<Dd> x c o m = x m a x θ _̄ 2 π (\ displaystyle x_ (com) = x_ (max) (\ frac (\ overline (\ theta)) (2 \ pi))) </Dd> <P> The process can be repeated for all dimensions of the system to determine the complete center of mass . The utility of the algorithm is that it allows the mathematics to determine where the "best" center of mass is, instead of guessing or using cluster analysis to "unfold" a cluster straddling the periodic boundaries . It must be noted that if both average values are zero, (ξ _̄, ζ _̄) = (0, 0) (\ displaystyle ((\ overline (\ xi)), (\ overline (\ zeta))) = (0, 0)), then θ _̄ (\ displaystyle (\ overline (\ theta))) is undefined . This is a correct result, because it only occurs when all particles are exactly evenly spaced . In that condition, their x coordinates are mathematically identical in a periodic system . </P> <P> A body's center of gravity is the point around which the resultant torque due to gravity forces vanishes . Where a gravity field can be considered to be uniform, the mass - center and the center - of - gravity will be the same . However, for satellites in orbit around a planet, in the absence of other torques being applied to a satellite, the slight variation (gradient) in gravitational field between closer - to (stronger) and further - from (weaker) the planet can lead to a torque that will tend to align the satellite such that its long axis is vertical . In such a case, it is important to make the distinction between the center - of - gravity and the mass - center . Any horizontal offset between the two will result in an applied torque . </P> <P> It is useful to note that the mass - center is a fixed property for a given rigid body (e.g. with no slosh or articulation), whereas the center - of - gravity may, in addition, depend upon its orientation in a non-uniform gravitational field . In the latter case, the center - of - gravity will always be located somewhat closer to the main attractive body as compared to the mass - center, and thus will change its position in the body of interest as its orientation is changed . </P>

Where is the center of gravity located on the body