Two metal spheres of same material and each of radius r are in contact with each other. The gravitational force of attraction between the spheres is proportional to

Options

A. `d^(2)r^(6)`
B. `d^(2)r^(4)`
C. `(d^(2))/(r^(4))`
D. `(r^(4))/(d^(2))`

Answer

To solve the problem of finding the gravitational force of attraction between two metal spheres of the same material and radius \( r \), we will follow these steps:

### Step 1: Determine the Mass of Each Sphere
The mass of a sphere can be expressed in terms of its volume and density. The volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
Let the density of the material be \( \rho \). Therefore, the mass \( m \) of each sphere can be calculated as:
\[ m = \text{density} \times \text{volume} = \rho \times \frac{4}{3} \pi r^3 \]

### Step 2: Write the Gravitational Force Formula
The gravitational force \( F \) between two masses \( m_1 \) and \( m_2 \) separated by a distance \( d \) is given by Newton's law of gravitation:
\[ F = G \frac{m_1 m_2}{d^2} \]
In this case, since both spheres are of the same mass \( m \), we can write:
\[ F = G \frac{m^2}{d^2} \]

### Step 3: Determine the Distance Between the Centers of the Spheres
Since the spheres are in contact with each other, the distance \( d \) between their centers is equal to twice the radius \( r \):
\[ d = 2r \]

### Step 4: Substitute the Mass and Distance into the Gravitational Force Formula
Substituting the mass \( m \) from Step 1 and the distance \( d \) from Step 3 into the gravitational force formula:
\[ F = G \frac{(\rho \frac{4}{3} \pi r^3)^2}{(2r)^2} \]

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

### Conclusion
From the final expression, we can see that the gravitational force \( F \) is proportional to \( r^4 \):
\[ F \propto r^4 \]

### Final Answer
The gravitational force of attraction between the spheres is proportional to \( r^4 \).

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