CENTROID. Finding the horizontal coordinate of the common centroid is analogical and the general formula for a figure consisting of n shapes is: 1 n
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1 ENTROD Te followig plr spe c e regrded (Fig) t cosists of spes For ec spe te positio of te cetroid d te re re kow Te res re ssiged wit,, d teir cetroids re,, respectivel For te prticulr exmple coordite sstem - will e used t will e demostrted ow to fid te verticl coordite of te commo cetroid Te verticl coordites of te seprte spes re, d (Fig) Te followig formul is used for determiig te coordite : Fig f te figure cosists of spes it c e writte: i i i i i i i i i i Fidig te oriotl coordite of te commo cetroid is logicl d te geerl formul for figure cosistig of spes is: i i i i Te followig rules ve to e mided we te positio of te cetroid s to e determied: if te figure s oe xis of smmetr te commo cetroid is lig o tis xis; if te figure s two or more xis of smmetr (s for istce rectgle or circle) te positio of te commo cetroid is i te cross poit of tese xis i
2 MOMENTS OF NERT Momets of ierti ccout ow re is situted wit respect to xis Teir psicl meig is to represet ow costructio c resist edig Te igger te momet of ierti of te cross-sectio is te igger re te trsverse lods tt c e pplied to costructio s for istce te em sow o Fig () c re igger force F t te oe from Fig () toug te re of te cross-sectio is oe d te sme () F F L L () Fig () Tt c e explied te fct tt te cross-sectio of te first costructio s igger momet of ierti ccordig to te oriotl xis (te edig is roud tis xis) wic is ovious from te formuls for te momet of ierti of rectgle give i te tle elow > ecuse > Momets of ierti wit respect to te xis pssig troug te cetroid for some simple spes Rectgle Squre ircle d ; π d 6
3 f te momet of ierti wit respect to xis wic is ot pssig troug te cetroid s to e determied, te followig formuls re used: * + e, * * + f, e were is te re of te spe, e is te distce etwee xis d *, f is te oe etwee d * Tis is te teorem of Steier Te terms wic re dded to te momets of ierti re clled Steier s dditios f s for istce for rectgulr spe (Fig) c e writte * * + e + e (), Fig e * * + f + f () * f Fig
4 EXMPLE For te spe sow o Fig5 determie te positio of te cetroid d te priciple momets of ierti 8 Solutio: Te spe s oe xis of smmetr tt mes tt te cetroid is lig o it For te purpose of fidig te cetroid te spe s to e represeted s sum of simple spes (Fig6) i te prticulr exmple two rectgles d oe squre Lter it s to e 0 mided tt ecuse te squre is ollow it s to e sutrcted we determiig te positio of te cetroid d te momet of ierti Te cetroid of ec spe i (i,,) is used s origi of locl coordite sstem 6 (Fig6) Te positio of te commo cetroid s to e determied wit respect to some uxilir Fig 5 coordite sstem Tis coordite sstem c e plced were ut for coveiece it is 8 recommeded tt if te spe s xis of smmetr oe of te xis of te coordite sstem s to coicide wit it tis exmple tis will e te verticl xis ' Te oriotl oe ' will coicide wit te lowest lie of te spe Now te positio of te cetroid will e foud ccordig to te uxilir sstem '' 0 (Fig6) Ol te verticl coordite of te cetroid will e serced ecuse of te xis of ' smmetr (te cetroid is lig o it) Te followig formul is used: ' ' + ' ' 6 ' + Fig 6 Tere is mius i frot of ecuse te squre is ollow Te res of te seprte spes re 8, 86, Te verticl coordites of te cetroids i (i,,) wit respect to '' uxilir coordite sstem re ' 9, ', ' Sustitutio i te formul for ' will led to
5 9 ( 8 ) + (8 6) ( ) ' 5, Now we te positio of te cetroid is k ow oe c determie te priciple momets of ierti wit respect to d xis (Fig7) Te followig c e writte for te momet met of ierti : + - 5,7 Becuse t e xis,, d re c o icidig (te re ll lig o te xis of smmetr) te Steier s dditios re equl 6 to ero Te momets of ierti of te seprte spes wit respect to i xis (i,,) re Fig 7 (8 ) (6 ) 8 ( ),, Sustitutig i te expressio for will led to (8 ) (6 ) 8 ( ) ( ) 8 Before clcultig te momet of ierti it s to e tke ito cosidertio tt te xis,, d re ot coicidig Te distce etwee tese xis re s follows:,5,, 7,, 7 Te momets of ierti of te seprte spes wit respect to i xis (i,,) re ( ) 8, (8 ) 6 ( ), Usig tis d kowig te res of te seprte spes wic were clculted erlier te expressio for cquires te view ( ( ) ) ( ) ( ) ( ( ) ) ( ) 8 (8 ) 6 ( ) + (,5) ( 8 ) + + (, 7) ( 8 6 ) + (, 7) ( ) ( ) ( ) ( ) 5, 99, 56 0,7, 8, 0,7 59,7 9,
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