A $250-$N child is in a swing that is attached to a pair of ropes $1\mathrm{.}90$ m long. Find the gravitational potential energy of the child$$Earth system relative to the child's lowest position at the following times.

$($a$)$ when the ropes are horizontal

$474\mathrm{.}2$ J

$($b$)$ when the ropes make a $27\mathrm{.}0$ angle with the vertical

$51\mathrm{.}74$ J

$($c$)$ when the child is at the bottom of the circular arc