If the distance between two bodies is dobled and mass of any one of the bodies is quadrupled then the percentage increase in their force of attraction will be.

Options

A. `5%`
B. `10%`
C. `25%`
D. Zero

Answer

To solve the problem, we will use the formula for gravitational force of attraction between two bodies, which is given by:

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

where:
- \( F \) is the gravitational force,
- \( G \) is the gravitational constant,
- \( m_1 \) and \( m_2 \) are the masses of the two bodies,
- \( r \) is the distance between the two bodies.

### Step 1: Write the initial force of attraction
Let the initial masses of the two bodies be \( m_1 \) and \( m_2 \), and the initial distance between them be \( r \). The initial force of attraction \( F_1 \) is:

\[ F_1 = \frac{G \cdot m_1 \cdot m_2}{r^2} \]

### Step 2: Modify the masses and distance
According to the problem:
- The mass of one body is quadrupled, so we can assume \( m_1 \) becomes \( 4m_1 \).
- The distance between the two bodies is doubled, so \( r \) becomes \( 2r \).

### Step 3: Write the new force of attraction
Now, the new force of attraction \( F_2 \) can be calculated as follows:

\[ F_2 = \frac{G \cdot (4m_1) \cdot m_2}{(2r)^2} \]

### Step 4: Simplify the expression for \( F_2 \)
Now, simplify the expression for \( F_2 \):

\[ F_2 = \frac{G \cdot 4m_1 \cdot m_2}{4r^2} \]

Notice that \( (2r)^2 = 4r^2 \).

### Step 5: Cancel out the common terms
Now we can simplify \( F_2 \):

\[ F_2 = \frac{G \cdot m_1 \cdot m_2}{r^2} \]

### Step 6: Compare the initial and new forces
From the above simplifications, we can see that:

\[ F_2 = F_1 \]

This means that the new force of attraction \( F_2 \) is equal to the initial force of attraction \( F_1 \).

### Step 7: Calculate the percentage change
The percentage change in force can be calculated using the formula:

\[ \text{Percentage Change} = \left( \frac{F_2 - F_1}{F_1} \right) \times 100\% \]

Substituting \( F_2 = F_1 \):

\[ \text{Percentage Change} = \left( \frac{F_1 - F_1}{F_1} \right) \times 100\% = 0\% \]

### Conclusion
Thus, the percentage increase in their force of attraction is **0%**.

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