Two identical spheres each of mass M and radius R are separated by a distance 3R. The force of attraction between them is proportional to

Options

A. `(1)/(R^(2))`
B. `R^(4)`
C. `R^(2)`
D. `(1)/(R^(4))`

Explanation

To solve the problem, we will use Newton's law of universal gravitation, which states that the gravitational force \( F \) between two masses \( M_1 \) and \( M_2 \) separated by a distance \( r \) is given by the formula:

\[ F = G \frac{M_1 M_2}{r^2} \]

where \( G \) is the gravitational constant.

### Step-by-Step Solution:

1. **Identify the Masses and Distance**:
- We have two identical spheres, each with mass \( M \).
- The distance between the centers of the two spheres is given as \( 3R \).

2. **Apply the Gravitational Force Formula**:
- According to the formula, the gravitational force \( F \) between the two spheres can be expressed as:
\[ F = G \frac{M \cdot M}{(3R)^2} \]

3. **Simplify the Expression**:
- Substitute the masses into the equation:
\[ F = G \frac{M^2}{(3R)^2} \]
- Calculate \( (3R)^2 \):
\[ (3R)^2 = 9R^2 \]
- Therefore, the force becomes:
\[ F = G \frac{M^2}{9R^2} \]

4. **Determine the Proportionality**:
- From the expression \( F = G \frac{M^2}{9R^2} \), we can see that the force \( F \) is proportional to \( M^2 \) and inversely proportional to \( R^2 \).
- Thus, we can write:
\[ F \propto \frac{M^2}{R^2} \]

### Final Answer:
The force of attraction between the two spheres is proportional to \( \frac{M^2}{R^2} \).