▁m ▁separated ▁by ▁a ▁distance ▁r ▁is ▁given ▁by ▁Newton ' s ▁law ▁ ▁where ▁ ▁is ▁a ▁vector ▁of ▁length ▁ 1 ▁pointing ▁from ▁M ▁to ▁m ▁and ▁G ▁is ▁the ▁gravit ational ▁constant . ▁ ▁Let ▁the ▁mass ▁m ▁move ▁at ▁the ▁velocity ▁v ▁then ▁the ▁work ▁of ▁gravity ▁on ▁this ▁mass ▁as ▁it ▁moves ▁from ▁position ▁r ( t 1 ) ▁to ▁ ▁r ( t 2 ) ▁is ▁given ▁by ▁ ▁The ▁position ▁and ▁velocity ▁of ▁the ▁mass ▁m ▁are ▁given ▁by ▁ ▁where ▁er ▁and ▁et ▁are ▁the ▁radial ▁and ▁tang ential ▁unit ▁vectors ▁directed ▁relative ▁to ▁the ▁vector ▁from ▁M ▁to ▁m . ▁ ▁Use ▁this ▁to ▁simplify ▁the ▁formula ▁for ▁work ▁of ▁gravity ▁to , ▁ ▁This ▁calculation ▁uses ▁the ▁fact ▁that ▁ ▁Pot ential ▁energy ▁for ▁electro static ▁forces ▁between ▁two ▁bodies ▁The ▁electro static ▁force ▁ex ert ed ▁by ▁a ▁charge ▁Q ▁on ▁another ▁charge ▁q ▁separated ▁by ▁a ▁distance ▁r ▁is ▁given ▁by ▁C oul omb ' s ▁Law ▁ ▁where ▁ ▁is ▁a ▁vector ▁of ▁length ▁ 1 ▁pointing ▁from ▁Q ▁to ▁q ▁and ▁ ε 0 ▁is ▁the ▁vac u um ▁perm itt ivity . ▁This ▁may ▁also ▁be ▁written ▁using ▁C oul omb ▁constant ▁. ▁ ▁The ▁work ▁W ▁required ▁to ▁move ▁q ▁from ▁A ▁to ▁any ▁point ▁B ▁in ▁the ▁electro static ▁force ▁field ▁is ▁given ▁by ▁the ▁potential ▁function ▁ ▁Reference ▁level ▁The ▁potential ▁energy ▁is ▁a ▁function ▁of ▁the ▁state ▁a ▁system ▁is ▁in , ▁and ▁is