Two masses A and B are separated by a distance of 1m. To increase the force of attraction by 2 times, the distance between them should be ___________m.

Options

A. ` 0.5`
B. ` (2)/(3)`
C. ` 2`
D. ` (1)/(sqrt2)`

Explanation

To solve the problem of how to increase the gravitational force of attraction between two masses A and B by 2 times, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Gravitational Force Formula**:
The gravitational force (F) between two masses A and B separated by a distance r is given by the formula:
\[ F = \frac{G \cdot m_A \cdot m_B}{r^2} \]
where G is the gravitational constant.

2. **Initial Condition**:
We know that the initial distance (r) between the two masses A and B is 1 meter. So, we can express the initial force (F) as:
\[ F = \frac{G \cdot m_A \cdot m_B}{1^2} = G \cdot m_A \cdot m_B \]

3. **Desired Condition**:
We want to increase the force of attraction to 2 times the initial force. Therefore, we set up the equation:
\[ 2F = \frac{G \cdot m_A \cdot m_B}{r'^2} \]
where \( r' \) is the new distance we need to find.

4. **Set Up the Equation**:
From the initial force, we know:
\[ 2 \cdot (G \cdot m_A \cdot m_B) = \frac{G \cdot m_A \cdot m_B}{r'^2} \]

5. **Cancel Common Terms**:
We can cancel \( G \cdot m_A \cdot m_B \) from both sides of the equation:
\[ 2 = \frac{1}{r'^2} \]

6. **Rearranging the Equation**:
Rearranging gives us:
\[ r'^2 = \frac{1}{2} \]

7. **Taking the Square Root**:
To find \( r' \), we take the square root of both sides:
\[ r' = \frac{1}{\sqrt{2}} \approx 0.707 \text{ m} \]

### Final Answer:
To increase the force of attraction by 2 times, the distance between the masses should be approximately **0.707 m**.

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