Two identical thin ring each of radius `R` are co-axially placed at a distance `R`. If the ring have a uniform mass distribution and each has mass `m_(1)` and `m_(2)` respectively, then the work done in moving a mass `m` from the centre of one ring to that of the other is :

Options

A. `(Gm)/(m_(2)R)(sqrt(2)+1)m`
B. `(Gm(m_(1)-m_(2)))/(sqrt(2)R)(sqrt(2)-1)`
C. `(Gmsqrt(2))/(R)(m_(1)+m_(2))`
D. zero

Explanation

To find the work done in moving a mass \( m \) from the center of one ring to the center of the other ring, we can follow these steps:

### Step 1: Understand the Setup
We have two identical thin rings of radius \( R \) placed coaxially at a distance \( R \) apart. Each ring has a mass \( m_1 \) and \( m_2 \) respectively. We need to calculate the gravitational potential at the center of each ring.

### Step 2: Calculate the Gravitational Potential at Point A (Center of Ring 1)
The gravitational potential \( V_A \) at the center of Ring 1 due to Ring 2 can be calculated using the formula:
\[ V_A = -\frac{G m_2}{R} \]
where \( G \) is the gravitational constant.

### Step 3: Calculate the Gravitational Potential at Point B (Center of Ring 2)
Similarly, the gravitational potential \( V_B \) at the center of Ring 2 due to Ring 1 can be calculated using the distance \( \sqrt{2}R \) (the distance from the center of Ring 1 to the center of Ring 2):
\[ V_B = -\frac{G m_1}{\sqrt{2}R} \]

### Step 4: Calculate the Change in Potential Energy
The work done \( W \) in moving the mass \( m \) from the center of Ring 1 to the center of Ring 2 is equal to the change in gravitational potential energy:
\[ W = m (V_B - V_A) \]
Substituting the values of \( V_A \) and \( V_B \):
\[ W = m \left(-\frac{G m_1}{\sqrt{2}R} - \left(-\frac{G m_2}{R}\right)\right) \]
This simplifies to:
\[ W = m \left(-\frac{G m_1}{\sqrt{2}R} + \frac{G m_2}{R}\right) \]

### Step 5: Simplify the Expression
Factoring out \( G \) and \( m \):
\[ W = \frac{G m}{R} \left(-\frac{m_1}{\sqrt{2}} + m_2\right) \]

### Final Result
Thus, the work done in moving the mass \( m \) from the center of one ring to the center of the other is:
\[ W = \frac{G m}{R} \left(m_2 - \frac{m_1}{\sqrt{2}}\right) \]