The full procedure for drawing a free-body diagram which isolates a body or system consists of the following steps. 8 Chapter 3 Equilibrium

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1 8 Chapter 3 Equilibriu all effect on a rigid bod as forces of equal agnitude and direction applied b direct eternal contact. Eaple 9 illustrates the action of a linear elastic spring and of a nonlinear spring with either hardening or softening characteristics. The force eerted b a linear spring, in tension or copression, is given b F k, where k is the stiffness of the spring and is its deforation easured fro the neutral or undefored position. The representations in Fig. 3/1 are not free-bod diagras, but are erel eleents used to construct free-bod diagras. Stud these nine conditions and identif the in the proble work so that ou can draw the correct free-bod diagras. Construction of Free-od Diagras The full procedure for drawing a free-bod diagra which isolates a bod or sste consists of the following steps. Step 1. Decide which sste to isolate. The sste chosen should usuall involve one or ore of the desired unknown quantities. Step 2. Net isolate the chosen sste b drawing a diagra which represents its coplete eternal boundar. This boundar defines the isolation of the sste fro all other attracting or contacting bodies, which are considered reoved. This step is often the ost crucial of all. ake certain that ou have copletel isolated the sste before proceeding with the net step. Step 3. Identif all forces which act on the isolated sste as applied b the reoved contacting and attracting bodies, and represent the in their proper positions on the diagra of the isolated sste. ake a ssteatic traverse of the entire boundar to identif all contact forces. Include bod forces such as weights, where appreciable. Represent all known forces b vector arrows, each with its proper agnitude, direction, and sense indicated. Each unknown force should be represented b a vector arrow with the unknown agnitude or direction indicated b sbol. If the sense of the vector is also unknown, ou ust arbitraril assign a sense. The subsequent calculations with the equilibriu equations will ield a positive quantit if the correct sense was assued and a negative quantit if the incorrect sense was assued. It is necessar to be consistent with the assigned characteristics of unknown forces throughout all of the calculations. If ou are consistent, the solution of the equilibriu equations will reveal the correct senses. Step 4. Show the choice of coordinate aes directl on the diagra. ertinent diensions a also be represented for convenience. Note, however, that the free-bod diagra serves the purpose of focusing attention on the action of the eternal forces, and therefore the diagra should not be cluttered with ecessive p a_ch03 d_8 Tuesda ar :21 UG Job nuber: 6191a ublisher: Wile uthor: eria Title: Engineering Statics, 5/e t

2 rticle 3/2 Sste Isolation and the Free-od Diagra 9 etraneous inforation. Clearl distinguish force arrows fro arrows representing quantities other than forces. For this purpose a colored pencil a be used. Copletion of the foregoing four steps will produce a correct free-bod diagra to use in appling the governing equations, both in statics and in dnaics. e careful not to oit fro the free-bod diagra certain forces which a not appear at first glance to be needed in the calculations. It is onl through coplete isolation and a ssteatic representation of all eternal forces that a reliable accounting of the effects of all applied and reactive forces can be ade. Ver often a force which at first glance a not appear to influence a desired result does indeed have an influence. Thus, the onl safe procedure is to include on the freebod diagra all forces whose agnitudes are not obviousl negligible. The free-bod ethod is etreel iportant in echanics because it ensures an accurate definition of a echanical sste and focuses attention on the eact eaning and application of the force laws of statics and dnaics. Review the foregoing four steps for constructing a free-bod diagra while studing the saple free-bod diagras shown in Fig. 3/2 and the Saple robles which appear at the end of the net article. Eaples of Free-od Diagras Figure 3/2 gives four eaples of echaniss and structures together with their correct free-bod diagras. Diensions and agnitudes are oitted for clarit. In each case we treat the entire sste as a single bod, so that the internal forces are not shown. The characteristics of the various tpes of contact forces illustrated in Fig. 3/1 are used in the four eaples as the appl. In Eaple 1 the truss is coposed of structural eleents which, taken all together, constitute a rigid fraework. Thus, we a reove the entire truss fro its supporting foundation and treat it as a single rigid bod. In addition to the applied eternal load, the free-bod diagra ust include the reactions on the truss at and. The rocker at can support a vertical force onl, and this force is transitted to the structure at (Eaple 4 of Fig. 3/1). The pin connection at (Eaple 6 of Fig. 3/1) is capable of suppling both a horizontal and a vertical force coponent to the truss. If the total weight of the truss ebers is appreciable copared with and the forces at and, then the weights of the ebers ust be included on the free-bod diagra as eternal forces. In this relativel siple eaple it is clear that the vertical coponent ust be directed down to prevent the truss fro rotating clockwise about. lso, the horizontal coponent will be to the left to keep the truss fro oving to the right under the influence of the horizontal coponent of. Thus,incon- p a_ch03 d_9 Tuesda ar :21 UG Job nuber: 6191a ublisher: Wile uthor: eria Title: Engineering Statics, 5/e t

3 10 Chapter 3 Equilibriu SLE FREE ODY DIGRS echanical Sste Free od Diagra of Isolated od 1. lane truss Weight of truss assued negligible copared with 2. Cantilever bea F 3 F 2 F 1 V F 3 F 2 F 1 3. ea ass F W = g Sooth surface contact at. ass N W = g 4. Rigid sste of interconnected bodies analzed as a single unit Weight of echanis neglected W = g Figure 3/2 structing the free-bod diagra for this siple truss, we can easil perceive the correct sense of each of the coponents of force eerted on the truss b the foundation at and can, therefore, represent its correct phsical sense on the diagra. When the correct phsical sense of a force or its coponent is not easil recognized b direct observation, it ust be assigned arbitraril, and the correctness of or error in the assignent is deterined b the algebraic sign of its calculated value. In Eaple 2 the cantilever bea is secured to the wall and subjected to three applied loads. When we isolate that part of the bea to the right of the section at, we ust include the reactive forces applied to the bea b the wall. The resultants of these reactive forces are shown acting on the section of the bea (E- p a_ch03 d_10 Tuesda ar :21 UG Job nuber: 6191a ublisher: Wile uthor: eria Title: Engineering Statics, 5/e t

4 rticle 3/2 Sste Isolation and the Free-od Diagra 11 aple 7 of Fig. 3/1). vertical force V to counteract the ecess of downward applied force is shown, and a tension F to balance the ecess of applied force to the right ust also be included. Then, to prevent the bea fro rotating about, a counterclockwise couple is also required. The weight g of the bea ust be represented through the ass center (Eaple 8 of Fig. 3/1). In the free-bod diagra of Eaple 2, we have represented the soewhat cople sste of forces which actuall act on the cut section of the bea b the equivalent force couple sste in which the force is broken down into its vertical coponent V (shear force) and its horizontal coponent F (tensile force). The couple is the bending oent in the bea. The free-bod diagra is now coplete and shows the bea in equilibriu under the action of si forces and one couple. In Eaple 3 the weight W g is shown acting through the center of ass of the bea, whose location is assued known (Eaple 8 of Fig. 3/1). The force eerted b the corner on the bea is noral to the sooth surface of the bea (Eaple 2 of Fig. 3/1). To perceive this action ore clearl, visualize an enlargeent of the contact point, which would appear soewhat rounded, and consider the force eerted b this rounded corner on the straight surface of the bea, which is assued to be sooth. If the contacting surfaces at the corner were not sooth, a tangential frictional coponent of force could eist. In addition to the applied force and couple, there is the pin connection at, which eerts both an - and a -coponent of force on the bea. The positive senses of these coponents are assigned arbitraril. In Eaple 4 the free-bod diagra of the entire isolated echanis contains three unknown forces if the loads g and are known. n one of an internal configurations for securing the cable leading fro the ass would be possible without affecting the eternal response of the echanis as a whole, and this fact is brought out b the free-bod diagra. This hpothetical eaple is used to show that the forces internal to a rigid assebl of ebers do not influence the values of the eternal reactions. We use the free-bod diagra in writing the equilibriu equations, which are discussed in the net article. When these equations are solved, soe of the calculated force agnitudes a be zero. This would indicate that the assued force does not eist. In Eaple 1 of Fig. 3/2, an of the reactions,,or can be zero for specific values of the truss geoetr and of the agnitude, direction, and sense of the applied load. zero reaction force is often difficult to identif b inspection, but can be deterined b solving the equilibriu equations. Siilar coents appl to calculated force agnitudes which are negative. Such a result indicates that the actual sense is the opposite of the assued sense. The assued positive senses of and in Eaple 3 and in Eaple 4 are shown on the free-bod diagras. The correctness of these assuptions is p a_ch03 d_11 Tuesda ar :21 UG Job nuber: 6191a ublisher: Wile uthor: eria Title: Engineering Statics, 5/e t

5 12 Chapter 3 Equilibriu proved or disproved according to whether the algebraic signs of the coputed forces are plus or inus when the calculations are carried out in an actual proble. The isolation of the echanical sste under consideration is a crucial step in the forulation of the atheatical odel. The ost iportant aspect to the correct construction of the alliportant free-bod diagra is the clear-cut and unabiguous decision as to what is included and what is ecluded. This decision becoes unabiguous onl when the boundar of the freebod diagra represents a coplete traverse of the bod or sste of bodies to be isolated, starting at soe arbitrar point on the boundar and returning to that sae point. The sste within this closed boundar is the isolated free bod, and all contact forces and all bod forces transitted to the sste across the boundar ust be accounted for. The following eercises provide practice with drawing freebod diagras. This practice is helpful before using such diagras in the application of the principles of force equilibriu in the net article. p a_ch03 d_12 Tuesda ar :21 UG Job nuber: 6191a ublisher: Wile uthor: eria Title: Engineering Statics, 5/e t

6 rticle 3/2 Free-od Diagra Eercises 13 FREE-ODY DIGR EXERCISES 3/ In each of the five following eaples, the bod to be isolated is shown in the left-hand diagra, and an incoplete free-bod diagra (FD) of the isolated bod is shown on the right. dd whatever forces are nec- essar in each case to for a coplete free-bod diagra. The weights of the bodies are negligible unless otherwise indicated. Diensions and nuerical values are oitted for siplicit. od Incoplete FD 1. ell crank supporting ass with pin support at. Fleible cable T g ull 2. Control lever appling torque to shaft at O. O F O 3. oo O, of negligible ass copared with ass. oo hinged at O and supported b hoisting cable at. O T O g 4. Unifor crate of ass leaning against sooth vertical wall and supported on a rough horizontal surface. g 5. Loaded bracket supported b pin connection at and fied pin in sooth slot at. Load L L Figure 3/ p a_ch03 d_13 Tuesda ar :21 UG Job nuber: 6191a ublisher: Wile uthor: eria Title: Engineering Statics, 5/e t

7 14 Chapter 3 Equilibriu 3/ In each of the five following eaples, the bod to be isolated is shown in the left-hand diagra, and either a wrong or an incoplete free-bod diagra (FD) is shown on the right. ake whatever changes or addi- tions are necessar in each case to for a correct and coplete free-bod diagra. The weights of the bodies are negligible unless otherwise indicated. Diensions and nuerical values are oitted for siplicit. od Wrong or Incoplete FD 1. Lawn roller of ass being pushed up incline θ. 2. r bar lifting bod having sooth horizontal surface. ar rests on horizontal rough surface. θ R N g N 3. Unifor pole of ass being hoisted into position b winch. Horizontal supporting surface notched to prevent slipping of pole. Notch T R g F 4. Supporting angle bracket for frae. in joints F F 5. ent rod welded to support at and subjected to two forces and couple. Figure 3/ p a_ch03 d_14 Tuesda ar :21 UG Job nuber: 6191a ublisher: Wile uthor: eria Title: Engineering Statics, 5/e t

8 rticle 3/2 Free-od Diagra Eercises 15 3/C Draw a coplete and correct free-bod diagra of each of the bodies designated in the stateents. The weights of the bodies are significant onl if the ass is stated. ll forces, known and unknown, should be labeled. (Note: The sense of soe reaction coponents cannot alwas be deterined without nuerical calculation.) 1. Unifor horizontal bar of ass suspended b vertical cable at and supported b rough inclined surface at. 5. Unifor grooved wheel of ass supported b a rough surface and b action of horizontal cable. 2. Wheel of ass on verge of being rolled over curb b pull. 6. ar, initiall horizontal but deflected under load L. inned to rigid support at each end. L 3. Loaded truss supported b pin joint at and b cable at. 7. Unifor heav plate of ass supported in vertical plane b cable C and hinge. C L 4. Unifor bar of ass and roller of ass 0 taken together. Subjected to couple and supported as shown. Roller is free to turn Entire frae, pulles, and contacting cable to be isolated as a single unit. L Figure 3/C p a_ch03 d_15 Tuesda ar :21 UG Job nuber: 6191a ublisher: Wile uthor: eria Title: Engineering Statics, 5/e t