Two metal spheres of same material and radius ‘r. are in contact with each other. The gravitational force of attraction between the spheres is given by (k in a constant).

Options

A. `F = Kr^4`
B. `F = k//r^2`
C. `F = k//4r^2`
D. `F = kr^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 can follow these steps:

### Step 1: Calculate the Mass of Each Sphere
The mass \( m \) of each metal sphere can be calculated using the formula for the volume of a sphere and the density of the material. The volume \( V \) of a sphere is given by:

\[ V = \frac{4}{3} \pi r^3 \]

Since density \( \rho \) is defined as mass per unit volume, we can express the mass as:

\[ m = V \cdot \rho = \frac{4}{3} \pi r^3 \rho \]

### Step 2: Apply 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:

\[ F = \frac{G m_1 m_2}{d^2} \]

In our case, both spheres have the same mass \( m \), and they are in contact with each other, so the distance \( d \) between their centers is \( 2r \) (the radius of one sphere plus the radius of the other).

Thus, we can write:

\[ F = \frac{G m^2}{(2r)^2} \]

### Step 3: Substitute the Mass into the Force Equation
Now, substituting the expression for mass \( m \) from Step 1 into the force equation:

\[ F = \frac{G \left(\frac{4}{3} \pi r^3 \rho\right)^2}{(2r)^2} \]

### Step 4: 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:

\[ F = \frac{4G \pi^2 \rho^2 r^4}{9} \]

### Step 5: Define the Constant \( k \)
To express the force in terms of a constant \( k \), we can define:

\[ k = \frac{4G \pi^2 \rho^2}{9} \]

Thus, the final expression for the gravitational force \( F \) becomes:

\[ F = k r^4 \]

### Final Answer
The gravitational force of attraction between the two spheres is given by:

\[ F = k r^4 \]