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?

Question

Lily Adams · Accepted Answer

For a mass of #m# potential energy of a test mass at distance r is given as #(G*m/r)# Total potential energy of the system is #(Gm^2)/l ×4+ 2*(Gm^2)/(sqrt(2)l)# or,#4 (Gm^2)/l+sqrt(2)(Gm^2)/l# ( Due to #4# mass at the vertices and 2 diagonally present,length of diagonal is #sqrt(2)l#) Distance of each mass from the centre is =length of diagonal/2 i.e #l/sqrt(2)# So,potential energy at the centre for 4 such mass is #((Gm)/(l/(sqrt(2)))*4)# i.e #4sqrt(2)(Gm)/l#

Benjamin Asiedu · Answer

undefined 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.