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 a system 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 with a mass of 1 kg, placed at the vertices of a square. The side length of the square is given as 1 m.

### Step 2: Identify the Distances
The distances between the particles are crucial for calculating the gravitational potential energy. The distances between adjacent particles (e.g., A and B) is 1 m, while the distances between diagonal particles (e.g., A and C) can be calculated using the Pythagorean theorem.

For a square:
- Distance between adjacent vertices (e.g., A and B): \( d = 1 \, \text{m} \)
- Distance between diagonal vertices (e.g., A and C):
\[ d_{AC} = \sqrt{(1^2 + 1^2)} = \sqrt{2} \, \text{m} \]

### Step 3: Calculate the Number of Unique Pairs
To find the total potential energy, we need to consider all unique pairs of particles. The number of ways to choose 2 particles from 4 is given by the combination formula \( \binom{n}{r} \):
\[ \text{Number of pairs} = \binom{4}{2} = \frac{4!}{2!(4-2)!} = 6 \]

### Step 4: Calculate the Gravitational Potential Energy for Each Pair
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 universal gravitational constant.

For our case:
- For pairs at distance 1 m (4 pairs: AB, BC, CD, DA):
\[ U_{1} = -\frac{G \cdot 1 \cdot 1}{1} = -G \]
Total for these pairs: \( 4 \times (-G) = -4G \)

- For pairs at distance \( \sqrt{2} \) m (2 pairs: AC, BD):
\[ U_{2} = -\frac{G \cdot 1 \cdot 1}{\sqrt{2}} = -\frac{G}{\sqrt{2}} \]
Total for these pairs: \( 2 \times \left(-\frac{G}{\sqrt{2}}\right) = -\frac{2G}{\sqrt{2}} = -\sqrt{2}G \)

### Step 5: Combine the Total Potential Energy
Now, we can combine the potential energies from both sets of pairs:
\[ U_{\text{total}} = -4G - \sqrt{2}G \]
This can be simplified to:
\[ U_{\text{total}} = -\left(4 + \sqrt{2}\right)G \]

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