The mass of a ball is four times the mass of another ball. When these balls are separated by a distance of `10 cm`, the gravitational force between them is `6.67xx10^(-7) N`. The masses of the two balls are in `kg`.

Options

A. `10,20`
B. `5,20`
C. `20,30`
D. `20,40`

Explanation

To find the masses of the two balls, we can use the formula for gravitational force:

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

Where:
- \( F \) is the gravitational force between the two masses,
- \( G \) is the gravitational constant (\( 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \)),
- \( m_1 \) and \( m_2 \) are the masses of the two balls,
- \( r \) is the distance between the centers of the two masses.

### Step 1: Define the masses
Let the mass of the first ball be \( m_1 \) and the mass of the second ball be \( m_2 = 4m_1 \) (since the mass of the second ball is four times that of the first).

### Step 2: Convert distance to meters
The distance \( r \) is given as \( 10 \, \text{cm} \). We need to convert this to meters:
\[ r = 10 \, \text{cm} = 0.1 \, \text{m} \]

### Step 3: Substitute values into the gravitational force equation
We know the gravitational force \( F = 6.67 \times 10^{-7} \, \text{N} \). Substituting the values into the gravitational force equation:

\[ 6.67 \times 10^{-7} = \frac{6.67 \times 10^{-11} \cdot m_1 \cdot (4m_1)}{(0.1)^2} \]

### Step 4: Simplify the equation
This simplifies to:

\[ 6.67 \times 10^{-7} = \frac{6.67 \times 10^{-11} \cdot 4m_1^2}{0.01} \]

\[ 6.67 \times 10^{-7} = 6.67 \times 10^{-11} \cdot 4m_1^2 \cdot 100 \]

\[ 6.67 \times 10^{-7} = 6.67 \times 10^{-9} \cdot m_1^2 \]

### Step 5: Solve for \( m_1^2 \)
Dividing both sides by \( 6.67 \times 10^{-9} \):

\[ m_1^2 = \frac{6.67 \times 10^{-7}}{6.67 \times 10^{-9}} = 100 \]

### Step 6: Calculate \( m_1 \)
Taking the square root of both sides:

\[ m_1 = \sqrt{100} = 10 \, \text{kg} \]

### Step 7: Calculate \( m_2 \)
Now, substitute back to find \( m_2 \):

\[ m_2 = 4m_1 = 4 \times 10 = 40 \, \text{kg} \]

### Final Answer
The masses of the two balls are:
- \( m_1 = 10 \, \text{kg} \)
- \( m_2 = 40 \, \text{kg} \)