

Mathematically it can be expressed as F = GmMr2

M and m are masses, r is the distance, and G is the gravitational constant.

What is Gravitational Constant 

We employ the gravitational constant G for calculating the gravitational effects. It is different from g, which denotes acceleration because of gravity. We mostly use gravitational constant in the equation, F = (G x m1 x m2) / r². 

Here, F denotes the force of gravity, 

m1 = mass of the first object, 

m2 = mass of the second object, 

r = separation between the two masses, 

and G =gravitational constant.

Like all the other constants of physics, this constant is also an empirical value, proven through multiple experiments and subsequent observations. Issac Newton initially introduced the gravitational constant in Philosophiae Naturalis Principia Mathematica (1687), but it came into operation only after 1798. 

Henry Cavendish, the famous physicist, was the first to measure the value of the gravitational constant. He measured the force in between two lead masses with the torsion balance. The value of G is so minimal that when multiplied with other quantities, it gives a small resultant force. The value if expanded is close to 0.00000000006673 N m2 kg-2. 

Dimensional Formula of Gravitational constant

We know that,

From the above equation, we can easily derive the formula for gravitational constantGravitational constant = G = F ×r2 mM

         = [Force][Distance]2/[Mass]2

• The dimensional of force is [M1T-2L1]

                    = [M1T-2L1] × [L2M0T0] / [M2]

                    = [M1+0T-2+0L1+0] / [M2]

                    = [M-1T-2L3]

Therefore the dimensional formula of the Gravitational constant is given by [M-1T-2L3]

Dimensional formula of Gravitational constant importance:

• The Dimensional formula of the Gravitational constant helps us understand the physical correctness of any equation involving force.
• It helps us to understand the relationship between different physical quantities involving it.
• It helps us in converting units from one physical quantity to another.
• In any relationship, the constant dimensions can be found using this analysis.

Dimensions in units and measurements

The dimensions can be written as the powers of the fundamental units of length, mass, and time. It depicts their nature and does not show their magnitude. 

Example of writing dimensions:

Let’s take the formula of the area of the rectangle:

Area of the rectangle = length x breadth

= l x l ( where breadth is also showing the length of the side)

= [L1] X [L1]

= [L2]

Here, we can see the length to the power of 2, and we cannot find the dimension of mass and time.

Hence, the dimension of the area of a rectangle is written as [M0 L2 T0]

Dimensional formula

The dimensional formula depicts the dependency of physical quantity with fundamental physical quantity and the powers.

Example:

Let’s take the formula of speed:

Speed = Distance / Time

The distance can be written in length [L]

Time can be written as [T]

The dimensional formula would be [ M0 L1 T-1]

Hence, we can conclude that the speed is dependent on only length and time, not mass.

Dimensional equation

The physical quantity is equated with the dimensional formula to get the dimensional equation. 

Example:

Velocity = [ M0 L1 T-1]

Here, velocity is the physical quantity, equating to the dimensional formula.

Numericals on Gravitational Constant

Example 1: 

What will be the gravitational force between Earth (m = 5.98 x 1024 kg) and an 80 kg person if he is standing at sea level with a distance of 6.35 x 106m from the centre of the Earth? 

From the question we know, m1 = 5.98 x 1024 kg, m2 = 80 kg, r = 6.35 x 106 m. 

Substituting the values in the formula: 

F = (G x m1 x m2) / r²

 where G = 6.674×10−11 m3⋅kg−1⋅s−2 gives, 

F = ( 6.674×10−11 m3⋅kg−1⋅s−2) X (5.98 x 1024 kg) X (80 kg) / (6.35 x 106m)²

F = 791.35 N

Example 2: 

What will be the gravitational force between Earth (m = 5.98 x 1024 kg) and a 60 kg person if he is at 35000 feet above the Earth’s surface in an aeroplane at a distance of 6.38 x 106 m from the centre of the Earth? 

From the question we know, m1 = 5.98 x 1024 kg, m2 = 60 kg, r = 6.38 x 106 m

Substituting the values in the formula: 

F = (G x m1 x m2) / r²

 where G = 6.674×10−11 m3⋅kg−1⋅s−2 gives, 

F = ( 6.674×10−11 m3⋅kg−1⋅s−2) X (5.98 x 1024 kg) X (60 kg) / (6.38 x 106 m)²

F = 587.95 N 

Conclusion:

This article explains the definition, the terms involved with it, and the dimensional importance of the gravitational constant. As per the law of universal gravitation, the gravitational force of the universe is considered, moving past the Earth’s gravitational force. Thus, it talks about the universality of gravity instead of only the Earth’s gravity. Newton has secured a place in the Gravity Hall of Fame because he discovered universal gravity and not just the gravitational force present on Earth.