To solve the problem, we will use the universal law of gravitation, which states that the gravitational force \( F \) between two masses \( m_1 \) and \( m_2 \) separated by a distance \( r \) is given by the formula:
\[
F = \frac{G \cdot m_1 \cdot m_2}{r^2}
\]
Where:
- \( G \) is the gravitational constant,
- \( m_1 \) and \( m_2 \) are the masses of the two spheres,
- \( r \) is the distance between the centers of the two spheres.
### Step 1: Identify the given values
- Mass of sphere 1, \( m_1 = 38 \, \text{kg} \)
- Mass of sphere 2, \( m_2 = 15 \, \text{kg} \)
- Distance between the centers, \( r = 20 \, \text{cm} = 0.20 \, \text{m} \)
- Force of attraction, \( F = 0.1 \, \text{mg} = 0.1 \times 10^{-6} \, \text{kg} \cdot g \)
To convert milligrams to Newtons, we use \( g \approx 9.81 \, \text{m/s}^2 \):
\[
F = 0.1 \times 10^{-6} \, \text{kg} \cdot 9.81 \, \text{m/s}^2 = 0.1 \times 9.81 \times 10^{-6} \, \text{N} = 9.81 \times 10^{-7} \, \text{N}
\]
### Step 2: Rearrange the formula to solve for \( G \)
From the gravitational force formula, we can rearrange it to find \( G \):
\[
G = \frac{F \cdot r^2}{m_1 \cdot m_2}
\]
### Step 3: Substitute the values into the equation
Now we substitute the known values into the equation:
\[
G = \frac{(9.81 \times 10^{-7} \, \text{N}) \cdot (0.20 \, \text{m})^2}{(38 \, \text{kg}) \cdot (15 \, \text{kg})}
\]
Calculating \( r^2 \):
\[
r^2 = (0.20)^2 = 0.04 \, \text{m}^2
\]
Now substituting \( r^2 \) into the equation:
\[
G = \frac{(9.81 \times 10^{-7}) \cdot (0.04)}{(38 \cdot 15)}
\]
Calculating the denominator:
\[
38 \cdot 15 = 570 \, \text{kg}^2
\]
### Step 4: Calculate \( G \)
Now we can calculate \( G \):
\[
G = \frac{(9.81 \times 10^{-7}) \cdot (0.04)}{570}
\]
\[
G = \frac{3.924 \times 10^{-8}}{570}
\]
\[
G \approx 6.88 \times 10^{-11} \, \text{N m}^2/\text{kg}^2
\]
### Final Answer
The gravitational constant \( G \) is approximately:
\[
G \approx 6.88 \times 10^{-11} \, \text{N m}^2/\text{kg}^2
\]
