CH. V ME256 STATICS Center of Gravity, Centroid, and Moment of Inertia CENTER OF GRAVITY AND CENTROID

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1 CH. ME56 STTICS Ceter of Gravt, Cetrod, ad Momet of Ierta CENTE OF GITY ND CENTOID 5. CENTE OF GITY ND CENTE OF MSS FO SYSTEM OF PTICES Ceter of Gravt. The ceter of gravt G s a pot whch locates the resultat weght of a sstem of partcles. If we cosder a sstem of partcles fed wth a rego of space as show fgure 5., t s evdet that the weghts of the partcles comprse a sstem of parallel forces whch ca be replaced b a sgle equvalet resultat weght havg the defed pot G of applcato. 3 G 5 Σ 4 Fgure 5.. Ceter of gravt G of a sstem of partcles space. s establshed prevousl, t s possble to obta a equvalet force of a sstem of forces b addg the vectors represetg each oe of the dvdual forces. lso, t was determed that the pot of applcato of that equvalet force s foud b dog summato of momets about a referece pot. Smlarl, a sstem of partcles wth weght (, ). The total weght s foud b addg up the dvdual weght of the partcles ad the locato at whch ths weght acts s determed b dog summato of momets about a pot of referece. Thus, the resultat of the weght s equal to the total weght of all partcles: The summato of the momets of the weghts of all the partcles about the,, ad aes s equal to the momet of the resultat weght about these aes. The, to determe the coordate of G, t s ecessar to do summato of momets about the as to obta

2 CH. ME56 STTICS Ceter of Gravt, Cetrod, ad Momet of Ierta I a smlar maer, to determe the coordate of G, calculato of summato of momets about the as s effectuated, The weghts do ot produce momet about as therefore, the coordate of G s calculated b rotatg the coordate sstem, wth the partcles attached to t, b 90 o about ether the or as, ad the proceed a smlar maer as before Thus, followg ths procedure s possble to determe the coordates of the ceter of gravt G of a sstem composed b partcles as (5.) Ceter of Mass. The determato of the pot kow as ceter of mass s ecessar to carr out the stud of problems cocerg to the moto of matter uder the fluece of a force. Provded that the accelerato due to the gravt g for ever partcle of the sstem s costat, the mg. I ths wa, Eq. (5.) ca be wrtte as m m m m m m (5.) 5. CENTE OF GITY, CENTE OF MSS, ND CENTOID FO BODY Ceter of Gravt. The determato of the ceter of gravt of a rgd bod s accomplshed b followg the same prcple dscussed the prevous secto. Sce a rgd bod s assumed to be cotuous, t s possble to epress ever oe of the coordate of the ceter of gravt as d d d d d d (5.3) To appl the above epressos s ecessar to have a relato betwee the weght of the rgd bod cosderato ad ts volume. If the dest of the elemet studed s costat, Eq. (5.3) become 9

3 CH. ME56 STTICS Ceter of Gravt, Cetrod, ad Momet of Ierta ρ gd ρ gd ρ gd ρ gd ρ gd ρ gd (5.4) I the above set of epressos g represets the accelerato of gravt (9.8 m/s, 3. ft/s ). Ceter of Mass. The ceter of mass of a rgd bod ca be determed usg the same prcples emploed to determe the ceter of gravt. Therefore, the ceter of mass of a bod ca be epressed as ρ d ρ d ρ d (5.5) ρ d ρ d ρ d Cetrod. The cetrod C s a pot whch defes the geometrc ceter of a object. Its locato ca be determed usg the same prcples emploed to determe the ceter of gravt of a bod. I the case where the materal composg a bod s uform or homogeeous, the dest or specfc weght wll be costat throughout the bod ad these values wll be factored out from the tegrals [Eqs. (5.4) ad (5.5)] smplfg the epressos for ceter of mass ad ceter of gravt. I ths specfc case, the ceters of mass, gravt ad geometr cocde. olume. he a bod s subdvded to volume elemets d, as show fgure 5., the locato of the cetrod C(,, ) for the volume of the object ca be determed b computg the momets of the elemets about each of the coordate aes. d d d d d d (5.6) (,, ) d (,, ) Fgure 5.. Calculato of the ceter of geometr a volume. 30

4 CH. ME56 STTICS Ceter of Gravt, Cetrod, ad Momet of Ierta rea. I a smlar maer, the cetrod for the surface area of a object (fgure 5.3) ca be determed b subdvdg the area to dfferetal elemets of area d ad the calculatg the momets of those ftesmal areas about each as of the coordate sstem. d C(,, ) Fgure 5.3. Calculato of the geometrc ceter of a area. pplcato of the same prcples used before to determe the ceter of gravt o a bod eld the followg epressos for geometrc ceter of a area d d d d d d (5.7) e. Fall, f the geometr of the studed bod s such as a th rod or wre, t ca be approached to a le. The balace of momets of the dfferetal elemets d about each coordate as results d d d d d d (5.8) Homework No. 5.: 9 4, 9 6, 9 9, 9 5, 9 38,

5 CH. ME56 STTICS Ceter of Gravt, Cetrod, ad Momet of Ierta d C(,, ) 5.3 COMPOSITE BODIES composte bod cossts of a seres of coected smpler shaped bodes, whch ma be rectagular, tragular, semcrcular, etc. Such a bod ca ofte be sectoed or dvded to ts composte parts ad f the weght ad locato of the ceter of gravt of each of these parts are kow, t s possble to determe the ceter of gravt of the whole bod from ths formato stead of carrg out the tegrato of the relevat equatos. The procedure to fd the ceter of gravt of a composte bod requres treatg each of the dfferet compoets of the bod as a partcle, ad the the applcato of the equatos developed prevousl for fdg the ceter of gravt of a set of dvdual partcles. he the bod has costat dest or specfc weght, the ceter of gravt cocdes wth the cetrod of the bod. The cetrod for composte les, areas, ad volumes ca be foud usg relatos aalogous to the equato preseted prevousl. Homework No. 5.: 9 45, 9 48, 9 60, 9 63, 9 75, ESUTNT OF GENE DISTIBUTED OD Fgure 5.4. Calculato of the geometrc ceter of a le segmet. Pressure dstrbuto o a surface. Cosder the plate show fgure 5.5. The plate s beg subjected to the loadg fucto P P(, ). Kowg ths fucto t s possble to determe the force df actg o the dfferetal area d. The magtude of ths force s df P(, ) d. The etre loadg o the plate represets the a sstem of parallel forces fte umber ad each actg o a separate dfferetal area d. 3

6 CH. ME56 STTICS Ceter of Gravt, Cetrod, ad Momet of Ierta P P(,, ) F (,, ) Fgure 5.5. Determato of the resultat of a dstrbuted load. Magtude of the resultat force. The magtude F s determed b summg each of the dfferetal forces df actg o the etre surface area of the plate. F P(, ) d (5.9) ocato of the resultat force. The locato of the cetrod (, ) of F s calculated b settg the momets of F equal to the momets of all the forces df about the respectve ad aes P(, ) d P(, ) d P(, ) d P(, ) d (5.0) The le of acto of the resultat force passes through the geometrc ceter or cetrod of the volume uder the dstrbuted loadg dagram. Homework No. 5.3: ssgmet problems wll be provded b structor. 5.5 EDUCTION OF SIMPE DISTIBUTED ODING I ma stuatos a large surface area of a bod ma be subjected to dstrbuted loadgs such as those caused b fluds, wd, or weght of materal supported b the surface of the bod. The test of these loadgs at each pot o the surface s defed as pressure p (force per ut area). The ut for pressure s pascal (Pa N/m ) or ps (poud per square ch). lso lb/ft s a ut frequetl used to defe pressure. The most commo case of a dstrbuted pressure loadg s the uform dstrbuted loadg. I ths case, the loadg s uform alog oe as of a flat rectagular bod upo whch the loadg s actg. Ths case s show schematcall fgure

7 CH. ME56 STTICS Ceter of Gravt, Cetrod, ad Momet of Ierta p p() p Fgure 5.6. Uform loadg o a plae surface. The drecto of the test of the pressure load s dcated b the arrows show o the load test dagram. The etre loadg o the plate s a sstem of parallel forces, fte umber ad each actg o a separate dfferetal area of the plate. I ths case, the loadg fucto, p p(), s ol fucto of sce the pressure s uform alog the as. Sce the loadg s uform alog the drecto, the dagram ca be preseted as w w() Fgure 5.7. Bdmesoal represetato of a uform loadg. Sce the pressure eerted b the loadg alog as s uform, t s possble to represet the loadg terms of force per ut legth b multplg the pressure, p(), b the wdth, a, of the plate. Thus, w() a p(). Ths loadg fucto show fgure 5.7, s a measure of load alog the plae of smmetr of the pressure load (le 0), ad t s a measure of the force per ut legth. Magtude of the esultat Force The calculato of the magtude of the resultat force F due to the dstrbuted loadg s obtaed b addg up all the forces the sstem. I ths case, the umber of forces s fte ad therefore the epresso represetg the load per ut legth must tegrated alog the as where the load s appled. F w( ) d (5.) 34

8 CH. ME56 STTICS Ceter of Gravt, Cetrod, ad Momet of Ierta s observed, the magtude of the total force s equal to the total area uder the loadg dagram. ocato of the esultat Force The determato of the pot at whch the resultat force s appled ca be foud b dog summato of momets for the resultat force about a referece pot ad equatg ths epresso to the summato of momets produced b each of the forces actg o the plate about the same pot. That s F X X w( ) d w( ) d F (5.) Ths equato represets the coordate for the geometrc ceter or the cetrod of the area uder the dstrbuted loadg dagram w(). Therefore, the resultat force has a le of acto whch passes through the cetrod C (geometrc ceter) of the area defed b the dstrbuted loadg dagram w(). Homework No. 5.4: 4 4, 4 47, 4 53, DEFINITION OF MOMENTS OF INETI FO ES I prevous sectos we leart to determe the cetrod for a area b calculatg the frst momet of the area about a as ths s, we evaluated the tegral of the form d tegral of the secod momet of a area, such as d s kow as momet of erta for the area. The momet of erta of a area s a quatt that relates the ormal stress σ or force per ut area, actg o the trasverse cross secto of a elastc beam to the appled eteral momet M, whch causes the bedg of the beam. The theor of mechacs of materal shows that the stress wth the beam vares learl wth the dstace from a as passg through the cetrod C of the beam s cross sectoal area (σ k), as show fgure 5.8. The magtude of the force actg o the area elemet d s df k d. Sce ths force s located at a dstace from the as, the momet of df about the as s dm df k d. The resultg momet of the etre stress dstrbuto s equal to the appled momet M. The 35

9 CH. ME56 STTICS Ceter of Gravt, Cetrod, ad Momet of Ierta M k Ths tegral represets the momet of erta of the area about the as. Ths kd of tegrals s commol foud mechacs of materals, structural mechacs, flud mechacs, ad mache desg. Momet of Ierta The momet of erta about as, for the dfferetal plaar area show fgure 5.8, whch les the plae, s di d whle the momet of erta about as for that dfferetal elemet of area s di d d d r O Fgure 5.8. Momet of erta for a area o the plae. Thus, the momets of erta for the area are determed b I d I d (5.3) The momet of erta about the pole O or as s also kow as polar momet of erta. Ths momet of erta s used to determe the torsoal stress a shaft. The polar momet of erta s defed as dj O r d where r s the perpedcular dstace from the pole to the elemet d. Thus, the etre polar momet of erta s J O r d I + I (5.4) 5.7 PE XIS THEOEM FO N E If the momet of erta for a area s kow about a as passg through ts cetrod, whch s ofte the case, t s coveet to determe the momet of erta of the area about a correspodg parallel as usg the parallel as theorem. 36

10 CH. ME56 STTICS Ceter of Gravt, Cetrod, ad Momet of Ierta as The momet of erta about as of the area show fgure 5.9 ca be epressed I ( d + ' ) d ' d + d ' d d + O d d C Fgure 5.9. Determato of the momet of erta of a area usg the parallel as theorem. d d ' d The frst tegral represets the momet of erta about the cetrod of the area. Ths tegral s commol kow ad preseted tables. The secod tegral s ero sce ' d ad 0 sce the arm of the momet s referred to the cetrod of the area. Fall the last tegral s the tegral of the area. Therefore smlar aalss wll lead to I I + d I I + d (5.5) (5.6) The polar momet of erta about a as perpedcular to the plae ad passg through the pole O ( as) J O J C + d (5.7) 5.8 DIUS OF GYTION OF N E The radus of grato of a plaar area has uts of legth ad s a quatt usuall emploed to desgg colums structural mechacs. If the momets of erta are kow, the rad of grato are determed from the formulas Homework No. 5.5: 0 5, 0 3, 0 9, 0 3. I I k k ko J O (5.8) 37

11 CH. ME56 STTICS Ceter of Gravt, Cetrod, ad Momet of Ierta 5.9 MOMENT OF INETI FO COMPOSITE ES composte area cossts of a seres f coected smpler parts or shapes. If the momet of erta of each of these parts s kow or ca be determed about a commo as, the the momet of erta of the composte area equals the algebrac summato of the momet of erta of all ts parts. Homework No. 5.6: 0 33, 0 37, 0 43, 0 46,