Two identical spheres of gold are in contact with each other. The gravitational force of attraction between them is

Options

A. directly proportional to the square of their radius
B. directly proportional to the cube of their radius
C. directly proportional to the fourth power of their radius
D. inversely proportional to the square of their radius.

Answer

To find the gravitational force of attraction between two identical spheres of gold that are in contact with each other, we can follow these steps:

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

### Step 2: Define the Masses
Since the spheres are identical, we can denote their masses as \( m_1 = m \) and \( m_2 = m \). Thus, the formula for gravitational force becomes:
\[ F = \frac{G \cdot m \cdot m}{r^2} = \frac{G \cdot m^2}{r^2} \]

### Step 3: Calculate the Mass of Each Sphere
To find the mass \( m \) of each sphere, we can use the formula for density:
\[ \rho = \frac{m}{V} \]
For a sphere, the volume \( V \) is given by:
\[ V = \frac{4}{3} \pi r^3 \]
Thus, the mass can be expressed as:
\[ m = \rho \cdot V = \rho \cdot \frac{4}{3} \pi r^3 \]

### Step 4: Substitute Mass into the Gravitational Force Equation
Substituting the expression for mass \( m \) into the gravitational force equation:
\[ F = \frac{G \cdot \left(\rho \cdot \frac{4}{3} \pi r^3\right)^2}{r^2} \]
This simplifies to:
\[ F = \frac{G \cdot \rho^2 \cdot \left(\frac{4}{3} \pi r^3\right)^2}{r^2} \]

### Step 5: Simplify the Expression
Now we can simplify the expression:
\[ F = \frac{G \cdot \rho^2 \cdot \frac{16}{9} \pi^2 r^6}{r^2} = G \cdot \rho^2 \cdot \frac{16}{9} \pi^2 r^4 \]
Thus, we have:
\[ F = \frac{16}{9} G \cdot \rho^2 \cdot \pi^2 r^4 \]

### Step 6: Conclusion
From the final expression, we can see that the gravitational force \( F \) is directly proportional to \( r^4 \). Therefore, the correct answer is that the gravitational force of attraction between the two identical spheres is directly proportional to the fourth power of their radius.