January : 2016 (CBCS)

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1 1 Engineeing Mecanics 1 st Semeste : ommon to all ances Janua : 16 (S) Note : Ma. maks : 6 (i) ttempt an five questions. (ii) ll questions ca equal maks. (iii) ssume suitable data o dimensions, if necessa, cleal mentioned it. Q.1 (a) Wat is engineeing mecanics? lassif te Engineeing mecanics and biefl eplain tem. ns. Engineeing mecanics : Engineeing mecanics descibes te beavio of a bod, in eite a beginning state of est o of motion of igid bod, subjected to te action of foces. Of couse, engineeing mecanics is an integal component of te education of enginees wose disciplines ae elated to te mecanical beavio of bodies o fluids. Suc beavio is of inteest to aeonautical, civil, cemical, electical, mecanical, metallugical, mining and tetile enginees. sound taining in engineeing mecanics continues to be one of te most impotant aspects of engineeing education due to te intedisciplina caacte of man engineeing applications (e.g. spacesip, obotic and manufactuing). It is appopiate to conclude tat te subject of engineeing mecanics is te coe of all engineeing analsis. lassification of engineeing mecanics : Engineeing mecanics ma be divided into following two main bances : 1. Statics. Dnamics Engineeing Mecanics Statics Dnamics Kinetics Kinematics 1. Static : It is tat banc of engineeing mecanics, wic deals wit te foces and tei effects, wile acting upon te bodies at est.. Dnamics : It is tat banc of engineeing mecanics, wic deals wit foces and tei effects, wile acting upon bod in motion. Dnamics ma be fute subdivided into te following sub bances : (i) Kinetics : It is tat banc of dnamics wic deals wit te bodies in motion due to te application of foces. (ii) Kinematics : It is tat banc of dnamics wic deals wit te bodies in motion. Witout an efeence to te foces wic ae esponsible fo te motion. Te stud of kinematics assimilates tems suc as displacement, velocit, acceleation, etadation etc. wic ae impotant to an enginee in te design of moving pat of a macine. Q.1 (b) Te esultant of te two foces, wen te act an angle of 6 is 14 N. If te same foces ae acting at igt angles, tei esultant is 1 N. Detemine te magnitude of two foces.

2 Engineeing Mecanics 1 st Semeste : ommon to all ances ns. Given : 1 14 N, 1 N 1 =14N 1 N =6 1 =9 1 ig.(a) Wen foce acts at 6 wit eac ote esultant is 14 N : ppling law of paallelogam, fom ig.(a), we get cos cos ig.(b) (i) Wen foce acts at 9 wit eac ote tei esultant is 1 N : ppling law of paallelogam, fom ig.(b), cos cos om equation (ii) and (i), we get (ii) (iii) lso, we can wite, ( ) om equation (ii) and (iii), we get ( ) ( 1) 48 ab a b ab ( ) 1 48 (iv) lso, we can wite, ( 1) 1 1 ( ab) a b ab ( ) ( 1) 4 (v) 1 4 dding equation (iv) and equation (v), we get N ns. om equation (iv), we get N ns. Q. (a) State te law of paallelogam of foces and sow tat te esultant, P Q wen te two foces P and Q ae acting at igt angles to eac ote. ind te value of if te angle between te foces is zeo.

3 Engineeing Mecanics 1 st Semeste : ommon to all ances ns. Law of paallelogam of foces : If two foce acting at a point ae epesented in magnitude and diection b te adjacent sides of a paallelogam ten te diagonal passing toug tei point of intesection epesents te esultant in bot magnitude and diection. Poof : Let te foces P and Q act at O. Let O and O epesent te foces P and Q acting at an angle. omplete te paallelogam O. Daw D pependicula to O. Let O, O denote te magnitude of te esultant and is te diection of esultant. Q om D, Now, cos D D cos D Q cos ig. OD O D O cos OD P Q cos [ O Q] lso, D sin Q sin esultant : i.e. Diection of esultant : O O OD D ( PQcos ) ( Qsin ) cos P Q PQ P Q PQ cos (i) D Qsin tan OD P Q cos P Equation (i) and equation (ii) give te equied magnitude and diection of te esultant. D (ii) Q. (b) Te fou foces 1,, and 4 of magnitudes 18 N, N, 15 N and N and diections, 45, 9, 1 (counte clock wise fom oizontal) espectivel acting at a point on a bod. Detemine te magnitude and diection of te foce fo equilibium condition of te bod. ns. Given :,, 1 and 4 of magnitudes 18 N, N, 15 N and N and diections 1 (counte clockwise fom oizontal)., 45, 9,

4 4 Engineeing Mecanics 1 st Semeste : ommon to all ances D : 15 N N N esolving te foces : 4 N sin N N cos cos N Magnitude and diection of te foce fo equilibium condition = Equilibant 18 cos 45 cos 6.161N 15 sin 45 sin N Magnitude of esultant foce () = Magnitude of equilibant (E) Diection of esultant, E N 1 tan N tan N sin So, diection of equilibant ns. Q. (a) Define and eplain te tems pincipal of equilibium, foces law of equilibium and moment law of equilibium. ns. Te bod is said to be in equilibium if te esultant of all foces acting on it is zeo. Tee ae two majo tpes of static equilibium, namel, tanslational equilibium i.e., (also known as, foces law of equilibium) and otational equilibium i.e. M (also known as, Moment law of equilibium) 18 E= Equilibant esultant 6.161N

5 5 Engineeing Mecanics 1 st Semeste : ommon to all ances 1. o coplana concuent foces (tanslational equilibium) :,. o coplana non concuent foces (tanslational and otational equilibium) :,, M Q. (b) lamp weiging 1 N is suspended fom te ceiling b a cain. It is pulled a side b a oizontal cod until te cain makes an angle of 6 wit te ceiling. ind te tension in te cain and te cod b appling Lami s teoem. ns. Given : Weigt of lamp = 1 N 6 ppling Lami s teoem : 1N T 1 T T T 1 sin9 sin15 sin1 1 1 sin1 T1 sin N T sin15 1 sin1 1 sin1 T sin N ns. ns. Q.4 (a) Eplain te vaious tpes of beams. Wat ae te diffeent tpes of suppot and loading on a beam eplain in bief? ns. 1N Tpes of suppots Hinged o pinned suppots olle suppots Knife edge suppots ied suppots (i) Hinged o Pinned suppot : Te essential featue of a pinned suppot is to pevent tanslation at te end of a beam but does not pevent otation. Tus, an end caing inged o pinned suppot cannot move in vetical o oizontal diection but, it can otate aound te inged. pin o tin suppot is capable of developing a eaction foce wit oizontal and vetical components ( and ), but cannot develop a moment eaction. Hence it can esist foce acting in an diection of plane but cannot esist otation.

6 6 Engineeing Mecanics 1 st Semeste : ommon to all ances eam Pin Hinged o pinned suppot ig.1 : Hinged o Pinned suppot (ii) olle suppot o Link suppot : olle suppot is capable of esisting foce onl in one specified line of action. eam eam eam D Link olle olles 9 (a) (b) ig. : olle suppot o Link suppot Te link sown in ig.(a) can esist foce onl along diection. olle in ig.(b) can esist foce onl in vetical diection. Weeas olle in ig.(c) can esist foces acting pependicula to line D. It sould be noted tat a olle suppot is capable of esisting a foce in eite diections (i.e. upwad and downwad) i.e. a beam is not allowed to lift off fom a suppot. (iii) Knife edge suppot : Te function of knife edge suppot is simila to olle suppot. It is also capable of esisting foce in onl one specified line of action, but ee beam can be lifted up o it ma leave te suppot. eam ig. : Knife edge suppot (iv) ied suppot : ied suppot is a suppot wic esists tanslation of beam as well as otation of beam. ig.4 : ied suppot Suppot eaction is te foce wic is offeed b te suppot, against te loads acting on te beam. Te suppot eaction fo vaious tpes of suppots is sown below. (i) Hinged o pinned suppot : ied end L ee end eam eam (esultant) ig.(a) : Hinged o pinned suppot

7 7 Engineeing Mecanics 1 st Semeste : ommon to all ances (ii) olle suppot : eam eam (iii) Knife edge suppot : ig.(b) : olle suppot eam eam ig.(c) : Knife edge suppot (iv) ied suppot : M eam eam ied end ig.(d) : ied suppot Note : Numbe of eaction at an suppot = Numbe of esticted motion. Eample : s inged suppot can esist foce acting at an diection in a given plane, teefoe it as two component of tat eaction foce ( and ). eam is classified into diffeent categoies depending upon te tpes of suppot as below : (i) Simpl suppoted beam : Simpl suppoted beam is a beam wic is suppoted at its two ends b eite two olle suppot o two inge suppot o one olle and one inged suppot. ig.(a) : Simpl suppoted beam (ii) Oveanging beam : n oveanging beam is a beam, simpl suppoted at an two point witin te span of beam but it also pojects beond one o bot te suppot. a L ig.(b) : Oveanging beam (iii) antileve beam : antileve beam is a beam wic is fied at one end and fee at ote. t fied suppot beam can neite tanslate no otate, weeas at fee end it ma do bot. ied ee end end L ig.(c) : antileve beam (iv) ied beam : Wen te bot ends of te beam ae built in o fied, suc beam is called as fied beam. ied end b ig.(d) : ied beam ied end

8 8 Engineeing Mecanics 1 st Semeste : ommon to all ances (v) ontinuous beam : It is a beam wic epands ove moe tan two suppots. eam ig.(e) : ontinuous beam (vi) Popped cantileve beam : cantileve beam wic as a suppot between fied end and fee end is called popped cantileve beam. ied end L ig.(f) : Popped cantileve beam Popped end Q.4 (b) beam of 6 m long in simpl suppoted at te ends and caies a unifoml distibuted load of 1.5 kn/m and tee concentated loads 1 kn, kn and kn acting espectivel at a distance of 1.5 m, m and 4.5 m fom te left end. Detemine te eacting at bot ends. ns. D : alculation fo eaction : ppling condition of equilibium, onside,,, and ae te eactions at suppot and espectivel. (No oizontal foce on beam) (i) Taking moments about suppot, we get, M om equation (i), we get kN ns kn ns. 1.5m 1kN kn kn D E m 4.5m 6m 1.5kN/m Q.5 (a) Deive an epession fo te moment of inetia of a tiangula section about an ais passing toug te.g. of te section and paallel to te base. ns. Moment of inetia of a tiangle wit base widt b and eigt is to be detemined about te base.

9 9 Engineeing Mecanics 1 st Semeste : ommon to all ances b d b onside an elemental stip at distance fom te base. Let d be te tickness of te stip and d its aea. Widt of tis stip is given b : b' b [ similait of tiangle] (i) Moment of inetia of tis stip about I om equation (i), we get d bd ' I bd I I I o bd 4 b 4 4 b 4 o o bd b I ns. 1 It is clea tat te centoidal ais will be paallel to base ence fom paallel ais teoem, I I Hee, is te distance between base and centoidal ais - and is equal to. b 1 b I b I 1 18 Moment of inetia about centoidal ais b b b I ns

10 1 Engineeing Mecanics 1 st Semeste : ommon to all ances Q.5 (b) Detemine te ente of gavit of te L section sown in ig.1. mm 1 mm mm ns. 6 mm ig.1 Te given composite figue can be divided into two ectangles. Selecting te efeence ais and as sown is figue. mm 8 mm 1 c 18.57, 8.57 coodinate of centoid : mm 6 mm mm, 1 1 mm, 1 6 mm 6 1 mm, mm, 1 mm coodinate of centoid : mm mm 16 1 Hence, te centoid is c(18.57, 8.57) as sown in figue above. Q.6 (a) State te Newton s law of motion and Gavitation, also eplain te vaious tems used in dnamics. ns. ccoding to Newton s law of gavitation, 1. Eve mass attacts eve ote mass.. ttaction is diectl popotional to te poduct of tei masses. g mm (i) 1. ttaction is invesel popotional to te squae of te distance between tei centes. g 1 (ii) d

11 11 Engineeing Mecanics 1 st Semeste : ommon to all ances m1 mm m 1 g G d Gavitational foce, mm G d 1 d Wee, G is known as univesal gavitational constant. 11 Te value of G is m /kg-s, mass of te eat 6 d m, we obtain g 9.84 m/s. 4 m kg and distance of te eat Newton s laws consist of te law of inetia, te law of motion, and te law of action and eaction. Newton s fist law (te law of inetia) : Te law states tat; bod will emain at est o in unifom motion in a staigt line unless it is compelled to cange tis state b foces impessed upon it. Te fist law depicts tat if tee is no etenal effect, an object must be still o moving at a constant velocit, wic leads to a concept of te net foce. Newton s second law (te law of motion) : Tis law states tat; bod acted upon b an etenal unbalanced foce will acceleate in popotion to te magnitude of tis foce in te diection in wic tis foce acts. Te second law quantifies te foce in tems of acceleation and mass of te object. ma Newton s tid law (te law of action and eaction) : Tis law states tat, o eve action (o foce) tee is an equal and opposite eaction (o foce). Te tid law illustates te eistence of te counte foce wic is elated to nomal foces, tension, etc. Vaious tems elated to dnamics : 1. Displacement : It is te sotest distance fom te initial to te final position of a point. It is a vecto quantit. Displacement = inal position Initial position Distance Pat taken ig. : Displacement. Distance tavelled : It is te lengt of pat tavelled b a paticle o bod. It is a scala quantit.. Velocit : Velocit is te ate of displacement of a bod wit espect to time. Velocit Displacement Time inteval 4. cceleation : cceleation is te ate of cange of velocit of a bod wit espect to time. cceleation Displacement Velocit Time inteval

12 1 Engineeing Mecanics 1 st Semeste : ommon to all ances 5. veage velocit : It is te aveage value of te given velocities. veage velocit is displacement ove total time. veage velocit t 6. Instantaneous velocit : It is te velocit at a paticula instant of time. It can be obtained fom te aveage velocit b coosing te time inteval t and te displacement. Instantaneous velocit, (o velocit) d v limit t t dt Te unit of velocit is m/s. 7. veage acceleation : Let v be te velocit of te paticle at an time t. If te velocit becomes ( v v) at a late time ( t t) ten, v veage acceleation t 8. Instantaneous acceleation : It is te acceleation of a paticle at a paticula instant of time and can be calculated b coosing te time inteval t and te velocit v. v dv d cceleation a limit. t t dt dt cceleation is position if te velocit is inceasing. Te unit of acceleation is dv d a as v dt dt d m/s. So, a dt dv dv d lso, a dt d dt and as d dt v dv So, a v d 9. Unifom motion : paticle is said to ave a unifom motion wen its acceleation is zeo and its velocit is constant wit espect to time. It also called unifom velocit. v v = constant dv a dt ig. : Unifom Motion 1. Unifoml acceleated motion : paticle moving wit a constant acceleation (a constant wit espect to time) is said to be in unifoml acceleated motion. v t dv a constant dt ig. : Unifoml cceleated Motion t

13 1 Engineeing Mecanics 1 st Semeste : ommon to all ances Q.6 (b) Detemine te foce in membes D, D and E of te tuss sown in ig.. 1 N ns. D : D 6 6 E 1m 1m 1m 1 N ig. 1 N N 1 N G 1 N alculation fo eactions : ppling condition of equilibium,, D 9 N X D 1 N D 6 6 E G E X 1m 1m 1m 4 cos 1 cos G, G 1 cos cos 1 cos N (i) G 1sin sin 1sin Taking moment about G, N cos 1 cos N om equation (i), we get G G Using metod of section : N Initiall conside all te membes unde tension. If te value of foces comes negative afte calculation, it means te foce in membes ae compessive, but we will not cange tei sense in an of te D (let it look like a tensile foce) duing solving, instead of doing tis we will put negative values of espective foces in te equations. utting a section passing toug D, D, E, conside left potion

14 14 Engineeing Mecanics 1 st Semeste : ommon to all ances In, sin 1sin 6m D sin 6 D sin D D cos 6 D E D cos 6 Taking moment about,,, D 1 D N (ompessive) ns. D D cos D cos 6 E D cos 6 E cos (ii) D cos 6 E D sin D sin sin D sin 6 ns. D om equation (ii), we get N(Tensile) ns. E Membe oce Magnitude (N) Natue D D ompessive D D E E Tensile Q.7 (a) Eplain and define te tem fee bod diagam. Daw te fee bod diagam of a ball of weigt W placed on a oizontal suface. ns. If a bod consist of moe tan one element and suppot, eac ten element and suppot can be isolated fom te sstem, suc diagam is called fee bod diagam. One of te most useful aids fo solving a statics poblem is te fee bod diagam (D). fee bod diagam is a gapic, demateialized, smbolic epesentation of te bod (stuctue, element o segment of an element) in wic all connecting ʺpiecesʺ ave been emoved. D is a convenient metod to model te stuctue, stuctual element, o segment tat is unde scutin. It is a wa to conceptualize te stuctue, and its composite elements, so tat an analsis ma be initialized. To daw D of a bod we emove all te suppots (like wall, floo, inge o an ote bod) and eplace tem b te eactions wic tese suppots eet on te bod.

15 15 Engineeing Mecanics 1 st Semeste : ommon to all ances Eample : onside a ball of weigt W esting on a oizontal plane as sown in ig.(a). Te steps to daw te D ae : W W ig.(a) : all esting on a plane ig.(b) : ee bod diagam of ball Step 1 : ist daw te object : Hee ou object is a ball. Step : emove all suppots : emove suface, ee suface will give eaction foce. Step : Sow all te foces : Sow bod foces as well as etenal foces as sown in ig. (b). Q.7 (b) Give te position of centoid of te following standad sections : (i) ectangle (ii) Tiangle (iii) Unifom od ns. entoids of vaious line segments : S. No. 1. Desciption Staigt line o unifom od asic line segment igue L/ L/ L (iv) Semicicle Lengt L Location of centoids ente of te line L. c of a cicle of adius and inscibed angle On te line of smmet at a sin distance of fom te cente of ac. sin. icle of adius ente of te cicle. 4. Semicicula ac of adius O On te line of smmet at a distance of fom te cente.

16 16 Engineeing Mecanics 1 st Semeste : ommon to all ances S. No. 5. Desciption asic line segment igue Lengt Quate cicula ac of adius Location of centoids distance fom cente along one adius and ten fom tat point pependicula to tat adius. entoids of aea of vaious sapes : Plane Sape ea c ectangle b ab b c Tiangle b b/ b/ igt angle b tiangle b Quatecicle O 4 Semi cicle d o 4

17 17 Engineeing Mecanics 1 st Semeste : ommon to all ances Plane Sape ea c c icula Secto O sin Paabola a 4 5 O a k n Geneal b ab cuve n 1 O a n 1 a n n1 b n 1 Hemispee 8 O igt cicula cone 1 4 O

Problem Set 5: Universal Law of Gravitation; Circular Planetary Orbits

Problem Set 5: Universal Law of Gravitation; Circular Planetary Orbits Poblem Set 5: Univesal Law of Gavitation; Cicula Planetay Obits Design Engineeing Callenge: Te Big Dig.007 Contest Evaluation of Scoing Concepts: Spinne vs. Plowe PROMBLEM 1: Daw a fee-body-diagam of a