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. R
B. `R^(2)`
C. `R^(-4)`
D. `R^(4)`

Answer

To solve the problem of finding the gravitational force between two balls placed in contact, we can follow these steps:

### Step 1: Understand the setup
We have two balls, each with a radius \( R \), equal mass \( m \), and equal density \( \rho \). They are in contact, meaning the distance between their centers is \( 2R \).

### Step 2: Write the formula for gravitational force
The gravitational force \( F \) between two masses is given by Newton's law of gravitation:
\[ F = \frac{G m_1 m_2}{r^2} \]
where \( G \) is the gravitational constant, \( m_1 \) and \( m_2 \) are the masses of the two balls, and \( r \) is the distance between their centers.

### Step 3: Substitute the values
Since both balls have equal mass \( m \), we can substitute \( m_1 = m_2 = m \) and \( r = 2R \):
\[ F = \frac{G m^2}{(2R)^2} = \frac{G m^2}{4R^2} \]

### Step 4: Express mass in terms of density and volume
The mass \( m \) of each ball can be expressed in terms of its 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, substituting for \( m \):
\[ m = \rho \left(\frac{4}{3} \pi R^3\right) \]

### Step 5: Substitute mass back into the force equation
Now, substituting \( m \) into the gravitational force equation:
\[ F = \frac{G \left(\rho \left(\frac{4}{3} \pi R^3\right)\right)^2}{4R^2} \]
\[ = \frac{G \left(\rho^2 \left(\frac{16}{9} \pi^2 R^6\right)\right)}{4R^2} \]
\[ = \frac{G \cdot 16 \pi^2 \rho^2 R^6}{36 R^2} \]

### Step 6: Simplify the expression
Now, simplify the expression:
\[ F = \frac{16 G \pi^2 \rho^2 R^4}{36} \]
This shows that the force \( F \) is proportional to \( R^4 \).

### Conclusion
Thus, the gravitational force between the two balls is proportional to \( R^4 \).

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

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