Rigid Bodies: Equivalent Systems of Forces
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1 Engneeng Statcs, ENGR 2301 Chapte 3 Rgd Bodes: Equvalent Sstems of oces
2 Intoducton Teatment of a bod as a sngle patcle s not alwas possble. In geneal, the se of the bod and the specfc ponts of applcaton of the foces must be consdeed. ost bodes n elementa mechancs ae assumed to be gd,.e., the actual defomatons ae small and do not affect the condtons of equlbum o moton of the bod. Cuent chapte descbes the effect of foces eeted on a gd bod and how to eplace a gven sstem of foces wth a smple equvalent sstem. moment of a foce about a pont moment of a foce about an as moment due to a couple An sstem of foces actng on a gd bod can be eplaced b an equvalent sstem consstng of one foce actng at a gven pont and one couple. 3-2
3 Etenal and Intenal oces oces actng on gd bodes ae dvded nto two goups: - Etenal foces - Intenal foces Etenal foces epesents the acton of othe bodes on the gd bod unde consdeaton. The ae entel esponsble fo the etenal behavo of the gd bod. If unopposed, each etenal foce can mpat a moton of tanslaton o otaton, o both. The ntenal foces ae the foces whch hold togethe the patcles fomng the gd bod. If the gd bod s stuctuall composed of seveal pats, the foce holdng the components pats togethe ae also defned as ntenal foces. 3-3
4 ncple of Tansmssblt: Equvalent oces ncple of Tansmssblt - Condtons of equlbum o moton ae not affected b tansmttng a foce along ts lne of acton. NOTE: and ae equvalent foces. ovng the pont of applcaton of the foce to the ea bumpe does not affect the moton o the othe foces actng on the tuc. ncple of tansmssblt ma not alwas appl n detemnng ntenal foces and defomatons. 3-4
5 ultplng vectos ultplng a vecto wth a vecto: Vecto (Coss) oduct The vecto poduct between two vectos a and b can be wtten as: The ght-hand ule allows us to fnd the decton of vecto c. The esult s a new vecto c, whch s: Hee a and b ae the magntudes of vectos a and b espectvel, and f s the smalle of the two angles between a and b vectos.
6 Vecto oduct of Two Vectos Concept of the moment of a foce about a pont s moe easl undestood though applcatons of the vecto poduct o coss poduct. Vecto poduct of two vectos and s defned as the vecto V whch satsfes the followng condtons: 1. Lne of acton of V s pependcula to plane contanng and. 2. agntude of V s V sn 3. Decton of V s obtaned fom the ght-hand ule. Vecto poducts: - ae not commutatve, - ae dstbutve, - ae not assocatve, S S 3-6
7 Vecto oducts: Rectangula Components 3-7 Vecto poducts of Catesan unt vectos, Vecto poducts n tems of ectangula coodnates V Refe to bottom of page 82 to smplf t futhe
8 oment of a oce About a ont A foce vecto s defned b ts magntude and decton. Its effect on the gd bod also depends on t pont of applcaton. The moment of about O s defned as O The moment vecto O s pependcula to the plane contanng O and the foce. agntude of O measues the tendenc of the foce to cause otaton of the bod about an as along O. O sn d The sense of the moment ma be detemned b the ght-hand ule. An foce that has the same magntude and decton as, s equvalent f t also has the same lne of acton and theefoe, poduces the same moment. 3-8
9 oment of a oce About a ont Two-dmensonal stuctues have length and beadth but neglgble depth and ae subected to foces contaned n the plane of the stuctue. The plane of the stuctue contans the pont O and the foce. O, the moment of the foce about O s pependcula to the plane. If the foce tends to otate the stuctue counteclocwse, the sense of the moment vecto s out of the plane of the stuctue and the magntude of the moment s postve. If the foce tends to otate the stuctue clocwse, the sense of the moment vecto s nto the plane of the stuctue and the magntude of the moment s negatve. 3-9
10 Vagnon s Theoem The moment about a gve pont O of the esultant of seveal concuent foces s equal to the sum of the moments of the vaous moments about the same pont O Vagon s Theoem maes t possble to eplace the dect detemnaton of the moment of a foce b the moments of two o moe component foces of. 3-10
11 Rectangula Components of the oment of a oce 3-11 O The moment of about O, O,
12 Sample oblem 3.1 A 100-lb vetcal foce s appled to the end of a leve whch s attached to a shaft at O. Detemne: a) moment about O, b) hoontal foce at A whch ceates the same moment, c) smallest foce at A whch poduces the same moment, d) locaton fo a 240-lb vetcal foce to poduce the same moment, e) whethe an of the foces fom b, c, and d s equvalent to the ognal foce. 3-12
13 Sample oblem 3.1 a) oment about O s equal to the poduct of the foce and the pependcula dstance between the lne of acton of the foce and O. Snce the foce tends to otate the leve clocwse, the moment vecto s nto the plane of the pape. O d O d 24n. cos lb12 n. 12 n. O 1200 lbn 3-13
14 Sample oblem 3.1 c) Hoontal foce at A that poduces the same moment, d O 24 n. d 1200 lbn n lbn n. sn n lb 3-14
15 Sample oblem 3.1 c) The smallest foce A to poduce the same moment occus when the pependcula dstance s a mamum o when s pependcula to OA. O 1200 lbn. d 24 n lbn. 24 n. 50 lb 3-15
16 Sample oblem 3.1 d) To detemne the pont of applcaton of a 240 lb foce to poduce the same moment, O 1200 lbn. d OB cos60 d 240 lb d 1200 lbn. 240 lb 5 n. 5 n. OB 10 n. 3-16
17 Sample oblem 3.1 e) Although each of the foces n pats b), c), and d) poduces the same moment as the 100 lb foce, none ae of the same magntude and sense, o on the same lne of acton. None of the foces s equvalent to the 100 lb foce. 3-17
18 Scala oduct of Two Vectos 3-18 The scala poduct o dot poduct between two vectos and s defned as esult scala cos Scala poducts: - ae commutatve, - ae dstbutve, - ae not assocatve, undefned S Scala poducts wth Catesan unt components,
19 Scala oduct of Two Vectos: Applcatons Angle between two vectos: cos cos oecton of a vecto on a gven as: OL cos poecton of along OL cos cos OL o an as defned b a unt vecto: OL cos cos cos 3-19
20 ed Tple oduct of Thee Vectos ed tple poduct of thee vectos, S scala esult The s med tple poducts fomed fom S,, and have equal magntudes but not the same sgn, S S S S S S Evaluatng S the med tple poduct, S S S S S S 3-20
21 oment of a Couple Two foces and - havng the same magntude, paallel lnes of acton, and opposte sense ae sad to fom a couple. oment of the couple, A B A B sn d The moment vecto of the couple s ndependent of the choce of the ogn of the coodnate aes,.e., t s a fee vecto that can be appled at an pont wth the same effect. 3-21
22 oment of a Couple Two couples wll have equal moments f 1 d1 2d 2 the two couples le n paallel planes, and the two couples have the same sense o the tendenc to cause otaton n the same decton. 3-22
23 Addton of Couples 3-23 Consde two ntesectng planes 1 and 2 wth each contanng a couple n plane n plane Resultants of the vectos also fom a couple 1 2 R B Vagon s theoem Sum of two couples s also a couple that s equal to the vecto sum of the two couples
24 Couples Can Be Repesented b Vectos A couple can be epesented b a vecto wth magntude and decton equal to the moment of the couple. Couple vectos obe the law of addton of vectos. Couple vectos ae fee vectos,.e., the pont of applcaton s not sgnfcant. Couple vectos ma be esolved nto component vectos. 3-24
25 Resoluton of a oce Into a oce at O and a Couple oce vecto can not be smpl moved to O wthout modfng ts acton on the bod. Attachng equal and opposte foce vectos at O poduces no net effect on the bod. The thee foces ma be eplaced b an equvalent foce vecto and couple vecto,.e, a foce-couple sstem. 3-25
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