Two uniform solid spheres of same materical and same radius are touching each other density is `'rho'` then find out gravitational force between them .

Answer

To find the gravitational force between two uniform solid spheres of the same material and radius that are touching each other, we can follow these steps:

### Step 1: Understand the Setup
We have two uniform solid spheres, each with radius \( r \) and density \( \rho \). They are touching each other.

### Step 2: Determine the Mass of Each Sphere
The mass \( m \) of each sphere can be calculated using the formula for mass:
\[ m = \text{density} \times \text{volume} \]
The volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
Thus, the mass of each sphere is:
\[ m = \rho \times \frac{4}{3} \pi r^3 \]

### Step 3: Calculate the Distance Between the Centers
Since the spheres are touching each other, the distance \( d \) between their centers is:
\[ d = 2r \]

### Step 4: Apply Newton's Law of Gravitation
According to Newton's law of gravitation, the gravitational force \( F \) between two masses \( m_1 \) and \( m_2 \) separated by a distance \( d \) is given by:
\[ F = G \frac{m_1 m_2}{d^2} \]
In our case, both spheres have the same mass \( m \), so we can write:
\[ F = G \frac{m \cdot m}{(2r)^2} \]

### Step 5: Substitute the Mass and Distance
Substituting the mass \( m \) and the distance \( d \):
\[ F = G \frac{\left(\rho \frac{4}{3} \pi r^3\right) \left(\rho \frac{4}{3} \pi r^3\right)}{(2r)^2} \]
This simplifies to:
\[ F = G \frac{\left(\rho^2 \left(\frac{4}{3} \pi r^3\right)^2\right)}{4r^2} \]

### Step 6: Simplify the Expression
Now, we simplify the expression:
\[ F = G \frac{\rho^2 \left(\frac{16}{9} \pi^2 r^6\right)}{4r^2} \]
\[ F = G \frac{16 \rho^2 \pi^2 r^6}{36 r^2} \]
\[ F = \frac{4}{9} G \pi^2 \rho^2 r^4 \]

### Final Answer
Thus, the gravitational force \( F \) between the two spheres is given by:
\[ F = \frac{4}{9} G \pi^2 \rho^2 r^4 \]