# Find the potential energy of a system of four particles.each of mass "m" place at the vertices of a square of side " l " also obtained the potential at the centre of the squqre?

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The potential energy of a system of four particles placed at the vertices of a square of side length ( l ) can be calculated using the formula for the potential energy of a system of point masses:

[ U = - G \frac{m_1 m_2}{r} ]

where ( G ) is the gravitational constant, ( m_1 ) and ( m_2 ) are the masses of the particles, and ( r ) is the distance between the particles.

For this system, each particle is at a distance ( l ) from the center of the square (which is also the distance between any two adjacent particles). Since all particles have the same mass ( m ), the potential energy between any two particles is:

[ U = - G \frac{m^2}{l} ]

Since there are four particles in the system, and each particle interacts with the other three, we need to consider all pairwise interactions. Therefore, the total potential energy ( U_{\text{total}} ) of the system is:

[ U_{\text{total}} = 4 \times U ]

[ U_{\text{total}} = - 4G \frac{m^2}{l} ]

The potential at the center of the square can be obtained by considering the distance from the center to each particle, which is ( \frac{l}{\sqrt{2}} ) (half the diagonal of the square). The potential at the center due to one particle is:

[ U_{\text{center}} = - G \frac{m^2}{\frac{l}{\sqrt{2}}} ]

Since there are four particles contributing to the potential at the center, the total potential at the center ( U_{\text{center total}} ) is:

[ U_{\text{center total}} = 4 \times U_{\text{center}} ]

[ U_{\text{center total}} = - 4G \frac{m^2}{\frac{l}{\sqrt{2}}} ]

So, the potential energy of the system of four particles and the potential energy at the center of the square can be calculated using these formulas.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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