1.1. WHAT IS MECHANICS?

Transcription

1 Mateials Engineeing Depatment Class: ist Date : Subect: Engineeing Mechanics Lectue: D. Emad AL-Hassani Lectue # WHAT IS MECHANICS? Mechanics can be defined as that science which descibes and pedicts the conditions of est o motion of bodies unde the action of foces. It is divided into thee pats: mechanics of igid bodies, mechanics of defomable bodies, and mechanics of fluids. The mechanics of igid bodies is subdivided into statics and dnamics, the fome dealing with bodies at est, the latte with bodies in motion. In this pat of the stud of mechanics, bodies ae assumed to be pefectl igid. Actual stuctues and machines, howeve, ae neve absolutel igid and defom unde the loads to which the ae subected. But these defomations ae usuall small and do not appeciabl affect the conditions of equilibium o motion of the stuctue unde consideation. The ae impotant, though, as fa as the esistance of the stuctue to failue is concened and ae studied in mechanics of mateials, which is a pat of the mechanics of defomable bodies. The thid division of mechanics, the mechanics of fluids, is subdivided into the stud of incompessible fluids and of compessible fluids. An impotant subdivision of the stud of incompessible fluids is hdaulics, which deals with poblems involving wate. Mechanics is a phsical science, since it deals with the stud of phsical phenomena. Howeve, some associate mechanics with mathematics, while man conside it as an engineeing subect. Both these views ae ustified in pat. Mechanics is the foundation of most engineeing sciences and is an indispensable peequisite to thei stud. 1

2 Howeve, it does not have the empiicism found in some engineeing sciences, i.e., it does not el on epeience o obsevation alone; b its igo and the emphasis it places on deductive easoning it esembles mathematics. But, again, it is not an abstact o even a pue science; mechanics is an applied science. The pupose of mechanics is to eplain and pedict phsical phenomena and thus to la the foundations fo engineeing applications UNDAMENTAL CONCEPTS AND PRINCIPLES The basic concepts used in mechanics ae space, time, mass, and foce. These concepts cannot be tul defined; the should be accepted on the basis of ou intuition and epeience and used as a mental fame of efeence fo ou stud of mechanics. The concept of space is associated with the notion of the position of a point P. The position of P can be defined b thee lengths measued fom a cetain efeence point, o oigin, in thee given diections. These lengths ae nown as the coodinates of P. To define an event, it is not sufficient to indicate its position in space. The time of the event should also be given. The concept of mass is used to chaacteie and compae bodies on the basis of cetain fundamental mechanical epeiments. Two bodies of the same mass, fo eample, will be attacted b the eath in the same manne; the will also offe the same esistance to a change in tanslational motion. A foce epesents the action of one bod on anothe. It can be eeted b actual contact o at a distance, as in the case of gavitational foces and magnetic foces. A foce is chaacteied b its point of 2

3 application, its magnitude, and its diection; a foce is epesented b a vecto. In Newtonian mechanics, space, time, and mass ae absolute concepts, independent of each othe. (This is not tue in elativistic mechanics, whee the time of an event depends upon its position, and whee the mass of a bod vaies with its velocit.) On the othe hand, the concept of foce is not independent of the othe thee. Indeed, one of the fundamental pinciples of Newtonian mechanics listed below indicates that the esultant foce acting on a bod is elated to the mass of the bod and to the manne in which its velocit vaies with time. You will stud the conditions of est o motion of paticles and igid bodies in tems of the fou basic concepts we have intoduced. B paticle we mean a ve small amount of matte which ma be assumed to occup a single point in space. A igid bod is a combination of a lage numbe of paticles occuping fied positions with espect to each othe. The stud of the mechanics of paticles is obviousl a peequisite to that of igid bodies. Besides, the esults obtained fo a paticle can be used diectl in a lage numbe of poblems dealing with the conditions of est o motion of actual bodies. The stud of elementa mechanics ests on si fundamental pinciples based on epeimental evidence. The Paallelogam Law fo the Addition of oces This states that two foces acting on a paticle ma be eplaced b a single foce, called thei esultant, obtained b dawing the diagonal of the paallelogam which has sides equal to the given foces. 3

4 The Pinciple of Tansmissibilit This states that the conditions of equilibium o of motion of a igid bod will emain unchanged if a foce acting at a given point of the igid bod is eplaced b a foce of the same magnitude and same diection, but acting at a diffeent point, povided that the two foces have the same line of action. Newton s Thee undamental Laws. omulated b Si Isaac Newton in the latte pat of the seventeenth centu, these laws can be stated as follows: IRST LAW If the esultant foce acting on a paticle is eo, the paticle will emain at est (if oiginall at est) o will move with constant speed in a staight line (if oiginall in motion). SECOND LAW If the esultant foce acting on a paticle is not eo, the paticle will have an acceleation popotional to the magnitude of the esultant and in the diection of this esultant foce., this law can be stated as: m a. (1.1) Whee, m, and a epesent, espectivel, the esultant foce acting on the paticle, the mass of the paticle, and the acceleation of the paticle, epessed in a consistent sstem of units. 4

5 THIRD LAW The foces of action and eaction between bodies in contact have the same magnitude, same line of action, and opposite sense. Newton s Law of Gavitation This states that two paticles of mass M and m ae mutuall attacted with equal and opposite foces and (ig. 1.1) of magnitude given b the fomula G (Mm/ 2 ).. (1.2) Whee distance between the two paticles G univesal constant called the constant of gavitation Newton s law of gavitation intoduces the idea of an action eeted at a distance and etends the ange of application of Newton s thid law: the action and the eaction in ig. 1.1 ae equal and opposite, and the have the same line of action. ig: (1.1) A paticula case of geat impotance is that of the attaction of the eath on a paticle located on its suface. The foce eeted b the eath 5

6 on the paticle is then defined as the weight W of the paticle. Taing M equal to the mass of the eath, m equal to the mass of the paticle, and equal to the adius R of the eath, and intoducing the constant g GM (1.3) The magnitude W of the weight of a paticle of mass m ma be epessed as W mg. (1.4) The value of R in fomula (1.3) depends upon the elevation of the point consideed; it also depends upon its latitude, since the eath is not tul spheical. The value of g theefoe vaies with the position of the point consideed. As long as the point actuall emains on the suface of the eath, it is sufficientl accuate in most engineeing computations to assume that g equals 9.81 m/s 2 o 32.2 ft/s SYSTEMS O UNITS With the fou fundamental concepts intoduced in the peceding section ae associated the so-called inetic units, i.e., the units of length, time, mass, and foce. These units cannot be chosen independentl if Eq. (1.1) is to be satisfied. Thee of the units ma be defined abitail; the ae then efeed to as basic units. The fouth unit, howeve, must be chosen in accodance with Eq. (1.1) and is efeed to as a deived unit. Kinetic units selected in this wa ae said to fom a consistent sstem of units. 6

7 Intenational Sstem of Units (SI Units) In this sstem, which will be in univesal use afte the United States has completed its convesion to SI units, the base units ae the units of length, mass, and time, and the ae called, espectivel, the mete (m), the ilogam (g), and the second (s). All thee ae abitail defined. The second, which was oiginall chosen to epesent 1/ of the mean sola da, is now defined as the duation of ccles of the adiation coesponding to the tansition between two levels of the fundamental state of the cesium-133 atom. The mete, oiginall defined as one ten-millionth of the distance fom the equato to eithe pole, is now defined as wavelengths of the oange-ed light coesponding to a cetain tansition in an atom of pton-86. The ilogam, which is appoimatel equal to the mass of m 3 of wate, is defined as the mass of a platinum-indium standad ept at the intenational Bueau of Weights and Measues at Seves, nea Pais, ance. The unit of foce is a deived unit. It is called the Newton (N) and is defined as the foce which gives an acceleation of 1 m/s 2 to a mass of 1 g (ig. 1.2). om Eq. (1.1) we wite 1 N (l g) (l m/s 2 ) l g.m/s 2.. (1.5) 7

8 The SI units ae said to fom an absolute sstem of units. This means that the thee base units chosen ae independent of the location whee measuements ae made. The mete, the ilogam, and the second ma be used anwhee on the eath; the ma even be used on anothe planet. The will alwas have the same significance. The weight of a bod, o the foce of gavit eeted on that bod, should, lie an othe foce, be epessed in newtons. om Eq. (1.4) it follows that the weight of a bod of mass 1 g (ig. 1.3) is W mg (1 g) (9.81 m/s 2) 9.81N Multiples and submultiples of the fundamental SI units ma be obtained though the use of the pefies defined in Table 1.1. The multiples and submultiples of the units of length, mass, and foce most fequentl used in engineeing ae, espectivel, the ilomete (m) and 8

9 the millimete (mm); the megagam (Mg) and the gam (g); and the ilonewton (N). Accoding to Table 1.1, we have 1 m 1000 m 1 mm m l Mg 1000g 1 g g 1N 1000 N The convesion of these units into metes, ilogams, and newtons, espectivel, can be effected b simpl moving the decimal point thee places to the ight o to the left. o eample, to convet 3.82 m into metes, one moves the decimal point thee places to the ight: 3.82 m 3820 m Similal, 47.2 mm is conveted into metes b moving the decimal point thee places to the left: 47.2 mm m 9

10 Table (1.1): SI Pefies Multiplication acto Petit Smbol tea T giga G mega M ilo hecto h dea da deci d centi c milli m mico µ nano n pico p femto f atto a Using scientific notation, one ma also wite 3.82 m m 47.2 mm The multiples of the unit of time ae the minute (min) and the hou (h). Since 1 min 60 s and 1 h 60 min 3600 s, these multiples cannot be conveted as eadil as the othes. 10

11 B using the appopiate multiple o submultiple of a given unit, one can avoid witing ve lage o ve small numbes. o eample, one usuall wites m athe than m, and 2.16 mm athe than m. Units of Aea and Volume The unit of aea is the squae mete (m 2 ), which epesents the aea of a squae of side 1 m; the unit of volume is the cubic mete (m 3 ), equal to the volume of a cube of side 1 m. In ode to avoid eceedingl small o lage numeical values in the computation of aeas and volumes, one uses sstems of subunits obtained b espectivel squaing and cubing not onl the millimete but also two intemediate submultiples of the mete, namel, the decimete (dm) and the centimete (cm). Since, b definition, 1 dm 0.1 m 10 1 m 1 cm 0.01 m 10 2 m 1 mm m 10 2 m The submultiples of the unit of aea ae 1dm 2 (1dm) 2 (10 1 ) m 2 1 cm 2 (1 cm) 2 (10 2 m) m 2 1 mm 2 (1 mm) 2 (10 3 m) m 2 and the submultiples of the unit of volume ae 1dm 3 (1 dm) 3 (10 1 m) m 3 11

12 1 cm3 (1 cm) 3 (10 2 m) m 3 1 mm 3 (1 mm) 3 (10 3 m) m 3 It should be noted that when the volume of a liquid is being measued, the cubic decimete (dm 3 ) is usuall efeed to as a lite (L). Othe deived SI units used to measue the moment of a foce, the wo of a foce, etc., ae shown in Table 1.2. While these units will be intoduced in late chaptes as the ae needed, we should note an impotant ule at this time: When a deived unit is obtained b dividing a base unit b anothe base unit, a pefi ma be used in the numeato of the deived unit but not in its denominato. o eample, the constant of a sping which stetches 20 mm unde a load of 100 N will be epessed as 100 N/20 mm l00n /0.020 m 5000 N/m O 5 N /m but neve as 5 N/mm. Table (1.2): Pincipal SI Units Used in Mechanics Quantit Unit Smbol omula Acceleation Mete pe second squaed. m/s 2 Angle Radian ad * Angula acceleation Radian pe second squaed. ad/s 2 Angula velocit Radian pe second. Rad/s 12

13 Aea Squae mete. m 2 Densit Kilogam pe cubic mete. g/m 3 Eneg Joule J N. m oce Newton N Kg. m/s 2 equenc Het H s 1 Impulse Length Mass Moment of a foce Newton-second Mete Kilogam Newton-mete.. Kg. m/s m g... N. m Powe Watt W J/s Pessue Stess Time Velocit Volume Solids Liquids Pascal Pascal Second Mete pe second Cubic mete Lite Pa N/m 2 Pa N/ m 2 s. m/s. m 3 L 10 3 m 3 Wo Joule J N. m *Supplementa unit (1 evolution 2π ad 360 ο ). 13

14 U.S. Customa Units Most pacticing Ameican enginees still commonl use a sstem in which the base units ae the units of length, foce, and time. These units ae, espectivel, the foot (ft), the pound (lb), and the second (s). The second is the same as the coesponding SI unit. The foot is defined as m. The pound is defined as the weight of a platinum standad, called the standad pound, which is ept at the National Institute of Standads and Technolog outside Washington, the mass of which is g. Since the weight of a bod depends upon the eath s gavitational attaction, which vaies with location, it is specified that the standad pound should be placed at sea level and at latitude of 45 to popel define a foce of 1 lb. Cleal the U.S. customa units do not fom an absolute sstem of units. Because of thei dependence upon the gavitational attaction of the eath, the fom a gavitational sstem of units. While the standad pound also seves as the unit of mass in commecial tansactions in the United Sates, it cannot be so used in engineeing computations, since such a unit would not be consistent with the base units defined in the peceding paagaph. Indeed, when acted upon b a foce of 1 lb, that is, when subected to the foce of gavit, the standad pound eceives the acceleation of gavit, g 32.2 ft/s 2 (ig. 1.4), not the unit acceleation equied b Eq. (1.1). The unit of mass consistent with the foot, the pound, and the second is the mass which eceives an acceleation of 1 ft/s 2 when a foce of 1 lb is applied to it (ig. 1.5). This unit, sometimes called a slug, can be deived fom the equation ma afte substituting 1 lb and 1 ft/s 2 fo and a, espectivel. We wite 14

15 ma 1 lb (1slug) (1ft/s 2 ) and obtain l slug l lb / l (ft/s 2 ) 1 lb.s 2 /ft.. (1.6) Compaing igs. 1.4 and 1.5, we conclude that the slug is a mass 32.2 times lage than the mass of the standad pound. The fact that in the U.S. customa sstem of units bodies ae chaacteied b thei weight in pounds athe than b thei mass in slugs will be a convenience in the stud of statics, whee one constantl deals with weights and othe foces and onl seldom with masses. Howeve, in the stud of dnamics, whee foces, masses, and acceleations ae involved, the mass m of a bod will be epessed in slugs when its weight W is given in pounds. Recalling Eq. (1.4), we wite m w/ g.. (1.7) Whee g is the acceleation of gavit (g 32.2 ft/s 2 ). 15

16 Othe U.S. customa units fequentl encounteed in engineeing poblems ae the mile (mi), equal to 5280 ft; the inch (in.), equal to (1/12) ft; and the ilo pound (ip), equal to a foce of 1000 lb. The ton is often used to epesent a mass of 2000 lb but, lie the pound, must be conveted into slugs in engineeing computations. The convesion into feet, pounds, and seconds of quantities epessed in othe U.S. customa units is geneall moe involved and equies geate attention than the coesponding opeation in SI units. If, fo eample, the magnitude of a velocit is given as v 30 mi/h, we convet it to ft/s as follows. ist we wite v 30 mi/h Since we want to get id of the unit miles and intoduce instead the unit feet, we should multipl the ight-hand membe of the equation b an epession containing miles in the denominato and feet in the numeato. But, since we do not want to change the value of the ight-hand membe, the epession used should have a value equal to unit. The quotient (5280 ft)/ (1 mi) is such an epession. Opeating in a simila wa to tansfom the unit hou into seconds, we wite v (30 mi/h) (5280ft/1mi) (1 h/3600s) Caing out the numeical computations and canceling out units which appea in both the numeato and the denominato, we obtain v 44 ft /s 16

17 1.4. CONVERSION ROM ONE SYSTEM O UNITS TO ANOTHER Thee ae man instances when an enginee wishes to convet into SI units a numeical esult obtained in U.S. customa units o vice vesa. Because the unit of time is the same in both sstems, onl two inetic base units need be conveted. Thus, since all othe inetic units can be deived fom these base units, onl two convesion factos need be emembeed. Units of Length B definition the U.S. customa unit of length is 1 ft m. (1.8) It follows that 1 mi 5280 ft 5280(0.3048m) 1609 m o 1 mi m (1.9) also 1 in. 1/12 ft 1/12 ( m) m O 1 in mm. (1.10) Units of oce Recalling that the U.S. customa unit of foce (pound) is defined as the weight of the standad pound (of mass g) at sea level and at a latitude of 45 (whee g m /s 2 ) and using Eq. (1.4), we wite W mg 1 lb ( g)(9.807 m/s 2 ) g m/s 2 17

18 o, ecalling Eq. (1.5), l Ib 4.448N (1.11) Units of Mass The U.S. customa unit of mass (slug) is a deived unit. Thus, using Eqs. (1.6), (1.8), and (1.11), we wite l slug 1lb.s 2 /ft 1 Ib/ 1 ft/s N/ m/s 2 and, ecalling Eq. (1.5), 1 slug 1 Ib. s 2 /ft g.. (1.12) Although it cannot be used as a consistent unit of mass, we ecall that the mass of the standad pound is, b definition, 1 pound mass g (1.13) This constant ma be used to detemine the mass in SI units (ilogams) of a bod which has been chaacteied b its weight in U.S. customa units (pounds). To convet a deived U.S. customa unit into SI units, one simpl multiplies o divides b the appopiate convesion factos. o eample, to convet the moment of a foce which was found to be M 47 lb. in. into SI units, we use fomulas (1.10) and (1.11) and wite M 47 lb. in. 47(4.448 N)(25.4 mm) 5310 N. mm 5.31 N. m 18

19 The convesion factos given in this section ma also be used to convet a numeical esult obtained in SI units into U.S. customa units. o eample, if the moment of a foce was found to be M 40 N. m, we wite, following the pocedue used in the last paagaph of Sec. 1.3, M 40N.m (40N.m)( 1 Ib/4.448 N) ( 1 ft / m ) Caing out the numeical computations and canceling out units which appea in both the numeato and the denominato, we obtain M 29.5 lb.ft The U.S. customa units most fequentl used in mechanics ae listed in Table 1.3 with thei SI equivalents. 19

20 20

21 Mateials Engineeing Depatment Class: ist Date : Subect: Engineeing Mechanics Lectue: D. Emad AL-Hassani Lectue # 2 Intoduction The obective fo the cuent chapte is to investigate the effects of foces on paticles: - eplacing multiple foces acting on a paticle with a single equivalent o esultant foce, Relations between foces acting on a paticle that is in a state of equilibium The focus on paticles does not impl a estiction to miniscule bodies. Rathe, the stud is esticted to analses in which the sie and shape of the bodies is not significant so that all foces ma be assumed to be applied at a single point. Resultant of Two oces.oce: action of one bod on anothe; chaacteied b its point of application, magnitude, line of action, and sense Epeimental evidence shows that the combined effect of two foces ma be epesented b a single esultant foce. The esultant is equivalent to the diagonal of a paallelogam which contains the two foces in adacent legs. oce is a vecto quantit 1

22 Vectos Vecto: paamete possessing magnitude and diection which add accoding to the paallelogam law. Eamples: displacements, velocities, acceleations. Scala: paamete possessing magnitude but not diection. Eamples: mass, volume, tempeatue Vecto classifications: - ied o bound vectos have well defined points of application that cannot be changed without affecting an analsis. - ee vectos ma be feel moved in space without changing thei effect on an analsis. - Sliding vectos ma be applied anwhee along thei line of action without affecting an analsis. Equal vectos have the same magnitude and diection. Negative vecto of a given vecto has the same magnitude and the opposite diection 2

23 Addition of Vectos Tapeoid ule fo vecto addition Tiangle ule fo vecto addition Law of cosines, R P Q 2PQ cos B R P Q B C C Law of sines, sin A Q sin B R sin C A B Vecto addition is commutative P Q Q P Vecto subtaction Addition of thee o moe vectos though epeated application of the tiangle ule 3

24 The polgon ule fo the addition of thee o moe vectos. Vecto addition is associative, P Q S ( P Q) S P ( Q S ) Multiplication of a vecto b a scala Resultant of Seveal Concuent oces Concuent foces: set of foces which all pass though the same point. A set of concuent foces applied to a paticle ma be eplaced b a single esultant foce which is the vecto sum of the applied foces. Vecto foce components: two o moe foce vectos which, togethe, have the same effect as a single foce vecto. 4

25 Sample Poblem The two foces act on a bolt at A. Detemine thei esultant. SOLUTION: Gaphical solution - constuct a paallelogam with sides in the same diection as P and Q and lengths in popotion. Gaphicall evaluate the esultant which is equivalent in diection and popotional in magnitude to the diagonal. Tigonometic solution - use the tiangle ule fo vecto addition in conunction with the law of cosines and law of sines to find the esultant. Gaphical solution - A paallelogam with sides equal to P and Q is dawn to scale. The magnitude and diection of the esultant o of the diagonal to the paallelogam ae measued, R 98 N 35 5

26 Gaphical solution - A tiangle is dawn with P and Q head-to-tail and to scale. The magnitude and diection of the esultant o of the thid side of the tiangle ae measued, R 98 N α 35 Tigonometic solution - Appl the tiangle ule. om the Law of Cosines, R 2 P 2 Q 2 2PQ cos B 2 2 ( 40N) ( 60N) 2( 40N)( 60N) cos155 R 97.73N om the Law of Sines, sin A Q sin A sin B R Q sin B R sin155 A α 20 A 60N 97.73N α

27 Rectangula Components of a oce: Unit Vectos Ma esolve a foce vecto into pependicula components so that the esulting paallelogam is a ectangle. and ae efeed to as ectangula vecto components and Define pependicula unit vectos i and which ae paallel to the and aes. Vecto components ma be epessed as poducts of the unit vectos with the scala magnitudes of the vecto components. and ae efeed to as the scala components of 7

28 8 Addition of oces b Summing Components Wish to find the esultant of 3 o moe concuent foces, Resolve each foce into ectangula components The scala components of the esultant ae equal to the sum of the coesponding scala components of the given foces and to find the esultant magnitude and diection, S Q P R ( ) ( ) S Q P i S Q P S i S Q Q i P P i R i R S Q P R S Q P R R R R R R tan θ

29 Sample Poblem ou foces act on bolt A as shown. Detemine the esultant of the foce on the bolt. SOLUTION: Resolve each foce into ectangula components. Detemine the components of the esultant b adding the coesponding foce components. Calculate the magnitude and diection of the esultant. SOLUTION: Resolve each foce into ectangula components foc mag comp comp Detemine the components of the esultant b adding the coesponding foce components. Calculate the magnitude and diection. R N tan α α 4. 1 R N α

30 R 14.3 N sin N Equilibium of a Paticle When the esultant of all foces acting on a paticle is eo, the paticle is in equilibium. Newton s ist Law: If the esultant foce on a paticle is eo, the paticle will emain at est o will continue at constant speed in a staight line. Paticle acted upon b two foces: - equal magnitude - same line of action - opposite sense Paticle acted upon b thee o moe foces: - gaphical solution ields a closed polgon - algebaic solution R

31 ee-bod Diagams Space Diagam: A setch showing the phsical conditions of the poblem. ee-bod Diagam: A setch showing onl the foces on the selected paticle Rectangula Components in Space The vecto is contained in the plane OBAC. Resolve into hoiontal and vetical components. h Resolve h into ectangula components h cosφ sinθ cosφ cosθ h sinφ sinθ sinθ sinφ 11

32 12 With the angles between and the aes, λ is a unit vecto along the line of action of and cosθ, cosθ, and cosθ ae the diection cosines fo Diection of the foce is defined b the location of two points ( ) i i i θ θ θ λ λ θ θ θ θ θ θ cos cos cos cos cos cos cos cos cos ( ) ( ) ,, and,, N M ( ) d d d d d d d d i d d d d d d d i d N M d 1 and vecto oining λ λ

33 Mateials Engineeing Depatment Class: ist Date : Subect: Engineeing Mechanics Lectue: D. Emad AL-Hassani Lectue # 3 Rigid Bodies: Equivalent Sstems of oces Intoduction Teatment of a bod as a single paticle is not alwas possible. In geneal, the sie of the bod and the specific points of application of the foces must be consideed. Most bodies in elementa mechanics ae assumed to be igid, i.e., the actual defomations ae small and do not affect the conditions of equilibium o motion of the bod. Cuent chapte descibes the effect of foces eeted on a igid bod and how to eplace a given sstem of foces with a simple equivalent sstem. - Moment of a foce about a point - Moment of a foce about an ais - Moment due to a couple An sstem of foces acting on a igid bod can be eplaced b an equivalent sstem consisting of one foce acting at a given point and one couple. Etenal and Intenal oces oces acting on igid bodies ae divided into two goups: - Etenal foces - Intenal foces Etenal foces ae shown in a fee-bod diagam. 1

34 If unopposed, each etenal foce can impat a motion of tanslation o otation, o both. Pinciple of Tansmissibilit: Equivalent oces Pinciple of Tansmissibilit - Conditions of equilibium o motion ae not affected b tansmitting a foce along its line of action. NOTE: and ae equivalent foces. Moving the point of application of the foce to the ea bumpe does not affect the motion o the othe foces acting on the tuc. Pinciple of tansmissibilit ma not alwas appl in detemining intenal foces and defomations. 2

35 3 Vecto Poduct of Two Vectos Concept of the moment of a foce about a point is moe easil undestood though applications of the vecto poduct o coss poduct. Vecto poduct of two vectos P and Q is defined as the vecto V which satisfies the following conditions: 1. Line of action of V is pependicula to plane containing P and Q. 2. Magnitude of V is V PQ sin θ 3. Diection of V is obtained fom the ighthand ule Vecto poducts: - ae not commutative, ( ) Q P P Q - ae distibutive, ( ) Q P Q P Q Q P -ae not associative, ( ) ( ) S Q P S Q P Vecto Poducts: Rectangula Components Vecto poducts of Catesian unit vectos, i i i i i i i i v

36 Vecto poducts in tems of ectangula coodinates V ( Pi P P ) ( Qi Q Q ) ( P Q P Q ) i ( P Q P Q ) ( P Q P Q ) i P Q P Q P Q Moment of a oce About a Point A foce vecto is defined b its magnitude and diection. Its effect on the igid bod also depends on it point of application. The moment of about O is defined as M O The moment vecto MO is pependicula to the plane containing O and the foce. Magnitude of MO measues the tendenc of the foce to cause otation of the bod about an ais along MO. M O sinθ d The sense of the moment ma be detemined b the ight-hand ule. 4

37 An foce that has the same magnitude and diection as, is equivalent if it also has the same line of action and theefoe, poduces the same moment. Two-dimensional stuctues have length and beadth but negligible depth and ae subected to foces contained in the plane of the stuctue. The plane of the stuctue contains the point O and the foce. MO, the moment of the foce about O is pependicula to the plane If the foce tends to otate the stuctue clocwise, the sense of the moment vecto is out of the plane of the stuctue and the magnitude of the moment is positive. If the foce tends to otate the stuctue counteclocwise, the sense of the moment vecto is into the plane of the stuctue and the magnitude of the moment is negative Vaignon s Theoem The moment about a give point O of the esultant of seveal concuent foces is equal to the sum of the moments of the vaious moments about the same point O. ( L) L Vaigon s Theoem maes it possible to eplace the diect detemination of the moment of a foce b the moments of two o moe component foces of. 5

38 6 Rectangula Components of the Moment of a oce The moment of about O, i i M O, ( ) ( ) ( ) i i M M i M M O The moment of about B, M B A B / ( ) ( ) ( ) i i B A B A B A B A B A / ( ) ( ) ( ) B A B A B A B i M

39 o two-dimensional stuctues M M O O ( ) M Z M M O O [( ) ( ) ] M A Z B ( A B ) ( A B ) A B Sample Poblem A 100-lb vetical foce is applied to the end of a leve which is attached to a shaft at O. Detemine: a) moment about O, b) hoiontal foce at A which ceates the same moment, c) smallest foce at A which poduces the same moment, d) location fo a 240-lb vetical foce to poduce the same moment, e) whethe an of the foces fom b, c, and d is equivalent to the oiginal foce 7

40 a) Moment about O is equal to the poduct of the foce and the pependicula distance between the line of action of the foce and O. Since the foce tends to otate the leve clocwise, the moment vecto is into the plane of the pape M O d d M O M O ( 24in. ) cos60 ( 100 lb)( 12 in. ) 1200 lb in 12 in. b) Hoiontal foce at A that poduces the same moment, d ( 24 in. ) sin in. M O d 1200 lb in. ( 20.8 in. ) 1200 lb in in lb c) The smallest foce A to poduce the same moment occus when the pependicula distance is a maimum o when is pependicula to OA. M O d 1200 lb in. ( 24 in. ) 1200 lb in. 24 in. 50 lb d) To detemine the point of application of a 240 lb foce to poduce the same moment, M O 1200 lb in. d ( 240 lb) 1200 lb in. d 5 in. 240 lb OB cos60 5 in. d OB 10 in. 8

41 e)although each of the foces in pats b), c), and d) poduces the same moment as the 100 lb foce, none ae of the same magnitude and sense, o on the same line of action. None of the foces is equivalent to the 100 lb foce. Sample Poblem The ectangula plate is suppoted b the bacets at A and B and b a wie CD. Knowing that the tension in the wie is 200 N, detemine the moment about A of the foce eeted b the wie at C. SOLUTION: The moment MA of the foce eeted b the wie is obtained b evaluating the vecto poduct, C A M A C C A A ( 0.3 m) i ( 0.08 m) 9

42 λ v M A ( 200 N) ( 200 N) C D C D ( 0.3 m) i ( 0.24 m) ( 0.32 m) ( 120 N) i ( 96 N) ( 128 N) 0.5 m ( 7.68 N m) i ( 28.8 N m) ( 28.8 N m) Scala Poduct of Two Vectos The scala poduct o dot poduct between two vectos P and Q is defined as P Q PQ cosθ Scala poducts: ( scala esult) P Q Q P -ae commutative, -ae distibutive, P ( Q1 Q2 ) P Q1 P Q2 P Q S -ae not associative ( ) undefined 10

43 11 Scala poducts with Catesian unit components ( ) ( ) Q Q Q i P P P i Q P i i i i v P P P P P P P Q P Q P Q Q P

44 Mateials Engineeing Depatment Class: ist Date : Subect: Engineeing Mechanics Lectue: D. Emad AL-Hassani Lectue # 4 Equilibium of Rigid Bodies Intoduction o a igid bod in static equilibium, the etenal foces and moments ae balanced and will impat no tanslational o otational motion to the bod. The necessa and sufficient condition fo the static equilibium of a bod ae that the esultant foce and couple fom all etenal foces fom a sstem equivalent to eo, 0 ( ) 0 M O Resolving each foce and moment into its ectangula components leads to 6 scala equations which also epess the conditions fo static equilibium, 0 M 0 0 M 0 0 M 0 ee-bod Diagam ist step in the static equilibium analsis of a igid bod is identification of all foces acting on the bod with a fee-bod diagam. Select the etent of the fee-bod and detach it fom the gound and all othe bodies 1

45 Indicate point of application, magnitude, and diection of etenal foces, including the igid bod weight Indicate point of application and assumed diection of unnown applied foces. These usuall consist of eactions though which the gound and othe bodies oppose the possible motion of the igid bod Include the dimensions necessa to compute the moments of the foces Reactions at Suppots and Connections fo a Two-Dimensional Stuctue Reactions equivalent to a foce with nown line of action. 2

46 Reactions equivalent to a foce of unnown diection and magnitude. Reactions equivalent to a foce of unnown diection and magnitude and a couple.of unnown magnitude Equilibium of a Rigid Bod in Two Dimensions o all foces and moments acting on a two-dimensional stuctue 0 M M 0 M M O 3

47 Equations of equilibium become 0 0 M A 0 whee A is an point in the plane of the stuctue The 3 equations can be solved fo no moe than 3 unnowns The 3 equations can not be augmented with additional equations, but the can be eplaced 0 M A 0 M B 0 Staticall Indeteminate Reactions Moe unnowns than equations ewe unnowns than equations, patiall constained Equal numbe unnowns and equations but impopel constained 4

48 Sample Poblem A fied cane has a mass of 1000 g and is used to lift a 2400 g cate. It is held in place b a pin at A and a oce at B. The cente of gavit of the cane is located at G. Detemine the components of the eactions at A and B. SOLUTION: Ceate a fee-bod diagam fo the cane Detemine B b solving the equation fo the sum of the moments of all foces about A. Note thee will be no contibution fom the unnown eactions at A. Detemine the eactions at A b solving the equations fo the sum of all hoiontal foce components and all vetical foce components Chec the values obtained fo the eactions b veifing that the sum of the moments about B of all foces is eo. Ceate the fee-bod diagam Detemine B b solving the equation fo the sum of the moments of all foces about A. M A 0 : B ( 1.5m) 9.81 N( 2m) ( ) N 6m B N 5

49 Detemine the eactions at A b solving the equations fo the sum of all hoiontal foces and all vetical foces 0 : A B 0 A 107.1N 0 : A 9.81 N 23.5 N 0 A 33.3 N Chec the values obtained Equilibium of a Rigid Bod in Thee Dimensions Si scala equations ae equied to epess the conditions fo the equilibium of a igid bod in the geneal thee dimensional case. 0 M 0 0 M 0 0 M 0 These equations can be solved fo no moe than 6 unnowns which geneall epesent eactions at suppots o connections The scala equations ae convenientl obtained b appling the vecto foms of the conditions fo equilibium 0 M 0 O ( ) 6

50 Reactions at Suppots and Connections fo a Thee-Dimensional Stuctue 7

51 8 Sample Poblem A sign of unifom densit weighs 270 lb and is suppoted b a ball-andsocet oint at A and b two cables. Detemine the tension in each cable and the eaction at A. SOLUTION: Ceate a fee-bod diagam fo the sign Appl the conditions fo static equilibium to develop equations fo the unnown eactions Ceate a fee-bod diagam fo the sign. Since thee ae onl 5 unnowns, the sign is patiall constain. It is fee to otate about the ais. It is, howeve, in equilibium fo the given loading. ( ) ( ) i T i T T T i T i T T T EC EC E C E C EC EC BD BD B D B D BD BD

52 Appl the conditions fo static equilibium to develop equations fo the unnown eactions. A TBD TEC ( 270 lb) i : A 3 TBD 7 TEC : A 3 TBD 7 TEC 270 lb : A T 0 3 BD T 7 EC M A B TBD E TEC : 5.333TBD 1.714TEC 0 : 2.667T 2.571T 1080 lb 0 BD EC ( 4 ft) i ( 270 lb) 0 Solve the 5 equations fo the 5 unnowns, TBD lb v A T EC 315 lb ( 338 lb) i ( lb) ( 22.5 lb) 9

53 Mateials Engineeing Depatment Class: ist Subect: Engineeing Mechanics Lectue: D. Emad AL-Hassani Centoids and Centes of Gavit Intoduction: The eath eets a gavitational foce on each of the paticles foming a bod. These foces can be eplace b a single equivalent foce equal to the weight of the bod and applied at the cente of gavit fo the bod The centoid of an aea is analogous to the cente of gavit of a bod. The concept of the fist moment of an aea is used to locate the centoid. Detemination of the aea of a suface of evolution and the volume of a bod of evolution ae accomplished with the Theoems of Pappus- Guldinus Cente of Gavit of a 2D Bod Cente of gavit of a plate M M W W W dw W dw Cente of gavit of a wie 1

54 Centoids and ist Moments of Aeas and Lines Centoid of an aea W ( γat) ( γt) A fist moment with espect to A da Q fist moment with espect to dw da da Q Centoid of a line W ( γ La) ( γ a) L L dw dl dl dl ist Moments of Aeas and Lines An aea is smmetic with espect to an ais BB if fo eve point P thee eists a point P such that PP is pependicula to BB and is divided into two equal pats b BB. 2

55 The fist moment of an aea with espect to a line of smmet is eo. If an aea possesses a line of smmet, its centoid lies on that ais If an aea possesses two lines of smmet, its centoid lies at thei intesection An aea is smmetic with espect to a cente O if fo eve element da at (,) thee eists an aea da of equal aea at (-,-). The centoid of the aea coincides with the cente of smmet 3

56 Centoids of Common Shapes of Aeas Centoids of Common Shapes of Lines 4

57 Composite Plates and Aeas Composite plates X Y W W W W Composite aea X Y A A A A Sample Poblem o the plane aea shown, detemine the fist moments with espect to the and aes and the location of the centoid. SOLUTION: Divide the aea into a tiangle, ectangle, and semicicle with a cicula cutout. Calculate the fist moments of each aea with espect to the aes ind the total aea and fist moments of the tiangle, ectangle, and semicicle. Subtact the aea and fist moment of the cicula cutout 5

58 Compute the coodinates of the aea centoid b dividing the fist moments b the total aea. ind the total aea and fist moments of the tiangle, ectangle, and semicicle. Subtact the aea and fist moment of the cicula cutout. Q Q mm mm 3 3 Compute the coodinates of the aea centoid b dividing the fist moments b the total aea X 3 A mm 3 2 A mm 3 X 54.8 mm Y A A mm mm 2 3 Y 36.6 mm 6

59 Mateials Engineeing Depatment Class: ist Subect: Engineeing Mechanics Lectue: D. Emad AL-Hassani iction Intoduction In peceding chaptes, it was assumed that sufaces in contact wee eithe fictionless (sufaces could move feel with espect to each othe) o ough (tangential foces pevent elative motion between sufaces). Actuall, no pefectl fictionless suface eists. o two sufaces in contact, tangential foces, called fiction foces, will develop if one attempts to move one elative to the othe. Howeve, the fiction foces ae limited in magnitude and will not pevent motion if sufficientl lage foces ae applied. The distinction between fictionless and ough is, theefoe, a matte of degee. Thee ae two tpes of fiction: d o Coulomb fiction and fluid fiction. luid fiction applies to lubicated mechanisms. The pesent discussion is limited to d fiction between nonlubicated sufaces. The Laws of D iction. Coefficients of iction Bloc of weight W placed on hoiontal suface. oces acting on bloc ae its weight and eaction of suface N. Small hoiontal foce P applied to bloc. o bloc to emain stationa, in equilibium, a hoiontal component of the suface eaction is equied. is a static-fiction foce. 1

60 As P inceases, the static-fiction foce inceases as well until it eaches a maimum value m. m µ s N uthe incease in P causes the bloc to begin to move as dops to a smalle inetic-fiction foce. µ N Maimum static-fiction foce: m µ s N Kinetic-fiction foce: µ N µ 0.75µ s 2

61 ou situations can occu when a igid bod is in contact with a hoiontal suface: No fiction, (P 0) No motion,(p < m) Motion impending, (P m) Motion, (P > m) 3

62 Angles of iction It is sometimes convenient to eplace nomal foce N and fiction foce b thei esultant R: No fiction No motion m µ s N tanφs N N tanφ µ s s Motion impending 4 tanφ tanφ Motion N µ µ N N

63 Conside bloc of weight W esting on boad with vaiable inclination angle θ. No fiction No motion Motion impending Motion 5

64 Poblems Involving D iction All applied foces nown Coefficient of static fiction is nown Detemine whethe bod will emain at est o slide All applied foces nown Motion is impending Detemine value of coefficient of static fiction. Coefficient of static fiction is nown Motion is impending Detemine magnitude o diection of one of the applied foces 6

65 Sample Poblem A 450 N foce acts as shown on a 1350 N bloc placed on an inclined plane. The coefficients of fiction between the bloc and plane ae ms 0.25 and m Detemine whethe the bloc is in equilibium and find the value of the fiction foce. SOLUTION: Detemine values of fiction foce and nomal eaction foce fom plane equied to maintain equilibium. Calculate maimum fiction foce and compae with fiction foce equied fo equilibium. If it is geate, bloc will not slide. If maimum fiction foce is less than fiction foce equied fo equilibium, bloc will slide. Calculate inetic-fiction foce. 0 : 3 50 N ( 1350 N ) N 0 : 4 N ( 1350 N) 0 5 N 1080 N µ m sn m ( ) 270 N N The bloc will slide down the plane N actual µ ( ) N actual 216 N 7