Spheres of the same material and same radius `r` are touching each other. Show that gravitational force between them is directly proportional to `r^(4)`.

Explanation

To show that the gravitational force between two spheres of the same material and radius \( r \) is directly proportional to \( r^4 \), we can follow these steps:

### Step 1: Calculate the mass of each sphere
The mass \( m \) of a sphere can be calculated using the formula:
\[ m = \frac{4}{3} \pi r^3 \rho \]
where \( r \) is the radius of the sphere and \( \rho \) is the density of the material.

Since both spheres are made of the same material and have the same radius \( r \), the mass of each sphere \( m_1 \) and \( m_2 \) is:
\[ m_1 = m_2 = \frac{4}{3} \pi r^3 \rho \]

### Step 2: Write the formula for gravitational force
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:
\[ F = \frac{G m_1 m_2}{d^2} \]
where \( G \) is the gravitational constant.

### Step 3: Determine the distance between the centers of the spheres
Since the spheres are touching each other, the distance \( d \) between their centers is equal to \( 2r \) (the sum of their radii).

### Step 4: Substitute the values into the gravitational force formula
Substituting \( m_1 \) and \( m_2 \) into the gravitational force equation, we have:
\[ F = \frac{G \left(\frac{4}{3} \pi r^3 \rho\right) \left(\frac{4}{3} \pi r^3 \rho\right)}{(2r)^2} \]

### Step 5: Simplify the equation
Calculating \( (2r)^2 \):
\[ (2r)^2 = 4r^2 \]
Now substituting this back into the force equation:
\[ F = \frac{G \left(\frac{4}{3} \pi r^3 \rho\right)^2}{4r^2} \]
\[ F = \frac{G \cdot \frac{16}{9} \pi^2 r^6 \rho^2}{4r^2} \]
\[ F = \frac{4G \pi^2 \rho^2 r^6}{9r^2} \]
\[ F = \frac{4G \pi^2 \rho^2}{9} r^4 \]

### Step 6: Conclude the relationship
From the final expression, we can see that:
\[ F \propto r^4 \]
This shows that the gravitational force between the two spheres is directly proportional to \( r^4 \).