Properties of Areas & Volumes

Properties of Areas & Volumes - ’i‘; APPENDIX...

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Unformatted text preview: ’i‘; APPENDIX Properties of Areas and Lines 3.1 Areas The coordinates of the centroid of the area A-‘are y / x dA /y dA _ A _ r A x : ' a y = ‘ / dA / dA A ‘ A ! i 1 i i The moment of inertia about the x axis Ix, the moment of inertia about the y axis 2 [y, and the product of inertia Ixy are g S Ix=/y2dA, [y=/x2dA, Ixy=/xydA. A A A The polar moment of inertia about 0 is J0: /r2dA = /(x2+y2)dA =Ix+1y. A A Area = bh 578 Appendix B Properties of Areas and Lines 1 Area = gbh Triangular area 1 A = —bh rea 2 Triangular area Circular area Semicircular area y l : 54 l 4R 317 Quarter-circular area 8.2 Lines 579 Area = aR2 1 1 1 1 [X = ZR4<0¢ M ism 2a), IVV ZR4<0¢ + Esin 201), 19th Ixy * 0 '__b2h2 2 A 1 b rea 2 ~ 47m 1 ' 1 1 Ix = Ewaba I = 167761319, [W = ~8-a2b2 n+1 Area 2 Cb n + 1 I ” 63b3n+1 H,cbn+3 I _ 62yb2n+2 x 9n+3’ n+3y xy 4n+4 f (n +1)b r x n + 2 i 16+!) 1' Spandrel 313 Lines Thelcoordinates of the centroid of the line L are y x dL /y dL /z dL 4 =1 gm / dL / dL / dL wj g- L L L ’y _ x Z W7 A ,R y /R Appendix A Review of Mathe D’PENDIX ‘Volumes and i, wus Objects ine coordinates of the centroid of the volume V are /de /de V V 3=”—> y: , /dV /dV V V The center of mass of a homogeneous object coincides with the centroid of its volume. ‘" [z axis " 0: Iy’axis : [z’axis — l _ ’ 2 Ix’ axis — Iy’ axis ‘ ZmR a Thin circular plate Appendix C Properties of Volumes and’Homogeneous Objects 581 l l Ixaxis : gmhz? [yaxis : _Mb2~ 1: axis 2 gm“; + kg) 1 1 h? 1’ 1 mbz I 1 m(b2 + h”) , . _‘—_ “In *5 , . : ——- a 1,. . : “- “ x axxs 12 y 3x15 12 ax1s 12 Thin rectangular plate m m [x axis : Elm 1}) axis : 21y, 12 axis ~ Ixaxis + 1y axis The terms Ix and [y are the moments of inertia of the plate’s cross-sectional area A about the x and y axes. he centroid Thin plate l 1 Ix’ axis - 1—2m(a2 " b2), [y’ axis _ Emu; + C2): 1' - ” —1~m(b2 -- 62) M z axrs 12 Rectangular prism I j Volume 2 WRZZ 1 l 1 IaniS : [yams : m<§lz + 1R2)? Izaxis : *mRZ 582 Appendix C Properties of Volumes and Homogeneous Objects 1 Volume = Eszh 3 3 3 IaniS _ [Vans 3 m ghz + 7—0’R2 ’ Izaxis = —mRZ .4 Circular cone 3 3 Ix’axis ’_ [y’axis — 7” gig]? + 36R , IZ,axis _ ._mR Volume = Err}? q A; = -. — _ 2 Ix’ axis Iy’ axis Iz’ axis 5 mR Volume = 377R3 2 2 : Iv axis : 133x13 2 “MR ‘ 5 Ix axis 83 Ix, = 1y] = 355W, Iz/ = gmzez Hemisphere ...
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Properties of Areas &amp;amp; Volumes - ’i‘; APPENDIX...

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