UNIT 1 COPLANAR AND NON-COPLANAR FORCES

Transcription

1 UNIT 1 COPLANA AND NON-COPLANA FOCES Cplanar and Nn-Cplanar Frces Structure 1.1 Intrductin Objectives 1. System f Frces 1.3 Cplanar Frce Law f Parallelgram f Frces 1.3. Law f Plygn f Frces eslutin and Cmpsitin 1.4 esultant f Cncurrent Frces esultant f Cplanar Cncurrent Frces 1.4. esultant f Nn-cplanar Cncurrent Frces 1.5 Mment f a Frce Mment f Cplanar Frces 1.5. Mment f a Frce abut a Pint and an Axis Cuples and their Prperties 1.6 esultant f Nn-cncurrent Frces 1.7 Summary 1.8 Answers t SAQs 1.1 INTODUCTION This unit seeks t intrduce yu the different systems f frces. The prerequisite fr this is the cncept f a frce and the varius frms f frces ccurring in nature. In additin t this, yu shuld have the basic knwledge f algebra. The study f this unit will enable yu t understand the effect f frces n different types f structures. Objectives After studying this unit, yu shuld be able t identify the different systems f frces, add cncurrent frces vectrially, reslve frces int cmpnents, add frces by cmpnents, find the mment f a frce, and find the resultant f nn-cncurrent frces. 1. SYSTEM OF FOCES Newtn s First Law f Mtin helps t define a frce as an external agency which tends t change the state f rest r f unifrm mtin f a bdy. Frce tends t prduce mtin in a bdy, changes the mtin f a bdy r checks the 5

2 Applied Mechanics mtin f a bdy. In simple wrds, the actin f ne bdy n any ther bdy can be called a frce. These actins may be f varius frms: Pull r push n a bdy, gravitatinal frce knwn as weight f a bdy, frce exerted by an elastic spring, frce exerted by a lcmtive n the train, resistance ffered by the track. T specify a frce, ne need t knw its magnitude, directin and the pint f applicatin. Its magnitude is expressed as Newtn in SI unit. Graphically, a frce is represented by drawing a line t scale, shwing its magnitude and an arrwhead indicating its directin. Such a straight line is called vectr. A cmbinatin f several frces acting n a bdy is called a system f frces r a frce system. Example 1.1 Slutin Cnsider a sphere f mass m suspended by means f a string resting against a smth wall, as shwn in Figure 1.1(a). What are the frces acting n it? Let us identify the frces acting n the sphere. These are as fllws : (i) (ii) Weight f the sphere, W mg, acting vertically dwnwards frm the centre f gravity f the sphere. Tensin, T, in the string. (iii) eactin,, ffered by the wall. Thus, the sphere is subjected t a system f three frces as shwn in Figure 1.1. (a) Figure 1.1 Example 1. Slutin Cnsider a dam retaining water as shwn in Figure 1.(a). What are the frces acting n it? The frces acting n the dam are as fllws : (i) (ii) Weight f the masnry acting vertically dwnwards thrugh the centrid f the crss-sectin, and The hrizntal water pressure, p, which ges n increasing as the depth f water increases. 6

3 Thus, the structure shwn is subjected t a system f frces as indicated in Figure 1. Cplanar and Nn-Cplanar Frces (a) Figure 1. Frce System The system f frces can be classified accrding t the arrangement f the lines f actin f the frces f the system. The frces (i.e., system f frces) may be classified as (i) (ii) cplanar r nn-cplanar, cncurrent r nn-cncurrent, and (iii) parallel r nn-parallel. All these systems f frces have been explained in subsequent paragraphs. Cplanar Frces Frces acting in the same plane are called cplanar frces. In Figure 1.3(a), frces a and b are acting in vertical plane ABCD. They are called cplanar frces. Frces c and d are als called cplanar frces as they are acting in ne plane. But frces a and c are nt cplanar frces as they are acting in tw different plane. If all frces acting n a bdy meet at a pint, they are called cncurrent frces. Frces a, b and c shwn in Figure 1.3 are cncurrent frces as they are meeting at pint O, whereas frces d, e and f are called as nn-cncurrent frces because all the three frces are nt meeting at a pint. Cplanar frces can als be classified as parallel frces and nn-parallel frces. If the line f actin f frces are parallel then the frces are called parallel frces. If the frces pint the same directin they are called parallel frces, and if they pint in ppsite directins, they are called unlike parallel frces, and the frces p and q are like parallel frces, whereas q and r r p and r are unlike parallel frces as shwn in Figure 1.3(c). Frces s, t and u are nn-parallel frces. Cncurrent frces are nn-parallel frces. But nn-parallel frces may be cncurrent r nncncurrent. 7

4 Applied Mechanics (a) Nn-Cplanar Frces (c) Figure 1.3 If the line f actin f varius frces d nt lie in the same plane then the frces are called nn-cplanar frces. These frces may be cncurrent r parallel as shwn in Figures 1.4(a) and 1.4, respectively. Frces l, m, n and are nn-cplanar cncurrent frces, and frces x, y and z are nncplanar parallel frces. SAQ 1 (a) Figure 1.4 Identify the system f frces in Figures 1.5(a), (c) and (d) and classify them. 8

5 Cplanar and Nn-Cplanar Frces (a) (c) Figure 1.5 (d) 1.3 COPLANA FOCES If a system f cplanar frces is acting n a bdy, its ttal effect is usually expressed in terms f its resultant. Frce is a vectr quantity. The resultant f the system f frces can be fund ut by using vectr algebra, e.g., if the resultant f tw frces is t be fund ut then the law f parallelgram f frces is used fr the purpse Law f Parallelgram f Frces If the tw cplanar frces meet at a pint, their resultant may be fund by the law f parallelgram f frces, which states that, If tw frces acting at a pint are such that they can be represented in magnitude and directin by the tw adjacent sides f parallelgram, the diagnal f the parallelgram passing thrugh their pint f intersectin gives the resultant in magnitude and directin. Cnsider tw frces P and Q acting at a pint O in the bdy as shwn in Figure 1.6(a). Their cmbined effect can be fund ut by cnstructing a parallelgram using vectr P and vectr Q as tw adjacent sides f the parallelgram as shwn in Figure 1.6. The diagnal passing thrugh O represents their resultant in magnitude and directin. Yu can prve by the gemetry f the figure that the magnitude f the resultant and the angle it makes with P are given by : P + Q + PQcsα, and Qsin α tan θ P + Qcsα r Psin α tanβ Q + Pcsα 9

6 Applied Mechanics where, α is the angle between P and Q θ is the angle between and P β is the angle between and Q. The abve tw frces can als be cmbined by using the law f triangle f frces which states that if the secnd frce is drawn frm the end f the first frce then the line jining the starting pint f first frce t the end f the secnd frce represents their resultant [Figure 1.6(c)]. (a) Figure 1.6 Frm the triangle f frces, by using trignmetric relatins, yu can find that, P + Q + PQcsα Q P and sinθ sinα, sin β sinα 10 Figure 1.6(c) 1.3. Law f Plygn f Frces If mre than tw frces are acting n a bdy, then their resultant can be fund by repeated applicatins f the parallelgram law r the triangle law fr ne pair f frces at a time. Yu may start with any tw frces and find their resultant first and then add vectrially t this resultant the remaining frces taking ne at a time. In the final frm a plygn wuld be cmpleted. Therefre, if mre than tw cplanar frces meet at a pint, their resultant may be fund by the law f plygn f frces, which states that, If number f frces acting at a pint are such that they can be represented in magnitude and directin by the sides f an pen plygn taken in rder, then their resultant is represented in magnitude and directin by the clsing side f the plygn but taken in the ppsite rder. Example 1.3

7 Cnsider five frces each f 80 N acting at O in a bdy. Draw the plygn f frces and shw the resultant f all the frces. Cplanar and Nn-Cplanar Frces Slutin (a) Figure 1.7 Let us cnstruct a plygn such that the frces A, B,C, D and E represent the sides f a plygn taken in rder (bviusly, each side will be equal in magnitude), each frce being drawn frm the end f earlier frce then their resultant is represented by the line jining the starting pint f the first frce A t the end f the last frce E (Figure 1.7). SAQ Determine the resultant in magnitude and directin f tw frces shwn in Figures 1.8(a) and using the parallelgram law and the triangle law. (a) Figure 1.8 SAQ 3 Fur frces are acting at O as shwn in Figure 1.9. Find the resultant in magnitude and directin by using (i) (ii) plygn law, and methd f reslutin f frces. 11

8 Applied Mechanics Figure eslutin and Cmpsitin In many engineering prblems, it is desirable t reslve a frce int rectangular cmpnents. This prcess f splitting the frce int cmpnents is called the reslutin f a frce, whereas the prcess f finding the resultant f any number f frces is called the cmpsitin f frces. The reslutin f frces helps in determining the resultant f a number f frces acting n a bdy as it reduces vectrial additin t algebraic additin. A frce F making an angle θ with respect t x-axis, as shwn in Figure 1.10, can be reslved int tw cmpnents Fx and Fy acting alng x and y axes, respectively. If i and j are the unit vectrs acting alng x and y axes, respectively then the frce F can be expressed as, F F i x + F y j where, F x and F y are the magnitude f the cmpnents alng x and y axes. eferring t Figure 1.10, F x and F y are determined as, F x F cs θ F y F sin θ, and tan θ F F y x (a) Figure 1.10 Nte : θ is measured in anticlckwise directin with respect t psitive x-axis. 1

9 The magnitude f the frce can als be expressed as F ( F x) + ( Fy ) Cplanar and Nn-Cplanar Frces Example 1.4 A frce f 10 N is exerted n a hk in the ceiling as shwn in Figure Determine the hrizntal and vertical cmpnents f the frce. Slutin Figure 1.11 As θ is t be measured in anticlckwise directin frm psitive x-axis, then we take θ 300. F x F cs θ 10 cs N. F y F sin θ 10 sin N. The vectr cmpnents f frce F are : F x (+ 60 N) i and Therefre, F can be expressed as Example 1.5 F + 60 i j F y ( N) j A frce f 80 N is acting n a blt as shwn in Figure 1.1. Find the hrizntal and vertical cmpnents f the frce. Slutin Figure

10 Applied Mechanics By principle f transmissibility f a frce, the frce can be cnsidered acting at any pint n the line f actin f the frce. θ with respect t psitive x-axis measured in anticlckwise directin (i.e. dtted line making with +ve x-line). F x F cs θ 80 cs N F y F sin θ 80 sin N F ( 40 N) i + ( 69.8 N) j 40i j ( ve signs indicating ve x and y directins). 1.4 ESULTANT OF CONCUENT FOCES The resultant f a cncurrent frce system can be defined as the simplest single frce which can replace the riginal system withut changing its external effect n a rigid bdy. Fr the nn-cncurrent frce system, the resultant will nt necessarily be a single frce but may be a frce system cmprising a frce r a cuple r a frce and a cuple tgether. The types f frce systems alng with their pssible resultants are given in Table 1.1 Table 1.1 Types f Frce System Cncurrent Cplanar, nn-cncurrent Parallel, nn-cplanar, nn-cncurrent Nn-parallel, nn-cplanar, nn-cncurrent Frce Pssible esultant Frce r a cuple Frce r a cuple Frce r a cuple r a frce and a cuple esultant f Cplanar Cncurrent Frces The technique f reslutin f a frce can be used t determine the resultant f cplanar cncurrent frces. If n cncurrent frces F 1, F, F 3,...,F n are acting at a pint in a bdy then each frce can be reslved int tw mutually perpendicular directins. Thus, we get n cmpnents in all. Each set f n cmpnents acts in ne directin nly. Therefre, we can algebraically add all these cmpnents t get the cmpnents f the resultant, x Σ F ix F 1x + F x + F 3x F nx y Σ F iy F 1y + F y + F 3y F ny Finally, cmbining these cmpnents x and y vectrially, we get the resultant. Thus, x i + y j 14 and ( x ) + ( y )

11 θ tan 1 y x Cplanar and Nn-Cplanar Frces where, θ is the angle f inclinatin f the resultant with respect t psitive x-axis. Example 1.6 Fur frces act n a bdy as shwn in Figure Determine the resultant f the system f frces. Figure 1.13 Slutin eslving all frces alng x-axis, we get : x Σ F x F 1 cs θ 1 + F cs θ + F 3 cs θ 3 + F 4 cs θ 4 40 cs cs cs cs 40 Nte : The angle made by 50 N frce is measured in anticlckwise directin frm psitive x-axis after making the frce act away frm O by principle f transmissibility f the frce. x 40 cs cs cs 0 0 cs 60 θ 90 may be chsen in apprpriate quadrant with prper signs as indicated abve. x x 30 N... (1.1) Similarly, reslving all frces alng y-axis, we get, y Σ F y F 1 sin θ 1 + F sin θ + F 3 sin θ 3 + F 4 sin θ 4 y 40 sin sin sin sin 40 y y 3.68 N... (1.) Thus, the resultant in vectr frm may be expressed by ( 30 N) i + ( 3.68 N) j The magnitude f the resultant is given by 15

12 Applied Mechanics ( x ) + ( y ) ( 30) + ( 3.68) N The directin θ can be wrked ut frm θ tan 1 y x tan The resultant has a magnitude f N and is acting in IVth quadrant making an angle f in anticlckwise directin frm psitive x-axis esultant f Nn-cplanar Cncurrent Frces In case f nn-cplanar frces system als, the technique f reslutin f frces can be used t determine the resultant. If three nn-cplanar frces F 1, F and F 3 are acting at a pint O in a bdy, the resultant 1 f the tw frces F 1 and F can be determined by law f parallelgram f frces. The frce 1 can next be cmbined with F 3 by means f the parallelgram, giving the resultant f three frces F 1, F and F 3 as. If there are mre frces in the system, the same prcess can be cntinued until all the frces have been cvered. Here, nte that the resultant f nn-cplanar frce system must pass thrugh the pint f cncurrence. The resultant f cncurrent frce system can als be determined as the vectr sum f the frces f the system. The vectr sum f the frces can be btained very easily if each frce is reslved int rectangular cmpnents. Thus, the vectr sum f a nn-cplanar system f cncurrent frces F 1, F and F 3. F F F3 which can be written in rectangular cmpnent frm as i j z k F1 x i + F1 y j + F1 z k + F x i + F y j + F x y z + F3 i + F3 y j + F x 3z ( 3 k F1 x + F x + F3 x ) i + ( F1 y + F y + F3 y ) j + ( F1 z + F z + F z ) k ( Σ Fx ) i + ( Σ Fy ) j + ( Σ Fz ) k Therefre, F + F + F ( Σ F ) x y z F F 1x x 3x x + F + F ( Σ F 1y y 3y y + F + F ) ( Σ F ) 1z z 3z z Finally, cmbining these cmpnents x, y and z vectrially, we get the resultant. Thus, i + j x y + z k k 16

13 and ( x ) + ( y ) ( z + ) Cplanar and Nn-Cplanar Frces r ( Σ Fx ) + ( Σ Fy ) + ( Σ Fz ) where θ, θ x cs θ axes respectively. y x and θ Σ F z x, cs θ y Σ F y, cs θ z Σ F are the angles which the resultant makes with x, y and z 1.5 MOMENT OF A FOCE If a frce F is acting n a bdy resting at O and the line f actin f frce des nt pass thrugh G, the centre f gravity f the bdy, it will nt give the bdy a straight line mtin, called the translatry mtin, but will try t rtate the bdy abut O as shwn in Figure The measure f this prperty f a frce by virtue f which it tends t rtate the bdy n which it acts is called the mment f a frce. The rtatin f the bdy may be happening either abut a pint r a line. z Figure 1.14 eferring t Figure 1.15, if F is the frce (in N) acting n the bdy alng AB and x is the perpendicular distance (in m) f any pint, O (t which the bdy may be pinned) frm AB, then, Mment f the frce F abut O M F x F OC Figure 1.15 Here, pint O is knwn as mment center r fulcrum and distance x is termed as mment arm. We can state in wrds, therefre : 17

14 Applied Mechanics Mment f Frce Frce Perpendicular distance between the line f actin f the frce and the pint r the axis abut which mment is required Mment is a vectr quantity and the vectr directin is alng the axis abut which the mment is taken. In terms f vectr algebra, we can define the mment M f a frce F with respect t pint O as the crss prduct f the perpendicular distance f pint O frm the line f actin f the frce F. If the mment f the frce abut a pint is zer, it means either the frce itself is zer r the perpendicular distance between the line f actin f the frce and the pint abut which mment is t be calculated is zer, i.e. the frce passes thrugh that pint. Varignn s Therem It states that the mment f a frce abut any pint is equal t the sum f the mments f its cmpnents abut the same pint. This principle is als knwn as principle f mments. Varignn s therem need nt be restricted t the case f nly tw cmpnents but applies equally well t a system f frces and its resultant. Fr this it can be slightly mdified as, the algebraic sum f the mments f a given system f frces abut a pint is equal t the mment f their resultant abut the same pint. This principle f mment may be extended t any frce system Mment f Cplanar Frces Let F 1, F and F 3 be the three cplanar frces acting n a bdy and let θ 1, θ and θ 3 be the angles which these frces make with psitive x-axis as shwn in Figure 1.16(a). (a) Figure 1.16 Nw, the magnitude and directin f resultant can be fund ut very easily by reslving all the frces hrizntally and vertically. Let the resultant makes an angle θ with psitive x-axis as shwn in Figure Nw by cmputatin f mment f frces, the psitin f resultant frce can be ascertained. T determine the pint f applicatin f the resultant, let it cut the hrizntal axis XOX at A at a perpendicular distance d frm O as shwn in Figure Fr pint O in Figure 1.16(a), let the algebraic sum f the mments f the given frces abut O be given by Σ M 0 (anticlckwise). 18 Then, Σ M 0 F 1 d 1 + F d + F 3 d 3

15 B Nw, by applying Varignn s therem, the psitin f resultant will be such that the mment f abut pint O, ( d) is equal t Σ M 0, and the directin f the mment due t abut mment centre O must be the same as f Σ M 0 due t given system f frces. Cplanar and Nn-Cplanar Frces Thus, d Σ M 0 The distance d is cmputed frm the abve relatin and, whse magnitude and directin have already been determined earlier, is nw cmpletely lcated Mment f a Frce abut a Pint and an Axis The mment f a frce can be determined with respect t (abut) a pint and als with respect t a line r any axis. The mment f a frce F with respect t a pint A is defined as a vectr with a magnitude equal t the prduct f the perpendicular distance frm A t F and the magnitude f the frce and with a directin perpendicular t the plane cntaining A and F. The sense f the mment vectr is given by the directin a right-hand screw wuld advance if turned abut A in the directin indicated by F as shwn in Figure Figure 1.17 The mment f a frce abut a line r axis perpendicular t a plane cntaining the frce is defined as a vectr with a magnitude equal t the prduct f the magnitude f the frce and the perpendicular distance frm the line t the frce and with a directin alng the line. Thus, it is the same as the mment f the frce abut the pint f intersectin f the plane and the mment axis. Since the mment f a frce abut an axis is a measure f its tendency t turn r rtate a bdy abut the axis, the frce parallel t an axis has n mment with respect t the axis, because it has n tendency t rtate the bdy abut the axis. The mment f a frce abut varius pints and axes is illustrated in Figure The mment f the hrizntal frce F abut pint A has a magnitude f Fd 1 in the directin shwn by M A (Figure 1.18(a)). Similarly, M B, the mment abut pint B, has a magnitude f Fd and is perpendicular t the plane determined by B and the frce F. The mment f frce F abut the line AB is the same as M A (as shwn in Figure 1.18(a)) r M AB (as shwn in Figure 1.18). The mment f frce F abut line BC can be btained by reslving frce F int cmpnents F 1 and F. Since F 1 is parallel t line BC, it has n mment abut BC. The resultant F is in a plane perpendicular t BC and its mment is F d 3 in the directin shwn. Similarly, the 19

16 Applied Mechanics mment abut line BD is F 1 d 3 as indicated. Here, yu can nte that M AB, M BC and M BD are the rthgnal cmpnents f M B. Example 1.7 (a) Figure 1.18 The side f a square ABCD is 1.60 m lng. Fur frces equal t 6, 5, 4 and 8 N act alng the line CB, BA, DA and DB, respectively. Find the mment f these frces abut O, the pint f intersectin f the diagnals f the square (Figure 1.19). Slutin Taking mments abut O, Figure 1.19 esultant mment M x 5x + 4x where x1, x and x3 are the perpendicular distances f the frces f 4, 6 and 5 N, respectively frm O, and frce f 8 N has zer mment abut O as its line f actin passes thrugh this pint. 1.6 Here, x 1 x x3 0.8m. M 6(0.8) 5(0.8) + 4(0.8) N-m r 5.6 N-m (clckwise) 0

17 SAQ 4 The side f a regular hexagn ABCDEF is 0.6m. Frces 1,, 3, 4, 5 and 6 N are acting alng the sides AB, CB, DC, DE, EF and FA, respectively. Find the algebraic sum f the mments abut A (Figure 1.0). Cplanar and Nn-Cplanar Frces Figure Cuples and their Prperties A cuple is a frce system cnsisting f tw equal, cplanar, parallel frces acting in ppsite directins. Since a cuple cnstitutes tw equal and parallel frces, their resultant is zer and hence a cuple has n tendency t prduce translatry mtin but prduces rtatin in the bdy n which it acts. Figure 1.1 shws tw equal and ppsite frces, each equal t P and acting at A and B alng parallel lines, thus cnstituting a cuple. The perpendicular distance AB is called the arm f the cuple and is dented by p. Mment f a Cuple Figure 1.1 The mment f a cuple abut any pint in the plane cntaining the frces is cnstant and is measured by the prduct f any ne f the frces and the perpendicular distance between the lines f actin f the frces, i.e. M P p. 1

18 Applied Mechanics Prperties f Cuples The prperties which distinguish ne cuple frm every ther cuple are called its characteristics. A cuple whether psitive r negative, has the fllwing prperties/characteristics. (i) (ii) The algebraic sum f the frces cnstituting a cuple is zer. The algebraic sum f the mments f the frces frming a cuple is the same abut any pint in their plane (iii) The cuple can be balanced nly by anther cuple f the same mment but f the ppsite sense. (iv) The net effect f a number f cplanar cuple is equivalent t the algebraic sum f the effects f each f the cuples. A cuple is frequently indicated by a clckwise r cunter-clckwise arrw when cplanar frce systems are invlved instead f shwing tw separate frces. eplacement f a Frce by a Frce and Cuple Cnsider a frce F acting at a pint O 1. Imagine nw tw equal and ppsite frces F parallel t the given frce acting at O as shwn in Figure 1., as an additin t the system. Figure 1. These tw additinal frces d nt alter the system in its effect n a bdy. The new system is equivalent t a frce F acting at O, plus a cuple f mment M F.d. eplacement f a Cuple by tw Frces Cnsider a cuple f mment M, where the axis f the cuple is thrugh O perpendicular t the plane f paper as shwn in Figure 1.3. Figure 1.3

19 This cuple is equivalent t any tw parallel frces f magnitude F acting at a distance d apart such that F.d M and the directin f the frces chsen s as t give the crrect directin f M. Cplanar and Nn-Cplanar Frces 1.6 ESULTANT OF NON-CONCUENT FOCES As stated earlier, the resultant f a system f frces is the simplest frce system which can replace the riginal frces withut altering their external effect n a rigid bdy. The equilibrium f a bdy is the cnditin wherein the resultant f all the frces is zer. The prperties f frce, mment and cuple discussed in the preceding sectins will nw be used t determine the resultant f nn-cncurrent frce systems esultant f Cplanar Nn-cncurrent Frces The resultant f a system f cplanar nn-cncurrent frces can be btained by adding tw frces at a time and then cmbining their sums. The three frces F 1, F and F 3, shwn in Figure 1.4(a), can be cmbined by first adding any tw frces such as F and F 3. They can be mved alng their lines f actin t their pint f cncurrency A by the principle f transmissibility. (a) Figure 1.4 Their vectr sum 1 is frmed by the law f parallelgram f frce. The frce 1 may then be cmbined with F 1 by the parallelgram law at their pint f cncurrency B t btain the resultant f the three given frces. Here, the rder f cmbinatin f the frces is immaterial as may be verified by cmbining them in a different sequence. Nw, the frce may be applied at any pint n its established line f actin. Algebraically, the same result may be btained by frming (i.e. reslving the frces int) the rectangular cmpnents f the frces in any tw cnvenient perpendicular directins. In Figure 1.4, the x and y cmpnents f are seen t be the algebraic sums f the respective cmpnents f the three frces. Thus, in general, the rectangular cmpnents f the resultant f a cplanar system f frces may be expressed as Σ and y Σ F y x F x where, ( Σ Fx ) + ( Σ Fy ) 3

20 Applied Mechanics The angle made by with x-axis is given by : θ tan 1 Σ F Σ F y x The lcatin f the line f actin f may be cmputed with the help f Varignn s therem. The mment f, Figure 1.4(a), abut sme pint must equal t the sum f the mments f its tw cmpnents F 1 and 1 abut the same pint. The mment f 1, hwever, must equal t the sum f the mments f its cmpnents F and F 3 abut the same pint. It fllws that the mment f abut any pint equals the sum f the mments f F 1, F and F 3 abut this same pint. Applicatin f this principle f mments abut the pint O, shwn in Figure 1.5, gives the equatin d F 1 d 1 F d + F 3 d 3 Figure 1.5 Fr this system f frces where the clckwise directin has been taken as psitive, the distance d is cmputed frm this relatin, and, whse magnitude and directin have already been determined earlier, may nw be cmpletely lcated. In general, then, the mment arm d f the resultant is given by.d M 0 where M 0 stands fr the algebraic sum f the mment f the frces f the system abut any pint O. Figure 1.6 4

21 Fr a system f parallel frces, the magnitude f the resultant is the algebraic sum f the several frces, and the psitin f its line f actin may be btained frm the principle f mments. Nw, cnsider a frce system such as shwn in Figure 1.6, where the plygn f frces may be clsing and cnsequently there will be n resultant frce. Direct cmbinatin by the law f parallelgram shws that fr this case, the resultant is a cuple f magnitude F 3 d. The value f the cuple is equal t the mment sum abut any pint. Thus, it is seen that the resultant f a nn-cncurrent cplanar system f frces may be either a frce r a cuple. Cplanar and Nn-Cplanar Frces 1.7 SUMMAY In this unit, yu have studied the definitin f a frce and its nature. Different types f frce systems are explained, e.g. cplanar, cncurrent, parallel, nn-cplanar and nn-cncurrent frce systems etc. Cmpsitin f frces by laws f triangle f frces, parallelgram f frces and plygn f frces are discussed. Yu have als studied hw the frces can be reslved alng an rthgnal axes system. The translatinal and rtatinal tendencies f frces were als studied. With this knwledge base yu will be able t identify the frce system acting n a bdy and find its resultant. 1.8 ANSWES TO SAQs SAQ 1 (a) Cnsider frces acting at A. The gravity frce acting n the mass f the bdy will cause stretching f the tie member and shrtening f the jib member. Therefre, there are three frces acting at A : (i) (ii) Weight f the bdy acting vertically dwnwards, Tensin in the tie member, and (iii) Cmpressin in the jib member. These three frces pass thrugh the cmmn pint A. This is the system f cncurrent frces. Cnsider frces acting n the beam. If the string at P is cut, pint P will mve dwnward. Thus, the string is ffering a frce acting upwards t keep P in psitin shwn. Similarly, the string at Q is als ffering an upward frce. There are five frces acting n the beam (i) (ii) Weight W 1 acting vertically dwnward n the beam, Weight W acting vertically dwnward, (iii) Tensin in string at P acting vertically upward, (iv) Tensin in string at Q acting vertically upward, and (v) Weight f the beam acting vertically dwnward. The lines f actin f these frces are parallel t each ther. This is the system f parallel frces. 5

22 Applied Mechanics SAQ (c) Cnsider frces acting at C. (d) (a) There are fur frces acting at C. (i) (ii) Weight W acting vertically dwnward, Tensin in the tie, (iii) Cmpressin in left leg, and (iv) Cmpressin in right leg. The system is f cncurrent frces as all the frces pass thrugh the cmmn pint C. If frces acting n the rf truss are cnsidered, the lines f all frces d nt pass thrugh any cmmn pint. Neither the lines f actin f all frces are parallel. This is a system f nn-cncurrent nnparallel frces. Parallelgram Law Here P 60N and Q 80N epresent the frces P and Q in magnitude and directin by drawing lines OA and OB t scale and parallel t the line f actin f frces P and Q, respectively. Figure fr Answer t SAQ (a) : Parallelgram Law Cmplete the parallelgram f frces. Jin OC which is the diagnal f the parallelgram passing thrugh the pint f cncurrence O f P and Q. The diagnal OC, therefre, represents their resultant in magnitude and directin. Measure the length OC and get the magnitude f the resultant. Measure angle COA and get the directin f the resultant with respect t frce P. Using trignmetrical relatins, we get P + Q + PQ csα where and P 60 N and Q 80 N α angle between P and Q cs 105

23 N Q sin α tan θ P + Q csα Cplanar and Nn-Cplanar Frces 80 sin cs θ w.r.t. frce P. P sin α Als tan β Q + P csα sin cs 105 β w.r.t frce Q. Triangle Law Here p 60 N and Q 80 N epresent the frce P in magnitude and directin by drawing line OA t scale and parallel t the line f actin f frce P. Frm pint A, represent frce Q in magnitude and directin by drawing line AB t scale and parallel t the line f actin f frce Q. Then line OB jining the starting pint t the end pint B represents their resultant in magnitude and directin. Length OB gives the magnitude and angle BOA determines and directin f the resultant with respect t frce P. The resultant can als be fund by drawing Q as the first vectr and P as the secnd as shwn in the adjining diagram. Using trignmetric relatins, we get cs 105 sin θ N Q sin α 80 sin θ P Als sin β sin α sin 105 β (Nte : θ + β α 105 ) 7

24 Applied Mechanics Figure fr Answer t SAQ (a) : Triangle Law Parallelgram Law Here P 100 N and Q 10 N epresent frces P and Q in magnitude and directin by drawing lines OA and OB t scale and parallel t the line f actin f frces P and Q, respectively. Cmplete the parallelgram OABC and jin OC. As per the law f parallelgram f frces, OC represents their resultant in magnitude and directin. Measure the length t get the magnitude and angle COA t get the resultant with respect t P. Using trignmetrical relatins, we get where, P + Q + PQ cs α P 100 N and Q 10 N α angle between P and Q cs N Q sin α tan θ P + Q cs α 10 sin cs θ w.r.t. frce P. P sin α Als tan β Q + P cs α sin cs β w.r.t. frce Q.

25 Cplanar and Nn-Cplanar Frces Figure fr Answer t SAQ : Parallelgram Law Triangle Law Here P 100 N and Q 10 N epresent the frce P in magnitude and directin by drawing line OA t scale and parallel t the line f actin f frce P. Frm pint A, represent frce Q in magnitude and directin by drawing line AB t scale and parallel t the line f actin f frce Q. Then line OB jining the starting pint t the end pint B represents their resultant in magnitude and directin as per triangle law f frces. Length OB gives the magnitude and angle BOA determines the directin f the resultant with respect t P. The resultant can als be fund by drawing Q as the first vectr and P as the secnd vectr as shwn in the abve Figure fr Answer t SAQ. Figure fr Answer t SAQ : Triangle Law Using trignmetric relatins, we get cs N sin θ Q sin α sin 80 (sine ule) θ w.r.t frce P. Als sin β P sin α 9

26 Applied Mechanics 100 sin β w.r.t. frce Q. SAQ 3 (a) (Nte : θ + β α 80 ) We cnstruct a plygn such that the frces 60 N, 80 N, 40 N and 50 N represent the sides f a plygn taken in rder, each frce being drawn frm the end f earlier frce as shwn in Figure fr Answer t SAQ 3. Here, the scale t cnstruct the plygn has been taken as 1 cm 30 N. 30 (a) Figure fr Answer t SAQ 3 Nw, the resultant is represented by the line jining the starting pint f the first frce i.e. O t the end f the last frce i.e. D. This line OD measures 3.35 cm. Magnitude f resultant (Linear measurement f OD) (Scale f drawing) 3.35cm 30 N/cm N. T determine the directin f resultant, draw a c-rdinate system at pint D. We find that line OD makes 80 angle with psitive x axis measured in clckwise directin whereas θ is measured in anticlckwise directin frm psitive x axis. S, the directin f resultant θ 80 (measured in anticlckwise frm psitive x axis). eslving all the frces alng x-axis, we get 60 cs cs cs cs Similarly, reslving all frces alng y-axis, we get 60 sin sin sin sin 40

27 N Thus, the resultant in vectr frm may be expressed as : ( 17. N) i + (99.1 N) j The magnitude f the resultant is given by Cplanar and Nn-Cplanar Frces ( x ) + ( y ) ( 17.) + (99.1) N The directin θ can be wrked ut as θ tan 1 y x tan (clckwise) SAQ 4 θ (anticlckwise) The resultant has a magnitude f N and is acting in IVth quadrant making an angle f in anticlckwise directin frm psitive x-axis. In the hexagn ABCDEF shwn in Figure fr Answer t SAQ 4, the frces 1,, 3, 4, 5, and 6 N are acting alng the sides AB, CB, DC, DE, EF, and FA respectively. Figure fr Answer t SAQ 4 Let x1, x, x3, and x4 be the respective perpendicular distance f the frces f, 3, 4 and 5 N alng the sides CB, DC, DE, and EF respectively. Taking mment abut A, M A 1 0 x1 3x + 4x3 + 5x

28 Applied Mechanics As the frces f 1 N and 6 N acting alng AB and FA pass thrugh A, their mment abut this pint is zer. Als x 1 x4 AB sin m Cnsidering the Δ ABC and using csine frmula x3 AB + BC AB BC cs 10 x ( 0.5) 1.04 m M A ( 0.5) (3 1.04) + (4 1.04) + (5 0.5) N-m (anticlckwise) 3

29 EXTA SAQ 4 Cplanar and Nn-Cplanar Frces eslving all the frces alng x axis, we get, x Σ F x F + F + F 1 csθ1 csθ 3cs θ cs0 + 90cs cs N Similarly, reslving all frces alng y axis, we get, y Fy F + F + F 1 sinθ1 sinθ 3θ sin sin sin N Thus, the resultant in vectr frm may be expressed as (38.63N)i +( N)j The magnitude f the resultant is given by x + ( ) ( y ) 33

30 Applied Mechanics ( 38.63) + ( ) N The directin θ can be wrked ut as 1 θ tan y ( ) x tan 1 0 ( ) (clckwise) θ (anticlckwise) The resultant has a magnitude f N and is acting in Ivth quadrant making an angle f In anticlckwise directin frm psitive x axis. 34

31 APPLIED MECHANICS Cplanar and Nn-Cplanar Frces Mechanics is a physical science dealing with the study f frces and mtin f bdies when acted upn by external frces. Applied Mechanics, which frms the subject matter f this curse, is that branch f mechanics which deals with applicatin f the principles f mechanics t the slutin f practical engineering prblems. It prvides the pprtunity t develp lgical thinking, analytical capability, reasning and judgement, which are essential fr the slutin f great variety f prblems fr all branches f engineering and technlgy. This curse, cmprises six units, cvers Statics (Units 1,, 3 and 4) and Dynamics (Units 5 and 6). Statics is a branch f Applied Mechanics cncerned with the study f bdies at rest r in equilibrium under the actin f applied external frces. In Unit 1, yu will learn hw t determine the resultant effect f varius frce system acting n rigid bdies. Unit will enable yu t draw the free-bdy diagrams which are useful in analyzing and slving engineering prblems. Apart frm descriptin f varius trusses and their analysis, yu will als learn the static cnditins f equilibrium in this unit. Unit 3 intrduces the cncept f frictin and its rle in engineering situatins. It als includes the descriptin f different machines, which wrks n the cncepts cvered. Unit 4 discusses the gemetric prperties f bdies like centre f gravity and mment f inertia, which are required in the analysis f prblems. Dynamics is a branch f Applied Mechanics cncerned with the study f bdies in mtin. Unit 5 deals with rectilinear mtin, prjectiles and relative mtin f tw bdies. Unit 6 explains the laws f mtin. Apart frm describing Newtn s laws f mtin, it als discusses mtin n circular path and simple harmnic mtin. Twards the end, the unit includes the cncepts f wrk, pwer and energy. In each unit, yu will find a number f illustrative examples and SAQs (Self Assessment Questins) fr better understanding f the cncepts. Study the text and illustrative examples carefully. Attempt SAQs n yur wn and verify yur answers with thse given at the end f the unit. This will develp yur cnfidence in analysing and slving the practical prblems. At the end, we wish yu all the best fr yur all educatinal endeavurs. 35