Two identical solid spheres each of radius 7 are in contact with each other. If the gravitational attraction between them is F, then which of the following relations is correct?

Options

A. ` F prop (1)/ ( r ) `
B. ` F prop (1)/(r^(2))`
C. ` F prop (1)/(r^(3))`
D. ` F prop (1)/(r^(4))`

Answer

To solve the problem, we need to analyze the gravitational attraction between two identical solid spheres that are in contact with each other. Here are the steps to derive the correct relation:

### Step-by-Step Solution:

1. **Understanding the Problem**: We have two identical solid spheres, each with a radius of 7 units. They are in contact with each other. We need to determine how the gravitational force \( F \) between them relates to the distance \( R \) between their centers.

2. **Identifying the Distance**: Since the spheres are in contact, the distance \( R \) between their centers is equal to the sum of their radii. Therefore, if each sphere has a radius of 7 units, the distance \( R \) is:
\[ R = 7 + 7 = 14 \text{ units} \]

3. **Gravitational Force Formula**: The formula for gravitational force \( F \) between two masses \( m_1 \) and \( m_2 \) separated by a distance \( R \) is given by:
\[ F = \frac{G \cdot m_1 \cdot m_2}{R^2} \]
where \( G \) is the gravitational constant.

4. **Identifying Masses**: Since the spheres are identical, we can denote their masses as \( m \). Thus, the formula for the gravitational force becomes:
\[ F = \frac{G \cdot m \cdot m}{R^2} = \frac{G \cdot m^2}{R^2} \]

5. **Analyzing the Relationship**: In this equation, \( G \) and \( m^2 \) are constants (they do not change). The only variable that can change is \( R \). Therefore, we can express the gravitational force \( F \) in terms of \( R \):
\[ F \propto \frac{1}{R^2} \]
This indicates that the gravitational force is inversely proportional to the square of the distance between the centers of the two spheres.

6. **Conclusion**: The correct relation for the gravitational attraction \( F \) between the two spheres is:
\[ F \text{ is directly proportional to } \frac{1}{R^2} \]

### Final Answer:
The correct relation is:
\[ F \propto \frac{1}{R^2} \]