Two balls, each of radius R, equal mass and density are placed in contact, then the force of gravitation between them is proportional to

Options

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

Explanation

To determine the gravitational force between two balls of equal mass and density placed in contact, we can follow these steps:

### Step 1: Understand 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:
\[ F = G \frac{m_1 m_2}{r^2} \]
where \( G \) is the gravitational constant.

### Step 2: Identify the Mass of Each Ball
Since the balls have equal mass and density, we can denote the mass of each ball as \( m \). The mass can be expressed in terms of density \( \rho \) and volume \( V \):
\[ m = \rho V \]
The volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi R^3 \]
Thus, the mass of each ball becomes:
\[ m = \rho \left(\frac{4}{3} \pi R^3\right) \]

### Step 3: Calculate the Distance Between the Centers
When the two balls are in contact, the distance \( r \) between their centers is equal to the sum of their radii. Since both balls have radius \( R \):
\[ r = R + R = 2R \]

### Step 4: Substitute Values into the Gravitational Force Equation
Now, substituting \( m \) and \( r \) into the gravitational force formula:
\[ F = G \frac{m \cdot m}{(2R)^2} \]
Substituting \( m \):
\[ F = G \frac{\left(\rho \frac{4}{3} \pi R^3\right) \cdot \left(\rho \frac{4}{3} \pi R^3\right)}{(2R)^2} \]

### Step 5: Simplify the Expression
Simplifying the expression:
\[ F = G \frac{\left(\rho^2 \left(\frac{4}{3} \pi R^3\right)^2\right)}{4R^2} \]
\[ F = G \frac{\rho^2 \left(\frac{16}{9} \pi^2 R^6\right)}{4R^2} \]
\[ F = G \frac{4 \rho^2 \pi^2 R^6}{9R^2} \]
\[ F = \frac{4G \rho^2 \pi^2 R^4}{9} \]

### Conclusion
The force of gravitation between the two balls is proportional to \( R^4 \) (as well as \( \rho^2 \) and constants).

### Final Answer
Thus, the force of gravitation between the two balls is proportional to \( R^4 \).

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