The formula above is Newton's law of universal gratitation, where F is the attractive force, `gamma` is the gravitational constnat, `m_(1)and m_(2)` are the masses of the particles, and r is the distnace between their centres of mass. Which of the following givers r in terms of `F, gamma , m_(1),and m _(2)`?

Options

A. `r = sqrt(((m _(1))(m_(2)))/(gamma (m_(2))))`
B. `r=sqrt((F(m _(2)))/(gamma (m _(1))))`
C. `r=sqrt((gamma(m _(2)))/(F (m _(1))))`
D. `r=sqrt((gamma(m_(1))(m_(2)))/(F))`

Explanation

To solve for \( r \) in terms of \( F, \gamma, m_1, \) and \( m_2 \) using Newton's law of universal gravitation, we start with the formula:

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

### Step 1: Rearranging the equation
We want to isolate \( r^2 \) on one side of the equation. To do this, we can multiply both sides by \( r^2 \) and then divide both sides by \( F \):

\[ F \cdot r^2 = \gamma \cdot m_1 \cdot m_2 \]

### Step 2: Isolating \( r^2 \)
Next, we divide both sides by \( F \) to isolate \( r^2 \):

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

### Step 3: Taking the square root
To find \( r \), we take the square root of both sides:

\[ r = \sqrt{\frac{\gamma \cdot m_1 \cdot m_2}{F}} \]

### Conclusion
Thus, the expression for \( r \) in terms of \( F, \gamma, m_1, \) and \( m_2 \) is:

\[ r = \sqrt{\frac{\gamma \cdot m_1 \cdot m_2}{F}} \]

### Final Answer
The correct option is option 4:

\[ r = \sqrt{\frac{\gamma \cdot m_1 \cdot m_2}{F}} \]

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