Derive the Universal law of Gravitation F = GMm / d^2.

The Universal Law of Gravitation was formulated by Sir Isaac Newton in 1687. It states that every mass attracts every other mass in the universe with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. Here's how the law is derived:

### 1. **Proportionality of Force to Masses**
Consider two objects with masses \( M \) and \( m \), separated by a distance \( d \). The gravitational force \( F \) between them is directly proportional to the product of their masses:

\[
F \propto M \times m
\]

### 2. **Inversely Proportional to the Square of the Distance**
The force \( F \) is also inversely proportional to the square of the distance \( d \) between the two masses:

\[
F \propto \frac{1}{{d^2}}
\]

### 3. **Combining the Proportionalities**
By combining the two proportionalities, we get:

\[
F \propto \frac{M \times m}{{d^2}}
\]

### 4. **Introducing the Gravitational Constant \( G \)**
To convert this proportionality into an equation, we introduce a constant of proportionality, known as the gravitational constant \( G \):

\[
F = G \frac{{M \times m}}{{d^2}}
\]

### 5. **Final Form of the Universal Law of Gravitation**
The equation can now be written as:

\[
F = \frac{{G M m}}{{d^2}}
\]

Where:
- \( F \) is the gravitational force between the two objects,
- \( G \) is the gravitational constant (\( 6.674 \times 10^{-11} \, \text{Nm}^2\text{kg}^{-2} \)),
- \( M \) and \( m \) are the masses of the two objects,
- \( d \) is the distance between the centers of the two masses.

This equation describes the gravitational force between any two masses in the universe.