Finding the CenterDiscrete Center of MassDiscrete ProblemCenter of GravitySymmetryContinuous MassesFinding the CenterFinding the CenterDiscrete Center of MassDiscrete Center of MassDiscrete masses are treated Discrete masses are treated as separate objects.as separate objects.They can lie on a line or in a They can lie on a line or in a plane.plane. 4321443322111mmmmrmrmrmrmrMrmrCMNiiiCMm2m3m4m1rCMDiscrete ProblemDiscrete ProblemTwo trucks are on a barge. Two trucks are on a barge. Find the center of mass.Find the center of mass.•mm11 = 11 Mg, = 11 Mg, dd11 = -12 m = -12 m•mm22 = 23 Mg, = 23 Mg, dd22 = +12 m = +12 m•mmbb = 35 Mg = 35 Mgm2mbd1m1d20bbbCMmmmdmdmdmd212211•mm11 dd11 = -132 Mg m = -132 Mg m•mm22 dd22 = +276 Mg m = +276 Mg m•mmbb ddbb = 0 = 0•MM = 69 Mg = 69 Mg•ddCMCM = 2.1 m = 2.1 mCenter of GravityCenter of GravityAll forces act as if at the center of mass.All forces act as if at the center of mass.•This includes gravityThis includes gravity• Sometimes called the center of gravitySometimes called the center of gravitym2m1SymmetrySymmetryContinuous objects have a center of mass.Continuous objects have a center of mass.If an object is regular, the symmetry tells us the If an object is regular, the symmetry tells us the center along that direction.center along that direction.Continuous MassesContinuous MassesIn non-symmetrical In non-symmetrical directions, an integral is directions, an integral is needed.needed.•Example, the horizontal Example, the horizontal direction in the triangledirection in the
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