The Gravitational Constant

The gravitational constant, also referred to as the universal gravitational constant, Cavendish gravitational constant, and the Newtonian constant of gravitation, is denoted by the letter G. Its value equals 6.674×10−11 m3⋅kg−1⋅s−2. It is a physical constant used for calculating gravitational effects in the general theory of relativity by Einstein and the law of universal gravitation by Sir Issac Newton. In the Einstein equations, we use the gravitational constant for quantifying the relationship between the energy-momentum tensor and the geometry of spacetime. While in Newton’s law, we use the gravitational constant for connecting the product of masses of two bodies and the inverse square of their distance with the gravitational force between them. 

Understanding the Gravitational Constant 

We employ the gravitational constant G for calculating the gravitational effects. It is different from g that 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 denotes the mass of the first object, m2 denotes the mass of the second object, r denotes the separation between the two masses, and G denotes the gravitational constant. 

Just like all the other constants of physics, this constant is also an empirical value, that means it has been 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. 

What is Gravitational Force? 

One must know Newton’s law of universal gravitation to understand gravitational force. As per this law, we gather that every particle present in the universe attracts other particles through a powerful force directly proportional to the masses’ product and inversely proportional to the square of the separation distance between the two objects. 

We are surrounded by gravitational force. It is the mechanism behind a ball coming back to the ground when thrown in the air. It is responsible for determining the weight of every individual and how far the ball travels towards the sky before returning to the surface. 

We can understand the gravitational force of the earth as the force it exerts on us. It equals the weight of the person when at rest or near the surface of the earth. The gravitational force varies in different astronomical bodies. If you weigh yourself on other bodies like Mars or the moon, it will be different from that on earth. 

When objects get locked with the gravitational force, their gravitational force lies at the barycenter of the system and not at the centre of either of them. To understand this principle, we can look at the functioning see-saw. When two people having different weights sit on either side of the see-saw, the person having more weight must sit closer to the balance point to equalize the mass. 

For instance, if the weight of one person is twice that of the other, he must sit at a position that is half the distance from the fulcrum. The balance point in the swing acts as the centre of the mass of the see-saw, just like the barycenter acts as the balance point of the earth and moon system. 

Every system in the universe has a fixed barycenter. It is the gravitational pull in the universe that keeps everything in its place without things crashing and colliding. 

Solving Problems with The Law of Universal Gravitation by Newton 

As per the law of universal gravitation, the gravitational force of the universe is considered, as 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.

As per his law, since gravitational force stands directly proportional to the mass of the objects, more massive objects will have a greater gravitational force towards each other. Thus, with an increase in the mass of either object, the gravitational force between them will also rise. 

If the mass of either of the two objects is doubled, the gravitational force between the objects will also get doubled. If the mass of either of the two objects is tripled, the gravitational force between the objects will also get tripled. Lastly, if the mass of both the objects gets doubled, the resultant gravitational force between them will get multiplied by 4. 

Similarly, the force is inversely proportional to the square of the separation distance between the two objects. Thus, the more the separation, the weaker the gravitational force among the objects. As we keep increasing the distance between the objects, the gravitational force will decrease. 

For instance, if we double the separation distance between the objects, the gravitational force among them gets reduced by four times. If the separation distance between the objects gets tripled, then the gravitational force among them is reduced by nine times. 

Let us solve some examples to understand the application of the gravitational constant. 

Example 1

What will be the gravitational force between earth (m = 5.98 x 1024 kg) and a 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