Each of the four corners of a square with edge a is occupied by a point mass m. There is a fifth mass, also m, at the center of the square. To remove the mass from the center to a point far away the work that must be done by an external agent is given by:

Options

A. `4Gm^2//a`
B. `-4Gm^2//a`
C. `4sqrt2Gm^2//a`
D. `-4sqrt2Gm^2//a`

Explanation

To solve the problem of calculating the work done by an external agent to remove the mass \( m \) from the center of a square with four other masses \( m \) at the corners, we can follow these steps:

### Step 1: Understand the Configuration
We have a square with edge length \( a \). There are four point masses \( m \) located at each corner of the square and one mass \( m \) at the center of the square.

### Step 2: Calculate the Initial Potential Energy
The initial potential energy \( U_i \) of the system can be calculated by considering the gravitational potential energy between all pairs of masses.

1. **Pairs Involving the Center Mass**: The center mass interacts with each of the four corner masses. The distance from the center to any corner is given by:
\[ r = \frac{a}{\sqrt{2}} \]
The potential energy between the center mass and one corner mass is:
\[ U_{c} = -\frac{G m^2}{r} = -\frac{G m^2}{\frac{a}{\sqrt{2}}} = -\frac{G m^2 \sqrt{2}}{a} \]
Since there are four corner masses, the total potential energy due to these interactions is:
\[ U_{c\text{ total}} = 4 \left(-\frac{G m^2 \sqrt{2}}{a}\right) = -\frac{4G m^2 \sqrt{2}}{a} \]

2. **Pairs Among Corner Masses**: The potential energy between each pair of corner masses can be calculated. The distance between any two corner masses (adjacent corners) is \( a \) and for diagonal corners, it is \( \sqrt{2}a \). The total potential energy among the corner masses can be calculated as follows:
- For adjacent pairs (4 pairs):
\[ U_{adj} = 4 \left(-\frac{G m^2}{a}\right) = -\frac{4G m^2}{a} \]
- For diagonal pairs (2 pairs):
\[ U_{diag} = 2 \left(-\frac{G m^2}{\sqrt{2}a}\right) = -\frac{2G m^2}{\sqrt{2}a} \]

Combining these, the total initial potential energy \( U_i \) is:
\[ U_i = -\frac{4G m^2 \sqrt{2}}{a} - \frac{4G m^2}{a} - \frac{2G m^2}{\sqrt{2}a} \]

### Step 3: Calculate the Final Potential Energy
When the center mass \( m \) is moved to a point far away, it no longer interacts with the corner masses. Therefore, the final potential energy \( U_f \) is simply:
\[ U_f = -\frac{4G m^2}{a} - \frac{2G m^2}{\sqrt{2}a} \]

### Step 4: Calculate the Work Done
The work done \( W \) by the external agent is the change in potential energy:
\[ W = U_f - U_i \]
Substituting the values of \( U_f \) and \( U_i \) gives us the final expression for the work done.

### Final Expression
After simplifying the above expressions, we can find that the work done is:
\[ W = -\frac{4G m^2 \sqrt{2}}{a} + \frac{4G m^2}{a} + \frac{2G m^2}{\sqrt{2}a} \]
This will yield a specific numerical value based on the constants involved.