Two identical spheres of radius R, made of a material of density `rho`, are in contact with each other. If the gravitational attraction between them is F, then

Options

A. `F prop (rho^2)/(R^2)`
B. `F prop rho^2 R^4`
C. `F prop (rho^4)/(R^6)`
D. `F prop rho^4 R^4`

Answer

To solve the problem of finding the gravitational attraction \( F \) between two identical spheres of radius \( R \) and density \( \rho \), we can follow these steps:

### Step 1: Determine the mass of one sphere
The mass \( m \) of a sphere can be calculated using the formula:
\[ m = \text{Volume} \times \text{Density} \]
The volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi R^3 \]
Thus, the mass of one sphere becomes:
\[ m = \frac{4}{3} \pi R^3 \rho \]

### Step 2: Calculate the distance between the centers of the spheres
Since the two spheres are in contact, the distance \( d \) between their centers is equal to the sum of their radii:
\[ d = R + R = 2R \]

### Step 3: Apply Newton's law of gravitation
According to Newton's law of gravitation, the gravitational force \( F \) between two masses \( m_1 \) and \( m_2 \) separated by a distance \( d \) is given by:
\[ F = \frac{G m_1 m_2}{d^2} \]
For our two spheres, both have the same mass \( m \), so:
\[ F = \frac{G m^2}{(2R)^2} \]

### Step 4: Substitute the mass into the gravitational force equation
Substituting the expression for mass \( m \) into the gravitational force equation:
\[ F = \frac{G \left(\frac{4}{3} \pi R^3 \rho\right)^2}{(2R)^2} \]

### Step 5: Simplify the expression
Now, we simplify the expression:
\[ F = \frac{G \left(\frac{16}{9} \pi^2 R^6 \rho^2\right)}{4R^2} \]
This simplifies to:
\[ F = \frac{4G \pi^2 R^6 \rho^2}{9R^2} = \frac{4G \pi^2 R^4 \rho^2}{9} \]

### Step 6: Conclusion
From the final expression, we can conclude that the gravitational force \( F \) is directly proportional to \( R^4 \) and \( \rho^2 \):
\[ F \propto R^4 \rho^2 \]

### Final Answer
Thus, the gravitational attraction \( F \) between the two spheres is proportional to \( R^4 \) and \( \rho^2 \).

---