▁plan imeter ▁known ▁as ▁an ▁inte graph , ▁or ▁integer ometer , ▁can ▁be ▁used ▁to ▁establish ▁the ▁position ▁of ▁the ▁cent roid ▁or ▁center ▁of ▁mass ▁of ▁an ▁irregular ▁two - dimensional ▁shape . ▁This ▁method ▁can ▁be ▁applied ▁to ▁a ▁shape ▁with ▁an ▁irregular , ▁smooth ▁or ▁complex ▁boundary ▁where ▁other ▁methods ▁are ▁too ▁difficult . ▁It ▁was ▁regularly ▁used ▁by ▁ship ▁build ers ▁to ▁compare ▁with ▁the ▁required ▁disp lacement ▁and ▁center ▁of ▁bu oy ancy ▁ ▁of ▁a ▁ship , ▁and ▁ensure ▁it ▁would ▁not ▁cap size . ▁ ▁In ▁three ▁dimensions ▁An ▁experimental ▁method ▁to ▁locate ▁the ▁three - dimensional ▁coordinates ▁of ▁the ▁center ▁of ▁mass ▁begins ▁by ▁supporting ▁the ▁object ▁at ▁three ▁points ▁and ▁meas uring ▁the ▁forces , ▁F 1 , ▁F 2 , ▁and ▁F 3 ▁that ▁resist ▁the ▁weight ▁of ▁the ▁object , ▁ ▁( ▁is ▁the ▁unit ▁vector ▁in ▁the ▁vertical ▁direction ). ▁ ▁Let ▁r 1 , ▁r 2 , ▁and ▁r 3 ▁be ▁the ▁position ▁coordinates ▁of ▁the ▁support ▁points , ▁then ▁the ▁coordinates ▁R ▁of ▁the ▁center ▁of ▁mass ▁satisfy ▁the ▁condition ▁that ▁the ▁result ant ▁tor que ▁is ▁zero , ▁ ▁or ▁ ▁This ▁equation ▁yields ▁the ▁coordinates ▁of ▁the ▁center ▁of ▁mass ▁R * ▁in ▁the ▁horizontal ▁plane ▁as , ▁ ▁The ▁center ▁of ▁mass ▁lies ▁on ▁the ▁vertical ▁line ▁L , ▁given ▁by ▁ ▁The ▁three - dimensional ▁coordinates ▁of ▁the ▁center ▁of ▁mass ▁are ▁determined ▁by ▁performing ▁this ▁experiment ▁twice ▁with ▁the ▁object ▁position ed ▁so