318 Rotation of a Rigid Object About a Fixed AxisP10.82Conservation of energy between apex and the point wherethe grape leaves the surface:mg ymvImgRmvmRvRffff∆=+−=+FHGIKJFHGIKJ1212112122522222ωθcosafwhich gives gvRf17102−=FHGIKJcosθaf(1)Consider the radial forces acting on the grape:mgnmvRfcosθ−=2.At the point where the grape leaves the surface, n→0.Thus, mgmvRfcosθ=2or vRgf2=cosθ.Substituting this into Equation (1) givesggg−=coscosθθ710or cosθ=1017and θ=°54 0..R θi∆y = R—RcosθfnmgsinθmgcosθFIG. P10.82P10.83(a)There are not any horizontal forces acting on the rod, so the center of mass will not movehorizontally. Rather, the center of mass drops straight downward (distance h/2) with the rodrotating about the center of mass as it falls. From conservation of energy:
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