Two identical spheres are placed in contact with each other. The force of gravitation between the spheres will be proportional to ( R = radius of each sphere)

Options

A. `R`
B. `R^(2)`
C. `R^(4)`
D. None of these

Answer

To solve the problem, we need to determine how the gravitational force between two identical spheres is related to their radius \( R \).

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have two identical spheres in contact with each other.
- The distance between their centers is equal to the sum of their radii, which is \( 2R \) (since both spheres have radius \( R \)).

2. **Using the Gravitational Force Formula**:
- The gravitational force \( F \) between two masses \( m_1 \) and \( m_2 \) separated by a distance \( r \) is given by Newton's law of gravitation:
\[ F = \frac{G m_1 m_2}{r^2} \]
- In our case, both spheres are identical, so we can denote their mass as \( m \). The distance \( r \) between the centers of the spheres is \( 2R \).

3. **Substituting the Values**:
- Substituting \( m_1 = m_2 = m \) and \( r = 2R \) into the gravitational force formula:
\[ F = \frac{G m^2}{(2R)^2} \]
- This simplifies to:
\[ F = \frac{G m^2}{4R^2} \]

4. **Expressing Mass in Terms of Radius**:
- The mass \( m \) of a sphere can be expressed in terms of its density \( \rho \) and volume:
\[ m = \rho \cdot V = \rho \cdot \left(\frac{4}{3} \pi R^3\right) \]
- Therefore, substituting for \( m \):
\[ F = \frac{G (\rho \cdot \frac{4}{3} \pi R^3)^2}{4R^2} \]

5. **Simplifying the Expression**:
- Expanding the mass term:
\[ F = \frac{G \cdot \rho^2 \cdot \left(\frac{16}{9} \pi^2 R^6\right)}{4R^2} \]
- Simplifying further:
\[ F = \frac{G \cdot \rho^2 \cdot 4 \pi^2 R^6}{9R^2} \]
- This reduces to:
\[ F = \frac{G \cdot \rho^2 \cdot 4 \pi^2}{9} R^4 \]

6. **Conclusion**:
- From the final expression, we see that the gravitational force \( F \) is proportional to \( R^4 \):
\[ F \propto R^4 \]

### Final Answer:
The force of gravitation between the spheres will be proportional to \( R^4 \).