3) Energy Density in a Newtonian Gravitational Field. Consider assembling a system of N particles of mass MA assigned at

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3 Energy Density In A Newtonian Gravitational Field Consider Assembling A System Of N Particles Of Mass Ma Assigned At 1 (108.62 KiB) Viewed 11 times

3) Energy Density in a Newtonian Gravitational Field. Consider assembling a system of N particles of mass MA assigned at positions TA, A = 1, ..., N. The Newtonian potential energy of the system W is the potential energy of all the particles found by bringing them one by one from infinity into the potential of the particles already assembled. Show that this is 1 W Σ GMA MB 2 TA-TB A+B and that the corresponding formula for a continuum distribution of mass with density (T) is 1 1 GuT) (7') W = } /** [ 136 ' G :- 7) (). d’x u(T)$(7). T- W får -A(Z d Use the Newtonian field equation (3.18) to eliminate u(T) frm this expression and then the divergence theorem to write this as W Bata) a'r (2017)] = | ds excwe(7), F ° 1 87 G | dr = where eNewt (?) is the energy density of a Newtonian gravitational field. [40]