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# ENGR MECHANICS STATICS CVEN 221

Texas A&M

GPA 3.77

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This 13 page Class Notes was uploaded by Jaylan Rath on Wednesday October 21, 2015. The Class Notes belongs to CVEN 221 at Texas A&M University taught by Lee Lowery in Fall. Since its upload, it has received 61 views. For similar materials see /class/226114/cven-221-texas-a-m-university in Civil Engineering at Texas A&M University.

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Date Created: 10/21/15

XAMPLE PRSBLEM 53 Locate the ycoordinate of the centroid of the area of the quarter circle shown in Fig 58a 2 gr m3 H4 b Kain y xx 2 mW eehwxwmw We Er quot SOLUTION r 2 o a 4 Mg 2 Four different elements will be used to solve this problem 7 J j i V f A xquot METHOD 1 Double integral in rectangular coordinates V For the differential element shown in Fig 58b dA dy dx The element dA is at a distance y from the x axis therefore the moment of the area about the x axis is 9 i313 5 r rl x Mx J y d I f y dy dx Double integration is required to A 39 9 9 m solve the problem if a differential f d r 1721 362 dx 39 f element is chosen with sides dx 0 239 o 0 2 2 6 o 3 and dy at distances at and y from V the reference axes The princ1ple of moments states that AyslmequlW 3 33 3 W A 77724 37r Therefore 200 METHOD 2 Single integral using a horizontal strip Alternatively the element of area can be selected as shown in Fig 58c For this element which is located a distance y from the xaxis dA x dy r2 yz dy Therefore the moment of the area about the xaxis is r r2 232r 73 x2 2 if Mquot lay dA lay 1 l 3 o 3 The principle of moments states that Ayc f y dA MI Therefore 133 4r 71124 E Mx yc A METHOD 3 Single integral using a vertical strip The element of area could also be selected as shown in Fig 5 801 For this ele ment dA y dx r2 x2 dx however all parts of the element are at dif ferent distances 3 from the x axis For this type of element the results of Example Problem 52 can be used to compute a moment de which can be integrated to yield moment Mx Thus 11 23 fax FijiIf 24Av 234x 2d7 2 dx rz xz r2 cx3rr3 Mx lade la 239 dx 2 60 3 The principle of moments states that Ayc f y dA M Therefore is W3 4 W A 71124 37 METHOD 4 Double integral using polar coordinates Finally polar coordinates can be used to locate the centroid of the quarter cir cle With polar coordinates the element of area is dA p d6 Lip and the dis tance from the xaxis to the element is y p sin 0 as shown in Fig 586 Thus yVf wt MxjAydALfoagzsin65f6dp for p2 cos 61325110 2 Lrpz d The principle of moments states that Ayc f y dA M Therefore 4r quot A In a completely similar manner the xcoordinate of the centroid is obtained as xc 733 if A 77724 37239 The results are illustrated in Fig 58f The problem can be solved with a single integration if a differential element is chosen as a strip with width x that varies as a functidn ofy r 3 g A 5 o 2 a We 2 r 33 Sirn arly the problem can be solved with a single integration if a differential element is chosen as a strip with height 3 that varies as a function of 9 Frequently polar coordinates are more efficient when circular boundaries are involved 3 wk 201 quotm a 5 2 23 g Y Y 3 9Qquot EXAMPLE39PROBLEM 55 A circular arc of thin homogeneous wire is shown in Fig 54052 a Locate the x and ycoordinates of the mass center b Use the results of part a to determine the coordinates of the mass center for a half circle AMI M i af x KKK wig M iiquot x 4 13 B quot x a b Fig 510 SOLUTION a Therefore elements is dm p dV pA dL pAr d0 Therefore the total mass of the wire is quot3 myc f y m L r sin 0pAr d6 2pA1392cos B 21 cos B yc pAr7r 28 71 2B Since the length of wire is symmetric about the yaxis Ans xc 0 Ans b For the half circle 8 0 2r yc 7T Ans xc 0 ADS The wire can be assumed to consist of a large number of differential ele ments of length dL as shown in Fig 5 10b The mass of each of these mfdm r pArdapAr theprim 23 The distance y from the x axis to the element dm is y 139 sin 0 Thus w B fey pAr2 J sin 0 d6 pAT22 cos B 3 Polar coordinates are more ef cient for this determination since the shape of the Wire is a circular arc 5M 1 3 quot1 Rel 1 WM 1 m 039 it w W Qage i 203 acsz x 39 1va I 398Iz szx 217 zx zv xpsxf zx ZD x 273 zxz 0 39 w It A zx z DBOIT X ZU ll u x X Zx 39osz quot S c 727 fig W 9 HI xZ v Uf fo3991z 1443900 w 3x lynx 6 A zx Zn Zx v xp z D X D 0 U uS z7 Zx 1 1 Z zJ Hz 1 z XIX 27 Xp l U I z 223 f x X szx WZX S 9zx z I LZx 2quot 2 xp zx zquot EXJ quot512 z v 91 9I xm v K x4 v avg sz x v xf39zxz e z z xzv s z z I c z z z 9 zx zv 30571 Zx 2csz zv sxf 3911 I 8 t7 x IugszvZx zv x z D Zx zv xpzxzv zxf z 9zx Zml XI szx at x 3960Z zx Zn zx znVJ 8 I xpx iW zx zv V hwyW quotIx f m X D xp55 f w 15x 1 x zvz czx zv EJ xp x 2 Czxf 20 2x r A n m 17 Zac zv zv Czx zv x I Xp z 3 X I 3992 v z zx zv zv Ezx Zv 39LOZ x xp 39EZZ E zxzvz 2x quot z sxf Ivl z z v I IusE zx zv xzvg Ezx IraW xp Ezx Ito1903 x zquot I M 9617236 zv Xf 90 quotPJ 1 Zx zn 3 ZD zx zn 39I OZ Tex I 2x Z1 1 clac 217 J ugs x x X39pzx 1 5 2quot ME 5 zxquot dzquot quot ME X x 8 x MM sz zv aw mantles mourn macaw 57111191170 1w 228 Formulas for Uso in Trigonometry Relations of the Functions 1 sinx CSCX csc x 5m x 1 1 cosx secx secx cosx 1 sin x tanx smzx coszx 1 cotx cosx l tanzx seczx 1 cosx 2 2 cotx 1cotx csc x tan x sm x sinx 1 l coszx cosx 1 1 sinzx tanx 1 seczx l secx inaan 1 cotx i csc x 1 cscx 1 cotzx 1 sin x cos 90 x sin 180m x cosx sin 90 x cos180 x W tanx cot90 x tan 180 x cotx tan 90 x cot180 x x cscx cot cotx The sign in front ofradical depends on quadrant in which it falls Reduction Formulae sin a COSOl 90 sina 180 cosa 270 cosa sina 90 cosa 180 sina 270 tan a cota 90 tana 180 cota 270 cota tana 90quot cota 180 tanoz 270 seen CSC x 90 seca 180 csca 270 050a seca 90 cscoz 180 seca 270 FURTHER REDUCTION FORMULAE sin cos tan cot sec csc a sina cosa tana cota seca csca 90 a cosoz sin a cota tana csca seca 90 a cosa sin a cota tana CSCa seca 180 a sin a cosa tana c0ta seCa csca 180 a sin a cos a tan a cot a SCCa 8800 270quot a cos a sin a cot a tan a csca see a 270 a cos a sin a cot a tan a csc a see a 360 a sin a cos a tan a cot a sec at csc a 360 a sin a cos oz tan a cot a sec or csc oz The above table may be summarized and extended by the following easily remembered rule fia n90 1ga Z 05 is Formulas for Use it where n may be any integer positive negativ fis any one of the six trigonometric fu a may be any real Jangle measure lfn is even then g is the same function a off Sine and cosine tangent and cotangent se other 39 The second 1 sign is not necessarily the 5 follows For a given function f a given value the second 1 sign will be the same for all V check the sign for any one value of a and the 1 EXAMPLES tan a 270 1 cot 01 Since n3 is the sign assume a value of a in the rst quadrant where the tangent is negative st becomes tan a 270 cot 01 cos a 450 1 sin a Again assumin that 01 450 is in the fourth quadrant no minus sign is needed Hence the formi cos a 45C sec180 04 1 sec at Heren2 is ei mine the sign again assume a value ofa i second quadrant where the secant is neg of a is sec180 a Fundamental Identities Where a double sign appears in the follo39 quadrant in which the angle terminates Reciprocal relations 1 sec at 1 cos a ta sina cosa csca SCCOZ csc a sm a Product relations cos a cot a s cot a COS a c csc a sec 01 0 sin a tan a cos a tan a sin a see a sec 01 csc a tan a Quotient relations tan a cot a sma 005a t sec at csc a see a csc a csc a sec or tan a cot a a Figure 51 6 A hole in a composite area is treated as a negative area The first second and last terms are easily computed using known preperties of simple shapes Therefore the centroid of the composite body can be found from chc cht chr chh AC At Ar Ah But these equations are identical to Eq 5 13 if the area of the hole is considered to be a negative quantity Similar equations can be developed for composite lines volumes masses and weights The final results would show the A39s of Eqs 5 13 replaced with L s V39s m s and W39s respectively Tables 51 and 52 contain a listing of centroid locations for some common shapes 211 Z J CL CVEN 221 CLASS NUMBER DATE PAGE 5 Mg 0 am mj awasmgtgmg I I m FEW 2 5Qt V 3 03Q3m A ZOLff 2A 2 0067 h 735t51 CMgtltO 6 Z 28qWx2 a g af x ryz 39 g quot 39 g Akx la 332i KLJVAKV M g 92135 mg 31 WM WWWWMW W 1 12 a k W 5 43 W g fgmw jag56 a Li z w Q y ia 3 I 7 I V g K g m ww Ofng i f f im i C 117W MJE 31 O qmlOamp EEC E Maw4 lt23 1 W M May 123 6 mos Mia WW V2 553 Locate the center of gravity of the bracket shown 554 Locate the center of mass of the machine component a Fig P553 if the holes have 6in diameters shown in Fig P554 The brass p 8750 kgm3 disk C is mounted on the steel p 7870 kgm3 shaft B Fig P553 55 THEOREMS OF PAPPUS AND GULDINUS 5quot Two theorems stated by Pappus3 and Guldinus4 before the develop ment of calculus can be used to determine the surface area generated by revolving a plane curve or the volume generated by revolving an area about an axis that does not intersect any part of the plane curve or area Applications of the theorems require use of the equations pre Viously developed for locating the centroids of lines and areas X Theorem 1 The area A of a surface of revolution generated by revolving a plane curve of length L about any nonintersecting axis in its plane is equal to the product of the length of the curve and the length of the path traveled by the centroid of the curve If the curve AB of Fig 5 19 is revolved about the y axis the in crement of surface area dA generated by the element of length dL of line AB is 031 2m dL A jci A Thus the total surface area A generated by revolving the line AB about the yaxis is Figure 519 Determining the surface area of a body of revolution by Eq 5 11 as m M quot KN mg l g 3Pappus of Alexandria about AD 380 a Greek geometer 39 4Paul Guldin 1577 1643 a Swiss mathematician T v V t mmnwmmswwrm Cw szc vw m rlwmdm sqwigwaw W 221 Thus the surface area A generated by revolving the line AB about the 55 THEOREMS 0F pAppus y axis is WWMMW AND GULDINUS A ZwZCLL l 514 l V i Where 2772a is the distance ra i39i l d mtroid C of the line L in generating the surface area A The theorem is also valid if the line AB is rotated through an angle 0 other than 2w radians Thus for any an gle of rotation 0 0 S 6 S 277 the surface area A generated is A GZCLL 515 Surface areas of a wide variety of shapes can be determined by us ing Theorem 1 Examples include the surface area of the cylinder cone sphere torus ellipsoid and any other surface of revolution for which it if a g 7 the generating line can be described and the location of its centroid de 5 339 E g termined a c Theorem 2 The volume V of the solid of revolution generated by re 69 volving a plane area A about any nonintersecting axis the axis may be a boundary of area A but may not pass through the area in its plane is equal to the product of the area and the length of the path traveled by the cen troid of the area If the plane area A of Fig 520 is revolved about the yaxis the in crement of volume dV generated by the element of area dA is W ands Thus the total volume V generated by revolving the area A about the yaxis is Figure 520 Determining the volume of a body of revolution The coordinate zCA of the centroid C of an iea in the yzplane is given 39 w by Eq 510 as Ag c 39 1 If i A I U Thus the volume V generated by revolving the area A about the yaxis is V 2ch A 516 where 217201 is the distance traveled by the centroid C of the area A in generating the volume V The theorem is also valid if the area A is ro tated through an angle 0 other than 2 radians Thus for any angle of rotation 9 0 S 9 S 27 the volume V generated is V GZCA A 517 The following examples illustrate the procedure for determining surface areas and volumes for solids of revolution by using the theo rems of Pappus and Guldinus CVEN 221 CLAiS NUMBER DATE PAGE OiU 7 v i 1 L f W WW 3 f f g 5 I K 3 SA M H 3 an 1quot EQxESgN 35 1 Cf 39 I 12 jg 5 x rw kw gig a f QwL ggm I CVEN 221 CLASS NUMBER DATE PAGE I MJWM x L 39 I 1 a my A mm gwwimwhw 2 a 31 r is L x a A was X 1525 W x M a iwg 39 3 WM UM8 39Zm zggi mggcgagesia ZEEX b 533 V l39o f w h 353 5 M 5 Q CVEN 221 CLASS NUMBER DATE PAGE i r w w m cwCidw x M A lt3 a m aj 5 W E 39 e if W wayi lt nggg 0 g 5 u D 5 9 r A 5quot w M V g i v 3 2 V a 2 aw9A A A F v i p g E gage L gi f iwkwgi g We w W 3 W g2 me quot r 4 m 4 7i U M m W 3 MM CVEN 221 CLASS NUMBER DATE PAGE 3393 x 2 MMQ 200 832352 13 66mm gt quot Wf Ex w 363 E WEED Wg KGr0 2 WSSCDw fquot 3W 3 ngggwt 20 MW XQ W e88 M mq w 7qu if E2 LE Z ngfwj g 6 Mg We Ada L 250 S Q Q7j AID9 3 3 5073m H0g 01017 m 73 H 9137 M 7 39 2

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