9.3 Composite BodiesConsists of a series of connected “simpler”shaped bodies, which may be rectangular,triangular or semicircularA body can be sectioned or divided into itscomposite partsProvided the weight and location of the center ofgravity of each of these parts are known, theneed for integration to determine the center ofgravity for the entire body can be neglected
9.3 Composite BodiesAccounting for finite number of weights ∑ ~W x ∑ ~W y ∑ ~W z x= y= z= ∑W ∑W ∑WWherex, y, z represent the coordinates of the centerof gravity G of the composite body~, ~, ~x y z represent the coordinates of the centerof gravity at each composite part of the body∑W represent the sum of the weights of allthe composite parts of the body or total weight
9.3 Composite BodiesWhen the body has a constant density orspecified weight, the center of gravity coincideswith the centroid of the bodyThe centroid for composite lines, areas, andvolumes can be found using the equation ∑~W x ∑~W y ∑~W z x= y= z= ∑W ∑W ∑WHowever, the W’s are replaced by L’s, A’s and V’srespectively
9.3 Composite BodiesProcedure for AnalysisComposite Parts Using a sketch, divide the body or object into a finite number of composite parts that have simpler shapes If a composite part has a hole, or a geometric region having no material, consider it without the hole and treat the hole as an additional composite part having negative weight or size
9.3 Composite BodiesProcedure for AnalysisMoment Arms Establish the coordinate axes on the sketch and determine the coordinates of the center of gravity or centroid of each part
9.3 Composite BodiesProcedure for AnalysisSummations Determine the coordinates of the center of gravity by applying the center of gravity equations If an object is symmetrical about an axis, the centroid of the objects lies on the axis
9.3 Composite BodiesExample 9.9Locate the centroid of the wire.
9.3 Composite BodiesSolutionComposite PartsMoment Arms Location of the centroid for each piece is determined and indicated in the diagram
9.3 Composite Bodies Solution SummationsSegment L x (mm) y (mm) z (mm) xL yL zL (mm) (mm2) (mm2) (mm2) 1 188.5 60 -38.2 0 11 310 -7200 0 2 40 0 20 0 0 800 0 3 20 0 40 -10 0 800 -200 Sum 248.5 11 310 -5600 -200
9.3 Composite BodiesExample 9.11Locate the center of mass of thecomposite assembly. The conicalfrustum has a density ofρc = 8Mg/m3 and the hemispherehas a density of ρh = 4Mg/m3.There is a 25mm radiuscylindrical hole in the center.
9.3 Composite BodiesSolutionComposite Parts Assembly divided into 4 segments Area of 3 and 4 considered “negative”
9.3 Composite BodiesSolutionMoment Arm Location of the centroid for each piece is determined and indicated in the diagramSummations Because of symmetry, x = y=0
9.3 Composite BodiesSolutionSummations Segment m (kg) z (mm) zm (kg.mm) 1 4.189 50 209.440 2 1.047 -18.75 -19.635 3 -0.524 125 -65.450 4 -1.571 50 -78.540 Sum 3.141 45.815