When two identical spheres are kept in contact, the gravitational force between them is F. If two spheres of same material but with twice the radius are radius are kept in contact, then the gravitational force between them would be

Options

A. F/4
B. 4F
C. F/16
D. 16F

Answer

To solve the problem, we need to understand how the gravitational force between two spheres changes when their radius is doubled while keeping them made of the same material.

### Step-by-step Solution:

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

2. **Identifying the Mass of the Spheres**:
The mass \( m \) of a sphere is related to its volume and density. The volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
If the radius of the spheres is doubled (let's denote the new radius as \( R = 2r \)), the new volume \( V' \) becomes:
\[ V' = \frac{4}{3} \pi (2r)^3 = \frac{4}{3} \pi (8r^3) = 8 \cdot \frac{4}{3} \pi r^3 = 8V \]
Since the spheres are made of the same material, their mass will also double:
\[ m' = 8m \]

3. **Calculating the New Distance Between the Centers**:
When two spheres are in contact, the distance between their centers is equal to the sum of their radii. For the original spheres, the distance was \( r + r = 2r \). For the new spheres with radius \( 2r \), the distance becomes:
\[ R + R = 2R = 2(2r) = 4r \]

4. **Substituting into the Gravitational Force Formula**:
Now, we can substitute the new mass and distance into the gravitational force formula. The new gravitational force \( F' \) is given by:
\[ F' = \frac{G m' m'}{(4r)^2} = \frac{G (8m)(8m)}{(4r)^2} \]
Simplifying this:
\[ F' = \frac{G \cdot 64m^2}{16r^2} = \frac{64G m^2}{16r^2} = 4 \cdot \frac{G m^2}{r^2} = 4F \]

5. **Conclusion**:
Therefore, the gravitational force between the two larger spheres when they are kept in contact is \( 4F \).

### Final Answer:
The gravitational force between the two spheres with twice the radius is \( 4F \).

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