Find the total gravitational potential energy of this system?

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SUMMARY

The total gravitational potential energy (U) of a system with multiple masses can be calculated using the formula U = -G * Σ(m_i * m_j / r_ij), where G is the gravitational constant, m_i and m_j are the masses, and r_ij is the distance between each pair of masses. When all masses are doubled, the total gravitational potential energy also doubles. Conversely, if the dimensions of the rectangle containing the masses are halved, the potential energy increases due to the reduced distance between the masses, resulting in a higher gravitational interaction.

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  • Understanding of gravitational potential energy concepts
  • Familiarity with the gravitational constant (G)
  • Knowledge of distance calculations between point masses
  • Basic algebra for summing energy contributions
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  • Study the derivation of gravitational potential energy formulas
  • Learn about the gravitational constant (G) and its applications
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http://session.masteringphysics.com/problemAsset/1125570/2/Walker4e.ch12.Pr042.jpg
use this image for help

a. Consider the four masses shown in the figure . Find the total gravitational potential energy of this system.

b. How does your answer to part A change if all the masses in the system are doubled?

c. How does your answer to part A change if, instead, all the sides of the rectangle are halved in length?


I'm absolutely lost in this problem. I know that U=-GmMe/r
where Me is the mass of earth.
r= radius
However, I don't know how to apply that to so many different masses.
Please help and explain?!
 
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The total gravitational potential energy of the system is the total energy required to separate the constituent masses from each other ie the sum of the energy required to separate each pair of masses.
 

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