If the distance between two masses is doubled , the gravitational attraction between them

Options

A. Is doubled
B. Becomes four times
C. Is reduced to half
D. Is reduced to a quarter

Explanation

To solve the problem of how gravitational attraction changes when the distance between two masses is doubled, we can follow these steps:

### Step 1: Understand the Gravitational Force Formula
The gravitational force \( F \) between two masses \( m_1 \) and \( m_2 \) separated by a distance \( r \) is given by Newton's law of gravitation:
\[ F = \frac{G m_1 m_2}{r^2} \]
where \( G \) is the gravitational constant.

### Step 2: Calculate the Initial Gravitational Force
Let’s denote the initial distance between the two masses as \( r \). The initial gravitational force \( F \) can be expressed as:
\[ F = \frac{G m_1 m_2}{r^2} \]

### Step 3: Consider the New Distance
If the distance between the two masses is doubled, the new distance becomes \( 2r \).

### Step 4: Calculate the New Gravitational Force
Using the new distance \( 2r \), the new gravitational force \( F' \) can be calculated as:
\[ F' = \frac{G m_1 m_2}{(2r)^2} \]
This simplifies to:
\[ F' = \frac{G m_1 m_2}{4r^2} \]

### Step 5: Relate the New Force to the Initial Force
Now, we can relate the new force \( F' \) to the initial force \( F \):
\[ F' = \frac{1}{4} \cdot \frac{G m_1 m_2}{r^2} = \frac{F}{4} \]

### Step 6: Conclusion
Thus, when the distance between the two masses is doubled, the gravitational attraction between them is reduced to one-fourth of the original force.

### Final Answer
The gravitational attraction between the two masses is reduced to one-fourth (or \( \frac{1}{4} \)) of its original value.

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