The mass of a ball if 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`. Find the masses of the two balls.

Options

A. 0.5,2
B. 50,20
C. 2,5
D. 5,20

Answer

To solve the problem, we will 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 \( m \) and the mass of the second ball be \( 4m \) (since one ball is four times the mass of the other).

### Step 2: Substitute into the gravitational force formula
The gravitational force can be expressed as:

\[ F = \frac{G \cdot m \cdot (4m)}{r^2} \]

### Step 3: Simplify the equation
This simplifies to:

\[ F = \frac{4G \cdot m^2}{r^2} \]

### Step 4: Rearrange to find \( m \)
We can rearrange this equation to solve for \( m \):

\[ m^2 = \frac{F \cdot r^2}{4G} \]

### Step 5: Substitute the known values
We know:
- \( F = 6.67 \times 10^{-7} \, \text{N} \)
- \( r = 10 \, \text{cm} = 0.1 \, \text{m} \)
- \( G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \)

Now substituting these values into the equation:

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

### Step 6: Calculate \( m^2 \)
Calculating \( (0.1)^2 \):

\[ (0.1)^2 = 0.01 \]

Now substituting:

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

Calculating the numerator:

\[ 6.67 \times 10^{-7} \cdot 0.01 = 6.67 \times 10^{-9} \]

Calculating the denominator:

\[ 4 \cdot (6.67 \times 10^{-11}) = 2.668 \times 10^{-10} \]

Now substituting these into the equation:

\[ m^2 = \frac{6.67 \times 10^{-9}}{2.668 \times 10^{-10}} \approx 25 \]

### Step 7: Find \( m \)
Taking the square root:

\[ m = \sqrt{25} = 5 \, \text{kg} \]

### Step 8: Find the mass of the second ball
Since the mass of the second ball is \( 4m \):

\[ 4m = 4 \cdot 5 = 20 \, \text{kg} \]

### Final Answer
The masses of the two balls are:
- First ball: \( 5 \, \text{kg} \)
- Second ball: \( 20 \, \text{kg} \)

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