Answer
$\Delta U = -4.82\times 10^{-13}~J$
Work Step by Step
We can find the initial gravitational potential energy:
$U_i = -\frac{4G~m^2}{d}-\frac{2G~m^2}{\sqrt{2}~d}$
$U_i = -\frac{4G~m^2}{d}-\frac{\sqrt{2}~G~m^2}{d}$
$U_i = -\frac{(4+\sqrt{2})~G~m^2}{d}$
$U_i = -\frac{(4+\sqrt{2})~(6.67\times 10^{-11}~N~m^2/kg^2)~(20.0\times 10^{-3}~kg)^2}{0.600~m}$
$U_i = -2.4075\times 10^{-13}~J$
We can find the final gravitational potential energy:
$U_f = -\frac{4G~m^2}{d}-\frac{2G~m^2}{\sqrt{2}~d}$
$U_f = -\frac{4G~m^2}{d}-\frac{\sqrt{2}~G~m^2}{d}$
$U_f = -\frac{(4+\sqrt{2})~G~m^2}{d}$
$U_f = -\frac{(4+\sqrt{2})~(6.67\times 10^{-11}~N~m^2/kg^2)~(20.0\times 10^{-3}~kg)^2}{0.200~m}$
$U_f = -7.2226\times 10^{-13}~J$
We can find the change in gravitational potential energy:
$\Delta U = U_f-U_i$
$\Delta U = (-7.2226\times 10^{-13}~J)-(-2.4075\times 10^{-13}~J)$
$\Delta U = -4.82\times 10^{-13}~J$
