Two particles each of mass 1kg are seperated by a distance of 1m in a gravitational field. The work done to increase the separation Between them to 2m is

Options

A. G
B. G/2
C. G/4
D. 2G

Answer

To find the work done to increase the separation between two particles of mass 1 kg each from 1 meter to 2 meters, we can follow these steps:

### Step 1: Understand the formula for gravitational potential energy
The gravitational potential energy (U) between two masses (m1 and m2) separated by a distance (r) is given by the formula:
\[ U = -\frac{G \cdot m_1 \cdot m_2}{r} \]
where \( G \) is the gravitational constant.

### Step 2: Calculate the initial gravitational potential energy (U_i)
For the initial separation (r_i = 1 m):
- m1 = 1 kg
- m2 = 1 kg
- r_i = 1 m

Substituting these values into the formula:
\[ U_i = -\frac{G \cdot 1 \cdot 1}{1} = -G \]

### Step 3: Calculate the final gravitational potential energy (U_f)
For the final separation (r_f = 2 m):
- r_f = 2 m

Substituting this value into the formula:
\[ U_f = -\frac{G \cdot 1 \cdot 1}{2} = -\frac{G}{2} \]

### Step 4: Calculate the work done (W)
The work done to increase the separation is given by the change in gravitational potential energy:
\[ W = U_f - U_i \]

Substituting the values we calculated:
\[ W = \left(-\frac{G}{2}\right) - \left(-G\right) \]
\[ W = -\frac{G}{2} + G \]
\[ W = G - \frac{G}{2} = \frac{G}{2} \]

### Conclusion
Thus, the work done to increase the separation between the two particles from 1 meter to 2 meters is:
\[ W = \frac{G}{2} \]