Find the potential energy of 4-particles, each of mass 1 kg placed at the four vertices of a square of side length 1 m.

Options

A. `+4.0 G`
B. `7.5 G`
C. `-5.4 G`
D. `+6.3 G`

Explanation

To find the potential energy of four particles, each of mass 1 kg, placed at the vertices of a square with a side length of 1 m, we can follow these steps:

### Step 1: Understand the Configuration
We have four particles, each of mass \( m = 1 \, \text{kg} \), located at the vertices of a square. Let's label the vertices as A, B, C, and D. The distance between adjacent vertices (e.g., A and B) is the side length of the square, which is \( 1 \, \text{m} \).

### Step 2: Calculate the Distances
The distances between the particles are as follows:
- Distance between adjacent vertices (e.g., A and B): \( r_{AB} = 1 \, \text{m} \)
- Distance between opposite vertices (e.g., A and C): \( r_{AC} = \sqrt{2} \, \text{m} \) (using Pythagorean theorem)

### Step 3: Write the Formula for Gravitational Potential Energy
The gravitational potential energy \( U \) between two masses \( m_1 \) and \( m_2 \) separated by a distance \( r \) is given by:
\[ U = -\frac{G m_1 m_2}{r} \]
where \( G \) is the gravitational constant, \( G \approx 6.674 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \).

### Step 4: Calculate the Potential Energy Between Each Pair of Particles
We need to consider the potential energy between each pair of particles:
1. Between A and B:
\[ U_{AB} = -\frac{G \cdot 1 \cdot 1}{1} = -G \]
2. Between B and C:
\[ U_{BC} = -\frac{G \cdot 1 \cdot 1}{1} = -G \]
3. Between C and D:
\[ U_{CD} = -\frac{G \cdot 1 \cdot 1}{1} = -G \]
4. Between D and A:
\[ U_{DA} = -\frac{G \cdot 1 \cdot 1}{1} = -G \]
5. Between A and C:
\[ U_{AC} = -\frac{G \cdot 1 \cdot 1}{\sqrt{2}} = -\frac{G}{\sqrt{2}} \]
6. Between B and D:
\[ U_{BD} = -\frac{G \cdot 1 \cdot 1}{\sqrt{2}} = -\frac{G}{\sqrt{2}} \]

### Step 5: Sum the Potential Energies
Now, we sum all the potential energies:
\[ U_{\text{total}} = U_{AB} + U_{BC} + U_{CD} + U_{DA} + U_{AC} + U_{BD} \]
Substituting the values:
\[ U_{\text{total}} = -G - G - G - G - \frac{G}{\sqrt{2}} - \frac{G}{\sqrt{2}} \]
\[ U_{\text{total}} = -4G - 2\frac{G}{\sqrt{2}} = -4G - \sqrt{2}G \]

### Step 6: Final Expression
Thus, the total gravitational potential energy of the system is:
\[ U_{\text{total}} = -\left(4 + \sqrt{2}\right)G \]