G and g are two values that are frequently used while discussing gravitational force. Small g(g) is the acceleration due to gravity, and big g(G) is the universal gravitational constant. Though both are represented by the small alphabet, they are totally different. In this article, we look at the relation between g and G.
Acceleration due to gravity (g)
Small g is used to represent the acceleration due to the gravity of any object. This is generally used for massive objects, as tiny objects have very little gravitational force. Small g can be defined as the rate of change in velocity due to the gravitational force. This is a type of acceleration that is due to the gravitational force only. Since this is an acceleration, it has a unit of meters per second square(m/s2).
So when we drop any object the acceleration that it has is due to the gravitational force of the earth and we can say that this acceleration is the acceleration due to the gravity of the earth. g has a value of 9.8 m/s2 for planet Earth. It varies for different objects based on the mass and size of the object.
Universal gravitational constant (G)
Big g is the universal gravitational constant. It is a constant value and has a value of 6.67408 × 10⁻¹¹ m³ kg⁻¹ s⁻². Its value does not change from object to object but is a constant and has a unit of m³ kg⁻¹ s⁻² .
Derivation for the relationship between G and g
From Newton’s formula for gravity, we know gravitational force F=Gm1⋅m2/r2 –(1)
[Where m1 is the mass of the Earth(larger object), m2 is the mass of the tiny object, and r is the radius of the larger object]
We know the formula for force F= m*a
Here while considering the case of gravity, the acceleration will be due to gravitational force so let us assume a = g.
F = m2*g [mass = m2 since the force is exerted on the tiny object]–(2)
Equating both equations we get:
m2*g = G*m1*m2/r2
g = G*m1/r2 [relation between g and G in physics] —–(3)
What is the relation between g and G on planet Earth?
We derived the relation between g and G above as:
Now by putting the values of m1 and r for the Earth, we can get the acceleration due to gravity on Earth as 9.8 m/s2. Our planet Earth is an ellipsoid, which means the radius at the equator is greater than the radius at the poles. Since the acceleration due to gravity is inversely proportional to the square of the distance, the value of g at the equator is greater than the value of g at the poles. This variation is very slight, the value of g varies from approximately 9.78 9.8 m/s2 at the Equator to approximately 9.83 9.8 m/s2 at the poles. On the other hand, the value of G is considered to be constant at all locations in the universe.
You can learn more about the variations in the value of g on earth in this article.
Relation between g and G
| SnO | Basis for Comparision | Acceleration due to gravity(g) | Universal |
| 1 | Symbol | g | G |
| 2 | Definition | It is the acceleration caused by the gravitational force of a massive object | It is a constant in the formula for gravitational force. |
| 3 | Variation in values | Depends on the radius and mass of the object exerting gravitational force | It is a constant value |
| 4 | Unit | m/s2 | m³ kg⁻¹ s⁻² or Nm2/kg2 |
| 5 | Value on planet Earth | It has a value of approx 9.8 m/s2 | It has a value of 6.67408 × 10⁻¹¹ m³ kg⁻¹ s⁻² |
