DRAWING A FREE-BODY DIAGRAM

Transcription

1 H P T E R 6 RWING REE-OY IGRM The free-bod diagram is the most important tool in this book. It is a drawing of a sstem and the loads acting on it. reating a free-bod diagram involves mentall separating the sstem (the portion of the world ou re interested in) from its surroundings (the rest of the world), and then drawing a simplified representation of the sstem. Net ou identif all the loads (forces and moments) acting on the sstem and add them to the drawing. W 1 W 2 W 1 W 2 215

2 O J E T I V E S On completion of this chapter, ou will be able to: Isolate a sstem from its surroundings and identif the supports efine the loads associated with supports and represent these loads in terms of vectors Inspect a sstem and determine whether it can be modeled as a planar sstem Represent the sstem and the eternal loads acting on it in a diagram called a free-bod diagram s an eample of creating a free-bod diagram, consider the two friends leaning against each other in igure 6.1a. The free-bod diagram of riend 1 is shown in igure 6.1c. In going from igure 6.1a to 6.1c, we oomed in on and drew a boundar around the sstem (riend 1) to isolate him from his surroundings, as shown in igure 6.1b. This boundar is an imaginar surface and the sstem is (b definition) the stuff inside this imaginar surface. The sstem s surroundings are everthing else. You can think of the boundar as a shrink wrap around the sstem. fter drawing the boundar, we identified the eternal loads acting on the sstem either at or across this boundar and drew them at their points of application. These loads represent how the surroundings push, pull, and twist the sstem. In a free-bod diagram we draw the sstem somewhat realisticall and replace the surroundings with the loads the appl to the sstem. It is important to recognie that we are not ignoring the surroundings we simpl replace them with the loads the sstem eperiences because of them. or eample, in igure 6.1c we replace the back of riend 2 with the normal force he applies to the back of riend 1 ( normal, back 2 on back 1 ). This chapter is devoted eclusivel to creating free-bod diagrams. We build on the work in prior chapters on forces and moments and on the engineering analsis procedure presented in hapter 1. reating a free-bod diagram is part of the RW step in the analsis procedure. normal, back 2 on back 1 riend 1 riend 2 W friend 1 friction, floor loor normal, floor (c) igure 6.1 Two friends leaning against one another; isolate riend 1 b drawing a boundar. riend 1 is the sstem; (c) a free-bod diagram of riend 1 216

3 6.1 TYPES O EXTERNL LOS TING ON SYSTEMS TYPES O EXTERNL LOS TING ON SYSTEMS Some of the eternal loads acting on a sstem act across the sstem boundar; the principal eample of this tpe of load is gravit (which manifests itself as weight). nother eample is the magnetic force, which results from electromagnetic field interaction. The magnetic force is what turns a motor. Other eternal loads act directl on the boundar. onsider where: The boundar passes through a connection between the sstem and its surroundings, commonl referred to as a boundar support (or support for short). support ma be, for eample, a bolt, cable or a weld, or simpl where the sstem rests against its surroundings. We replace each support with the loads it applies to the sstem. These loads consist of the contact forces discussed in hapter 4 (normal contact, friction, tension, compression, and shear). Snonms for the term supports are reactions and boundar connections. The boundar separates the sstem from fluid surroundings. We refer to this as a fluid boundar. We replace the fluid with the loads the fluid applies to the sstem. This load consists of the pressure (force per unit area) of the fluid pressing on and/or moving along the boundar. In practice, a sstem ma be acted on b a combination of crossboundar loads (usuall gravit), loads at supports, and loads at fluid boundaries, as illustrated in igure 6.2. Notice that at some boundar locations no loads act. t other locations there are what are called known loads for eample, in igure 6.2, the 40-kN gravit force is a known load. epending on the nature of an eternal load acting on a sstem, when that load is drawn on a free-bod diagram it is represented either b a vector acting at a point of application or as a distributed load acting on an area. The load is given a unique variable label, and its magnitude (if it is a known load) is written net to the vector. able Tension (boundar support) W Pressure from water (fluid boundar) Weight of ship 40 kn igure 6.2 Isolate the ship b drawing a boundar. The ship is the sstem; a free-bod diagram of the ship able Tension (boundar support) E X E R I S E S The sstem to be considered is a coat rack with some items hanging off of it as shown in E In our mind draw a boundar around the sstem to isolate it from its surroundings. a. Make a sketch of the coat rack and the eternal loads acting on it. Show the loads as vectors and label them with variables, and where possible give word descriptions of the loads. b. List an uncertainties ou have about the free-bod diagram ou have created. E6.1.1

4 218 H 6 RWING REE-OY IGRM The sstem to be considered is defined as a mobile as shown in E In our mind draw a boundar around the sstem to isolate it from its surroundings at point E. a. Make a sketch of the mobile and the eternal loads acting on it. Show the loads as vectors and label them with variables, and where possible give word descriptions of the loads. b. List an uncertainties ou have about the free-bod diagram ou have created. E E The sstem to be considered is a person and a ladder, as shown in E In our mind draw a boundar around the sstem to isolate it from its surroundings. a. Make a sketch of the sstem and the eternal loads acting on it. Show the loads as vectors and label them with variables, and where possible give word descriptions of the loads. b. List an uncertainties ou have about the free-bod diagram ou have created. E Visit a weight room, and take a look at one of the eercise stations preferabl one that is in use! a. onsider where the person is standing, hanging, laing, and/or pushing on it. In our mind, draw a boundar around the person to define him or her as the sstem. Make a sketch of the sstem and the eternal loads acting on it, showing the loads as vectors with variable labels. Where possible give word descriptions of the loads. List an uncertainties ou have about the free-bod diagram ou have created. b. onsider where the person is standing, hanging, laing, and/or pushing on it. In our mind, draw a boundar around the eercise machine to define it as the sstem. Make a sketch of the sstem and the eternal loads acting on it, showing the loads as vectors with variable labels. Where possible give word descriptions of the loads. List an uncertainties ou have about the free-bod diagram ou have created Visit a local plaground near campus, and take a look at a jungle gm preferabl one that is in use! onsider where the children (or adults!) are standing/hanging. In our mind, draw a boundar around the jungle gm to define it as our sstem. Make a sketch of the sstem and the eternal loads acting on it, showing the loads as vectors with variable labels. List an uncertainties ou have about the free-bod diagram ou have created Visit a local pet store or oo and look at the fish tanks. a. In our mind, draw a boundar around the fish tank including the water to define it as our sstem. Make a sketch of the sstem and the eternal loads acting on it, showing the loads as vectors with variable labels. List an uncertainties ou have about the free-bod diagram ou have created. b. In our mind, draw a boundar around the fish tank ecluding the water to define it as our sstem. Make a sketch of the sstem and the eternal loads acting on it, showing the loads as vectors with variable labels. List an uncertainties ou have about the free-bod diagram ou have created. 6.2 PLNR SYSTEM SUPPORTS We now consider how to identif supports and represent the loads associated with them when working with planar sstems. sstem is planar if all the forces acting on it can be represented in a single plane and all moments are about an ais perpendicular to that plane. If a sstem is not planar it is a nonplanar sstem. In Section 6.4 we will la out guidelines for identifing planar and nonplanar sstems. or now, we assume that all of the sstems we are dealing with in this section are planar.

5 6.2 PLNR SYSTEM SUPPORTS 219 J J (c) (d) J J H E G igure 6.3 n object connected to its surroundings b various supports. The object can be modeled as a planar sstem onsider the sstems in igure 6.3 for which we want to draw feebod diagrams. Each sstem consists of a uniform bar of weight W, oriented so that gravit acts in the negative direction. However, each has different supports that connect it to its surroundings. or eample: In igure 6.3a, the supports consist of normal contact without friction at, and cable attached to the sstem at. In igure 6.3b, the supports consist of a spring attached to the sstem at, and normal contact with friction at. In igure 6.3c, the support consists of a sstem fied to its surroundings at E. In igure 6.3d, the supports consist of a sstem pinned to its surroundings at G, and a link attached to the sstem at H. t each support we consider whether the surroundings act on the sstem with a force and/or a moment. s a general rule, if a support prevents the translation of the sstem in a given direction, then a force acts on the sstem at the location of the support in the opposing direction. Likewise, if rotation is prevented, a moment opposing the rotation acts on the sstem at the location of the support. t Point (Normal ontact Without riction). t this support the sstem rests against a smooth, frictionless surface. normal force prevents the sstem from moving into the surface and is oriented so

6 220 H 6 RWING REE-OY IGRM (c), cable J W, normal J, normal (d ), spring W, friction H, link as to push on the sstem. ecause the supporting surface at is smooth, friction between the sstem and its surroundings is noneistent. Therefore there is no force component parallel to the surface. In igure 6.4a (the free-bod diagram of igure 6.3a) the force resulting from normal contact at is represented b,normal ; we know its direction is normal to the surface so as to push on the sstem. t Point (able). t this support a force acts on the sstem; the line of action of the force is along the cable. The force represents the cable pulling on the sstem because the cable can onl act in tension. In igure 6.4a the force from the cable at is represented b,cable ; we know its direction is along the cable ais in the direction that allows the cable to pull on the sstem. E ME J E W E J E W igure 6.4 ree-bod diagrams of the planar sstems shown in igure 6.3 t Point (Spring). t this support there is a force that either pushes or pulls on the sstem; the line of action of the force is along the ais of the spring. If the spring is etended b an amount, the spring is in tension and the force is oriented so as to pull on the sstem. If the spring is compressed b an amount, the force is oriented so as to push on the sstem. The magnitude of the force is proportional to the amount of spring etension or compression, and the proportionalit constant is the spring constant, k. In other words, the sie of the force is equal to the product of k and the spring etension or compression: k( ) (6.1) where the dimensions of k are force/length (e.g., N/mm). The value of in (6.1) will be positive when the spring is in tension (since will be positive) and negative when the spring is compressed (since will be negative). In igure 6.4b the spring force at is represented b,spring ; we know its direction is along the spring ais. If the spring is in tension, the force acts so as to pull on the sstem. If the spring is in compression, the force acts so as to push on the sstem. We have drawn the direction of,spring to indicate that the spring is in tension. We could equall well have chosen the direction of,spring to indicate that the spring is in compression, but as we will see in the net chapter, drawing the spring in tension will make interpreting numerical answers easier. t Point (Normal ontact with riction). t this support there are two forces acting on the sstem. One is a normal force (just like normal contact without friction). The second is due to friction and is parallel to the surface against which the sstem rests therefore it is perpendicular to the normal force. The force due to friction ( friction ) is related to and limited b normal contact force ( normal ) and the characteristics of the contact. Often the relationship between friction and normal is represented in terms of the oulomb riction Model. This model states that if friction static normal the sstem will not slide relative to its surroundings, where

7 6.2 PLNR SYSTEM SUPPORTS 221 static is the coefficient of static friction and tpicall ranges from 0.01 to 0.70, depending on the characteristics of the contact. If friction static normal, the condition is that of impending motion. In igure 6.4b the normal force at is represented b,normal ; we know its direction is normal to the surface so as to push on the sstem. The friction force is represented b,friction and is parallel to the surface and perpendicular to the normal force. We could have drawn,friction to point to the right or to the left; we arbitraril chose to draw it upward to the right. t Point E (Sstem ied to Surroundings, Referred to as a ied Support). t this support a force and a moment act on the sstem. To get a feeling for a fied boundar, consider the setup shown in igure 6.5 (better et, reproduce it ourself). The ruler is the sstem, and our hands are the surroundings. Grip one end of the ruler firml with our left hand and appl a force with a finger of our right hand, as shown in igure 6.5a. Notice that our left hand automaticall applies a load (consisting of a force and a moment) to the ruler in order to keep the gripped end from translating and rotating; this load fies the gripped end relative to our left hand. The load of our left hand acting on the ruler (a fied support) can be represented as a force of LH LH, i LH, j and a moment of M LH M LH, k that are the net effect of our left hand gripping the ruler (see igure 6.5b). Returning now to the sstem depicted in igure 6.3c, we can describe the loads acting at the fied support at E as E E i E j and M E M E k. These loads are shown in igure 6.4c. M LH LH, LH, igure 6.5 Illustrating a planar fied boundar connection: appling loads to a ruler; the resulting freebod diagram if the ruler is defined as the sstem. Note how the loads the lefthand applies to the ruler are depicted. t Point G (Sstem Pinned to Its Surroundings, Referred to as a Pin onnection). pin connection consists of a pin that is loosel fitted in a hole. t this support a force acts on the sstem. To get a feeling for the force at a pin connection consider the phsical setup in igure 6.6a. The ruler (which is the sstem) is ling on a flat surface in position 1. pencil, which is acting like a pin, is placed in the hole in the ruler and is gripped firml with our left hand. The pencil and our hands constitute the surroundings. Now load the sstem with our right hand as shown in igure 6.6a; notice how our left hand reacts with a force to counter the right-hand force. If ou net orient the ruler and right hand load as shown in igure 6.6b, again our left hand counters with a force. inall, load the ruler as shown in igure 6.6c, and notice that the ruler rotates because our left hand is unable to counter with an opposing moment. This eercise tells ou that there is a force ( LH ) acting on the sstem at the pin connection but no moment. The force LH lies in the plane perpendicular to the pencil s length. or the situation in igure 6.6, this means that LH can be written LH LH, i LH, j (igure 6.6d). or the sstem in igure 6.3d, we can describe the load acting at the pin connection at G as G G i G j, as shown in igure 6.4d. We have arbitraril chosen to draw both components in their respective positive direction.

8 222 H 6 RWING REE-OY IGRM Right-hand force Position 1 Position 2 Right-hand force Right-hand force What happens? Position 3 LH, LH, (c) (d) igure 6.6 Illustrating a planar pin connection: appling loads to a ruler (Position 1); appling loads to a ruler (Position 2); (c) appling loads to a ruler (Position 3); (d) loads acting on the ruler at the pin connection t Point H (Link). t this support there is a force that either pushes or pulls on the sstem; the line of action of the force is along the ais of the link. We shall have a lot more to sa about links in the net chapter for now we simpl sa that a link is a member with a pin connection at each end and no other loads acting on it. In igure 6.4d the force at H is represented b H,link acting along the long ais of the link. link ma either push or pull on the sstem, and here we have chosen to assume pulling. We could equall well have chosen the direction of H,link to indicate that the link is pushing, but as we will see in the net chapter, drawing the link as pulling will make interpreting numerical answers easier. The free-bod diagrams of the planar sstems in igure 6.3 are presented in igure 6.4. These diagrams include loads due to supports, as well as the load due to gravit acting at J. Each load is represented as a vector and is given a variable label. If the magnitude of a load is known, this value is included on the diagram. Summar Table 6.1 summaries the loads associated with the planar supports discussed, along with some other commonl found planar supports. on t feel that ou need to memorie all the supports in this table it is presented merel as a read reference. On the other hand, ou should be familiar with the loads associated with these standard planar supports.

9 6.2 PLNR SYSTEM SUPPORTS 223 Table 6.1 Standard Supports for Planar Sstems () () Loads to e Shown on Supports escription of Loads ree-od iagram 1. Normal contact without friction orce () oriented normal to surface on which sstem rests. irection is such that force pushes on sstem. Sstem Smooth surface 2. able, rope, wire orce () oriented along cable. irection is such that cable pulls on the sstem. Sstem able of negligible weight 3. Spring orce () oriented along long ais of spring. irection is such that spring pulls on sstem if spring is in tension, and pushes if spring is in compression. Sstem Etended spring ompressed spring 4. Normal contact with friction Two forces, one ( ) oriented normal to surface on which the sstem rests so as to Sstem push on sstem, other force ( ) is tangent to surface. Rough surface (normal) (friction) 5. ied support orce in plane represented in terms Weld of components and. Moment about ais (M ). M Sstem Sstem M

10 224 H 6 RWING REE-OY IGRM Table 6.1 (ont.) () () Loads to e Shown on Supports escription of Loads ree-od iagram 6. Pin connection orce perpendicular to pin represented in (pin or hole is part of sstem) terms of components and. Point of application is at center of pin. Pin Sstem 7. Link orce () oriented along link length; force can push or pull on the sstem. Sstem Link 8. Slot-on-pin orce () oriented normal to long ais of (slotted member is part of sstem) slot. irection is such that force can pull or push on sstem. Sstem 9. Pin-in-slot orce () oriented normal to long ais of (pin is part of sstem) slot. irection is such that force can pull or push on sstem. Sstem 10. Smooth collar on smooth shaft orce () oriented perpendicular to long ais of shaft. irection is such that force Sstem can pull or push on sstem. Moment (M ) about ais. M M 11. Roller or rocker orce () oriented normal to surface on which sstem rests. irection is such that Sstem Sstem Sstem force pushes on sstem. θ Roller θ Roller θ Rocker θ

11 6.2 PLNR SYSTEM SUPPORTS 225 EXMPLE 6.1 OMPLETE REE-OY IGRMS In igure 6.7 and igure 6.8, a block is supported at several points and the sstem is defined as the block. Gravit acts downward in the direction at the indicated center of gravit (G). In igure 6.7, the surface at is rough. s shown in igure 6.8, the magnitudes of and M are 10 lb and 40 in.-lb, respectivel. Eplain wh these figures are not free-bod diagrams. reate a free-bod diagram of each sstem. cable G α igure 6.7 Goal Eplain wh the two figures are not free-bod diagrams and create complete correct free-bod diagrams. Given We are given two sstems with specified loads and supports. ssume We assume that the sstem in each figure is planar because the known loads and the loads applied b supports all lie in a single plane. We also assume the slot-pin connection at in igure 6.8 is smooth (frictionless). raw or each sstem, isolate it b drawing a boundar around the block as is done in igure 6.9 and igure Then replace each support b its associated loads. Solution Neither figure is a free-bod diagram because the sstem (the block) has not been isolated from its surroundings at and each block is still shown connected to its surroundings. igure 6.7: We isolate the block using the boundar shown in igure 6.9, establish the coordinate sstem for the entire sstem and the ** coordinate sstem to simplif the representation of the support at. Then we note the following: t a pin connection attaches the sstem to its surroundings. ccording to Table 6.1, a pin connection applies a force to the sstem. s we do not know the direction or magnitude of this force, we represent it as two components, and, which we arbitraril draw in the positive and directions (igure 6.11). t the sstem rests against a surface inclined at angle relative to the horiontal. Since we know that the surface is rough, we must consider the friction between surface and sstem. ccording to Table 6.1, there will be a normal force,normal acting on the sstem, oriented perpendicular to the surface so as to push on the sstem, as shown in igure We do not know its magnitude. There is also the friction force,friction, perpendicular to,normal. We do not know the magnitude of,friction or whether it acts in the * or * direction, and so we arbitraril draw it in the * direction. t a cable pulls on the sstem, which we represent as a force of unknown magnitude but known direction ( ). t G (the center of gravit) a force W acts in the direction. M (40 in. lb) G (10 lb) igure 6.8 cable * α G * igure 6.9 M (40 in. lb) G (10 lb) igure 6.10

12 226 H 6 RWING REE-OY IGRM, normal * α, friction * * * W igure 6.11 M (40 in.-lb) W igure 6.12 (10 lb) The block presented in igure 6.11, with forces,,,normal,,friction,, and W each drawn at its point of application is a free-bod diagram of the sstem in igure 6.7. igure 6.8: We isolate the block using the boundar shown in igure 6.10, establish an coordinate sstem as shown, and then we note the following: t a pin connection attaches the sstem to its surroundings. ccording to Table 6.1, a pin connection applies a force to the sstem. s we do not know the direction or magnitude of this force, we represent it as two components, and, which we arbitraril draw in the positive and directions (igure 6.12). t a slot in the block is attached to a slider that allows the block to move in the direction but prevents movement in the direction. Therefore, we include a force, which we have arbitraril drawn in the positive direction. t there is a known force ( ) and a moment (M ). Known values are written net to the vectors. t G (the center of gravit) a force W acts in the direction. The block presented in igure 6.12, with forces,,, and,known, W, and moment M,known each drawn at its point of application constitutes a free-bod diagram of the sstem. Notice that this diagram includes the known magnitudes of and M. nswer igures 6.7 and 6.8 are not free-bod diagrams of sstems because each sstem (the block) has not been full separated from all supports. The free-bod diagrams of the sstems in igures 6.7 and 6.8 are shown in igures 6.11 and 6.12, respectivel. EXMPLE 6.2 EVLUTING THE ORRETNESS O REE-OY IGRMS onsider the description of each planar sstem in igures and determine whether the associated free-bod diagram is correct. Unless otherwise stated, assume that the weight of the sstem is negligible and therefore can be ignored. igure 6.13 P thin rod is supported b a smooth tube that is fied to the wall (igure 6.13a). The rod is touching the tube interior at and leaning on the tube end at. known force P is pushing on the rod at. ssume no friction on the surface of the tube.

13 6.2 PLNR SYSTEM SUPPORTS 227 nswer nswer Tube: The proposed free-bod diagram of the tube in igure 6.13b is correct. It accounts for the fied end at the left and the normal contact between the tube and the rod. Rod: The proposed free-bod diagram of the tube in igure 6.13b is not correct. The force,tube on rod in this diagram, representing the normal force of the tube pushing on the rod, should be in the other direction (Newton s third law). s shown, the tube is pulling on the rod which is not phsicall possible. beam, pinned at and resting against a roller at, is loaded b a 2-kN force and a 2.4-kN m moment (igure 6.14a). nswer The proposed free-bod diagram of the beam in igure 6.14b is not correct. The normal force,,normal, acting on the beam at should be oriented perpendicular to the inclined surface. (c) uniform beam weighing 200 lb is fied at. 400-lb and 1400-lb load are applied at and as shown (igure 6.15a). nswer The proposed free-bod diagram in igure 6.15b is correct. The weight of the beam is included at the 5 ft position since the beam is uniform. (d) door that weighs W hangs from hinges at and. Hinge acts like a pin connection, and hinge acts like a vertical slot support (igure 6.16a). nswer The proposed free-bod diagram in igure 6.16b is not correct. The hinge at does not appl a vertical force to the door, so this force should not be on the free-bod diagram. The vertical force component at ( ) is a thrust force., wall on tube eam M, wall on tube, wall on tube, tube on rod 2 kn, tube on rod, rod on tube 2 kn Tube Rod 2.4 kn m, rod on tube igure 6.13 (ont.) Proposed freebod diagram 2.4 kn m Inclined surface 400 lb igure 6.14, normal 1400 lb 3 ft 7 ft 400 lb oor M 200 lb 5 ft 1400 lb W igure 6.15 igure 6.16

14 228 H 6 RWING REE-OY IGRM Gman G table (e) man weighing 800 N sits at the picnic table halfwa between its two ends. His center of gravit (G man ) is noted. The table weighs 200 N, with a center of gravit at G table (igure 6.17a). ssume that an friction between the legs and the ground can be neglected. nswer The proposed free-bod diagram in igure 6.17b is not correct. The label for the normal force acting on the table at should be 2, (not ), because the force vector at represents the normal force for two legs. Table G man 800 N 2 igure 6.17 hatch G table 200 N (f) frame used to lift a hatch is pinned to the hatch at and. force is applied to the frame at (igure 6.18a). nswer This free-bod diagram in igure 6.18b is not correct. t each pin connection force components in the and directions should be shown. (g) frame consisting of members and supports the pulles, cable, and block L (igure 6.19a). nswer Whole frame: The proposed free-bod diagram in igure 6.19b is correct. It includes the forces at pins and, the cable tension cable pulling on the frame, and the gravit force from block L (W L ). Member : The proposed free-bod diagram in igure 6.19b is incorrect because the forces at the pin connection at have not been included Member : The proposed free-bod diagram in igure 6.19b is correct. Whole frame rame able E W L igure 6.18 E able Member Member E E E igure 6.19 L

15 6.2 PLNR SYSTEM SUPPORTS 229 (h) 50-kg roller is pulled up a 20 incline with a force P. s it is pulled over a smooth step, all of its weight rests against the step (igure 6.20a). P nswer The proposed free-bod diagram in igure 6.20b is not correct because the gravit force should be drawn in the negative * direction. The mass of the roller has been properl converted to weight on earth. The force is drawn correctl. (i) 1200-lb object is held up b a force T on a rope threaded through a sstem of frictionless pulles (igure 6.21a). nswer Pulle : The proposed free-bod diagram in igure 6.21b is correct. Since the pulles are frictionless, the force throughout the rope is constant (we will prove this in hapter 7). The tension on the rope is T wherever one cuts the rope. Pulle : The proposed free-bod diagram in igure 6.21b is correct. T T Pulle T T Pulle Roller igure * β θ P θ 20 * β W 490 N 1200 lb igure lb T T E X E R I S E S onsider the description of each planar sstem and determine whether the proposed free-bod diagram is correct. a. curved beam of weight W is supported at b a pin connection and at b a rocker, as shown in E6.2.1a. b. beam is pinned at and rests against a smooth incline at as shown in E6.2.1b. The total weight of the beam is W. c. forklift is lifting a crate of weight W 1 as shown in E6.2.1c. The weight of the forklift is W 2. The front wheels are free to turn and the rear wheels are locked. d. mobile hangs from the ceiling from a cord as shown in E6.2.1d. e. force acts on a brake pedal, as shown in E6.2.1e. f. child balances on the beam as shown in E6.2.1f. Planes and are smooth. The weight of the beam is negligible. g. beam is bolted to a wall at as shown in E6.2.1g. The weight of the beam is negligible. 500 N 200 N N 200 N W Proposed free-bod diagram. E6.2.1

16 230 H 6 RWING REE-OY IGRM (c) (d) E W 1 W 2 E 45 W 1 W 2 M E E Proposed free-bod diagram. Proposed free-bod diagram. Proposed free-bod diagram. (e) ( f) (g) k 200 N 300 N 350 N 20 L/2 φ L L/2 θ N E spring, tension 350 N foot L/2, normal W L/2 θ, normal M 200 N E 250 N 300 N Proposed free-bod diagram. Proposed free-bod diagram. Proposed free-bod diagram. E6.2.1 (ont.)

17 6.3 NONPLNR SYSTEM SUPPORTS The beam of uniform weight is fied at and rests against a smooth block at (E6.2.2). In addition, a 100-N weight hangs from point. ased on information in Table 6.1, what loads do ou epect to act on the beam at due to the fied condition? What loads do ou epect to act on the beam at where it rests on the smooth block? Present our answer in terms of a sketch of the beam that shows the loads acting on it at,, and. lso comment on whether the sketch ou created is or is not a free-bod diagram The belt-tensioning device is as shown in E Pulle is pinned to the L-arm at. ased on information in Table 6.1, what loads do ou epect to act on the pulle at? What loads do ou epect to act on the pulle due to the belt tension? Present our answer in terms of a sketch of the pulle that shows loads acting on it at and due to the belt tension. lso comment on whether the sketch ou created is or is not a free-bod diagram. T N E The truss is attached to the ground at with a rocker and at with a pin connection (E6.2.3). dditional loads acting on the beam are as shown. ased on information in Table 6.1, what loads do ou epect to act on the truss at due to the rocker connection? What loads do ou epect to act on the truss at due to the pin connection? Present our answer in terms of a sketch of the truss that shows the loads acting on it at and. lso comment on whether the sketch ou created is or is not a freebod diagram. Pulle O 30 L arm elt 30 T Mass E able passes over the small frictionless pulle without a change in tension and holds up the metal clinder (E6.2.5). ased on information in Table 6.1, what loads do ou epect to act on the clinder? Present our answer in terms of a sketch of the clinder that shows all the loads acting on it. lso comment on whether the sketch ou created is or is not a free-bod diagram E E N E NONPLNR SYSTEM SUPPORTS Now we consider how to identif nonplanar sstem supports and represent the loads associated with them. sstem is nonplanar if all the forces acting on it cannot be represented in a single plane or all moments acting on it are not about an ais perpendicular to that plane. You will see similarities to our discussion in the prior section on planar sstems and some important differences. s with planar sstems, if a boundar location prevents the translation of a nonplanar sstem in a given direction, then a force acts on the sstem at the location of the support in the opposite direction. Likewise, if rotation is prevented,

18 232 H 6 RWING REE-OY IGRM cable 2 spring 1 G 3 Normal contact without friction ied support 5 4 G a moment opposite the rotation acts on the sstem at the location of the support. onsider the sstems in igure 6.22, for which we want to draw freebod diagrams. The three nonplanar supports in igure 6.22a (normal contact without friction, cable, spring) are identical to their planar counterparts. ssociated with each support is a force acting along a known line of action, as depicted in igure 6.23a. The two nonplanar supports in igure 6.22b are similar to their planar counterparts. Normal contact with friction involves normal and friction forces, where the friction force is in the plane perpendicular to the normal force; these are represented in igure 6.22b as 4, 4, and 4, respectivel. The fied support of a nonplanar sstem is able to prevent the sstem from translating along and rotating about an ais therefore it involves a force with three components ( 5 5 i 5 j 5 k) and a moment with three components (M 5 M 5 i M 5 j M 5 k). The freebod diagram of the sstem in igure 6.22b is depicted in igure 6.23b. nother commonl found nonplanar support is a single hinge (igure 6.24a). It does not restrict rotation of the sstem about the hinge Normal contact with friction igure 6.22 plate connected to its surroundings b various supports. The plate must be modeled as a nonplanar sstem. cable M M 1 normal 2 G W spring 3 M M 5 5 M 5 W igure 6.23 ree-bod diagram of the nonplanar sstems shown in igure 6.22 igure 6.24 The forces acting on the sstem ( block) at a single-hinge connection; the forces acting on the sstem (a trap door) at multiple hinges; (c) hinges designed to prevent motion along the hinge ais. The force component is commonl referred to as a thrust force. (c)

19 6.3 NONPLNR SYSTEM SUPPORTS 233 pin. single hinge applies a force (with two components) and a moment (with two components) perpendicular to the ais of the hinge. or the eample in igure 6.24a, we represent the loads acting at the single hinge support as i j and moment M M i M j. If a hinge is one of several properl aligned hinges attached to a sstem, each hinge applies a force perpendicular to the hinge ais (see igure 6.24b) and no moment. epending on the design of a hinge (and regardless of whether it is a single hinge or one of several), it ma also appl a force along the ais of the pin ( in igure 6.24c). The eperiments outlined in Eample 6.3 are intended to illustrate the difference in the loads involved with single versus multiple hinges. Summar Table 6.2 summaries the loads associated with supports for nonplanar sstems. Other supports commonl found with nonplanar sstems are also included in the table. or eample, the ball-and-socket support restricts all translations of the sstem b appling a force to the sstem, but it does not restrict rotation of the sstem about an ais. n eample of a ball-and-socket support familiar to everone is the human hip joint (igure 6.25). s another eample, a journal bearing does not restrict sstem rotation about one ais, while restricting translation in a plane perpendicular to the ais (igure 6.25d). Take a few minutes to stud Table 6.2 and notice the similarities and differences between hinges, journal bearings, and thrust bearings. Table 6.2 is not an ehaustive list of nonplanar supports. It contains commonl found and representative eamples. If ou find ourself considering a support that is not neatl classified as one of these, remember that ou can alwas return to the basic characteristics associated with an support: If a support prevents the translation of the sstem in a given direction, then a force acts on the sstem in the opposing direction. If rotation is prevented, a moment opposing the rotation is eerted on the sstem. all Socket (c) earings igure 6.25 ball-and-socket connection; the forces that the socket applies to the ball; (c) the hip joint is a ball-and-socket connection; (d) isometric view of a journal bearing with a shaft running through it Shaft (sstem) (d)

20 234 H 6 RWING REE-OY IGRM Table 6.2 Standard Supports for Nonplanar Sstems () () Loads to e Shown in Support escription of oundar Loads ree-od iagram 1. Normal contact without friction orce () oriented normal to surface on which sstem rests. irection is such that Sstem force pushes on sstem. 2. able, rope, wire orce () oriented along cable. irection is such that force pulls on sstem. Sstem 3. Spring orce () oriented along long ais of spring. irection is such that force pulls on sstem if spring is in tension and pushes if spring is in compression. Spring Sstem Spring in tension + Spring in compression 4. Normal contact with friction Two forces, one ( ) oriented normal to surface so as to push on sstem, other force is tangent to surface on which the sstem rests and is represented in terms Sstem of its components ( ). 5. ied support orce represented in terms of components ( ). M M M Moment represented in terms of components (M M M ). Sstem M M M

21 6.3 NONPLNR SYSTEM SUPPORTS 235 () () Loads to e Shown in Support escription of oundar Loads ree-od iagram 6. Single hinge orce in plane perpendicular to shaft ais; (shaft and articulated collar) represented as and components ( ). M M Moment with components about aes Sstem perpendicular to shaft ais (M M ). OR epending on the hinge design, ma also appl force along ais of shaft, ( ). M M M M 6. Multiple Hinges orce in plane normal to shaft ais t hinge : (one of two or more properl represented in terms of components t hinge : aligned hinges) ( ). Point of application at center of shaft. OR Sstem epending on design, ma also appl force along ais of shaft ( ). 7. all and socket support orce represented as three components. (ball or socket as part of sstem) Sstem 8. Single journal bearing orce in plane perpendicular to shaft ais; (frictionless collar that holds represented as and components ( ). M M a shaft) Moment with components about aes perpendicular to shaft ais (M M ). Sstem M M (ontinued)

22 Table 6.2 (ont.) () () Loads to e Shown in Support escription of oundar Loads ree-od iagram 8. Multiple journal bearings orce in plane perpendicular to shaft t journal bearing : (two or more properl aligned ais represented in terms of components journal bearings holding a shaft) ( ). Point of application t journal bearing : at center of shaft. Sstem 9. Single thrust bearing orce represented in terms of three (journal bearing that also restricts motion along ais components ( ). omponent in direction of shaft ais ( ) is sometimes M M of shaft) referred to as the thrust force. Point of application is at center of shaft. M Moment with components perpendicular to Sstem shaft ais (M M ). Smaller diameter M 9. Multiple thrust bearings orce represented in terms of three t thrust bearing : (one of two or more properl components ( ). omponent aligned thrust bearings) in direction of shaft ais ( ) is sometimes Thrust bearing referred to as the thrust force. Point of Sstem application is at center of shaft. Journal bearing or thrust bearing Smaller diameter 10. levis: ollar on shaft with pin orce with components perpendicular (collar and shaft are part to shaft ais ( ). M of sstem) Moment with components perpendicular to ollar shaft ais (M ). Pin Round shaft (sstem) M 11. Smooth roller in guide orce represented as two components. Sstem One component ( ) normal to surface on which sstem rests; the other is perpendicular to rolling direction ( ).

23 6.3 NONPLNR SYSTEM SUPPORTS 237 EXMPLE 6.3 EXPLORING SINGLE N OULE ERINGS N HINGES or each situation described below, draw a free-bod diagram and describe the loads involved. To create the situations ourself, ou will need a ardstick, a rubber band, and a cand bar (to serve as a weight). Your hands will serve as models of bearings and hinges. When considering the sstem, ignore the weight of the ardstick. Gravit Situation 1: Hold the ardstick level as shown in igure 6.26a. The left hand is at 0 in. and the right hand is at 18 in. The cand bar is hanging at the far right. escription The left-hand fingers push down on the top of the ardstick. Notice that ou can move our left thumb awa from the stick because there is no load on it. The right thumb pushes up on the bottom of the ardstick. Notice that ou can move our right-hand fingers awa from the stick because there is no load on them. ree-od iagram for Situation 1 onsider how the hands appl loads to the ardstick. efining the ardstick as the sstem, draw these loads on the ardstick to create the free-bod diagram (igure 6.26b). Situation 2: Hold the ardstick level as shown in igure 6.27a. The left hand is at 9 in. and the right hand is at 18 in. The cand bar is hanging at the far right. Each hand acts like a bearing or hinge. escription The description for (1) still holds. The difference is, that in order to keep the ardstick level, the forces involved in pushing down with the left fingers and up with the right thumb are larger in magnitude. ree-od iagram for Situation 2 onsider how the hands appl loads to the ardstick. efining the ardstick as the sstem, draw these loads on the ardstick to create the free-bod diagram (igure 6.27b). Situation 3: Hold the ardstick level as shown in igure 6.28a. The right hand is at 18 in., and the cand bar is hanging at the far right. The right hand acts like a single bearing or hinge. escription The thumb pushes up on the bottom of the ardstick and in conjunction with the right-hand fingers works to prevent the stick from rotating; in doing this, the right hand applies a moment and pushes upward with a force. ree-od iagram for Situation 3 onsider how the hand applies loads to the ardstick. efining the ardstick as the sstem, draw these loads on the ardstick to create the free-bod diagram (igure 6.28b). Summar Situations 1 and 2 are analogous to sstems with two properl aligned bearings or hinges, with the hands plaing the role of bearings/hinges. lthough each hand applies onl a force, each does create an equivalent moment at a specified moment center (as introduced in hapter 5). or eample, if we call 18 in. the moment center (this is the point of 1 2 Left hand = 0 in left hand 2 = 0 in Left hand = 9 in left hand Right hand = 18 in Yard stick right hand W cand Right hand = 18 in right hand Yard stick and bar = 36 in. igure 6.26 Situation 1; freebod diagram of Situation and bar = 36 in. W cand igure 6.27 Situation 2; freebod diagram of Situation 2

24 238 H 6 RWING REE-OY IGRM M right hand Right hand = 18 in. and bar = 36 in. right hand W cand igure 6.28 Situation 3; free-bod diagram of Situation 3 application of right hand ) in Situation 1, left hand creates a counterclockwise equivalent moment of (18 in. right hand ) that counters the clockwise equivalent moment created b the dangling cand of (18 in. W cand ). In situation 2 the cand stas at the same position, creating the same clockwise equivalent moment of (18 in. W cand ) about 18 in. Since the left hand is placed at 9 in., it must eert a larger force to maintain the same counterclockwise moment. In situation 3 the right hand acts like a single bearing or hinge, and must appl a force and a moment to counter the clockwise moment created b the dangling cand that is 18 inches from the hand. Notice that in igure 6.28b the right hand applied both a force and a moment to the ard stick. EXMPLE 6.4 EVLUTING THE ORRETNESS O REE-OY IGRMS onsider the description of each nonplanar sstem in igures and determine whether the associated free-bod diagram is correct or not correct. Unless stated otherwise, assume that gravit forces can be ignored. G Situation : The triangular plate in igure 6.29a is supported b a ball and socket at, a roller at, and a cable at. The plate weighs 100 N. T cable Plate nswer The proposed free-bod diagram in igure 6.29b is not correct. ecause the cable is in tension, it will pull on the plate in the direction (not push on it in the direction, as shown). 100 N G Situation : bar is supported b three well-aligned journal bearings at,, and and supports a 200-N load (igure 6.30a). igure 6.29 nswer The proposed free-bod diagram in igure 6.30b is correct. s summaried in Table 6.2, because there is more than one journal bearing supporting the sstem, each bearing applies a force (and no moment) to the sstem.

25 6.3 NONPLNR SYSTEM SUPPORTS N ar 200 N igure 6.30 Situation : rod is supported b a thrust bearing at and a cable that etends from to. known force is applied as shown in igure 6.31a. nswer The proposed free-bod diagram in igure 6.31b is not correct. ecause there is a single journal bearing supporting the sstem, the bearing also applies moments about the and aes at. Situation : bar has built-in support at and loads applied at and as shown in igure 6.32a. Rod T cable igure 6.31 nswer The proposed free-bod diagram in igure 6.32b is correct. Situation E: The L-bar is supported at b a cable and at b a smooth square rod that just fits through the square hole of the collar. known vertical load is applied as shown in igure 6.33a. M nswer The proposed free-bod diagram in igure 6.33b is not correct. Since the rod is square and therefore prevents rotation of the collar, the connection at also applies a moment about the ais. Since the square rod is smooth, it cannot appl a force in the direction at. M M M ar T cable L-bar M igure 6.32 igure 6.33

26 150 N 30 N igure 6.34 Situation : The 150-N door is supported at and b hinges. Someone attempts to open the door b appling a force of 30 N to the handle, but because of a high spot in the floor at, the door won t open. oth hinges are able to appl forces along their pin ais (igure 6.34a). nswer The proposed free-bod diagram in igure 6.34b is correct. Notice that unlike Eample 6.3(), here the door must be treated as a nonplanar sstem, and therefore the hinge forces in the direction must be considered. E X E R I S E S onsider the description of each nonplanar sstem (part of each figure) and determine whether the proposed free-bod diagram (part of each figure) is correct. a. sign of weight W (500 N) with center of gravit as shown is supported b cables and a collar joint (E6.1.1a). b. crankshaft is supported b a journal bearing at and a thrust bearing at. Ignore the weight of the crank (E6.1.1b). c. pulle is used to lift a weight W. The shaft of the pulle is supported b a journal bearing, as shown. Ignore the weights of the pulle and the shaft (E6.1.1c). d. pole is fied at and tethered b a rope, as shown. Ignore the weight of the pole (E6.1.1d). e. triangular plate is supported b a rope at and a hinge at. Its weight of 400 N acts at the plate s center of gravit at, as shown (E6.1.1e). G H G G G G I E (G) (G) E E W E6.3.1

27 6.4 PLNR N NONPLNR SYSTEMS N (c) P W = 10 Ni 30 Nj 10 Nk 80 N X X Z Z M M P W M = 10 Ni 30 Nj 10 Nk (d) (5, 6, 1) m (e) W (3, 0, 4) m (5, 6, 1) m * M * * * * M * M * * (3, 0, 4) m * M M W = 400 N E6.3.1 (ont.)

28 242 H 6 RWING REE-OY IGRM tower crane is fied to the ground at as shown in E ased on information in Table 6.2, what loads do ou epect to act on the tower at? Present our answer in terms of a sketch of the tower that shows the loads acting on it at. lso comment on whether the sketch ou created is or is not a free-bod diagram The welded tubular frame in E6.3.4 is secured to the horiontal plane b a ball-and-socket connection at and receives support from a loose-fitting ring at. ased on information in Table 6.2, what loads do ou epect to act on the frame at? What loads do ou epect to act on the frame at? Present our answer in terms of a sketch of the frame that shows the loads acting on it at and. lso comment on whether the sketch ou created is or is not a free-bod diagram. (0, 4.5, 6.0) m = 840 N 2 kn E The uniform 7-m steel shaft in E is supported b a ball-and-socket connection at in the horiontal floor. The ball end rests against the smooth vertical walls, as shown. ased on information in Table 6.2, what loads do ou epect to act on the shaft at? What loads do ou epect to act on the shaft at? Present our answer in terms of a sketch of the shaft that shows the loads acting on it at and. lso comment on whether the sketch ou created is or is not a free-bod diagram bar is supported at b a hinge, and at it rests against a rough surface. The surface is defined as the ** plane. ased on information in Table 6.2, what loads do ou epect to act on the bar at? What loads do ou epect to act on the bar at? Present our answer in terms of a sketch that shows the loads acting on the bar at and. lso comment on whether the sketch ou created is or is not a free-bod diagram. E N/m 20 N/m * * E6.3.3 * E PLNR N NONPLNR SYSTEMS efining the loads at a sstem s boundar is simplified if we can classif the sstem as a planar sstem, which is one in which all the forces acting on the sstem lie in the same plane and all moments are about an ais perpendicular to that plane. In this case, cross-boundar loads (e.g., gravit), known loads, fluid boundar loads, and supports are all in a

29 6.4 PLNR N NONPLNR SYSTEMS 243 single plane. igure 6.35a shows an eample of a sstem that can be classified as a planar sstem planar because the gravit force and supports and are all in a single plane. Planar sstems are referred to as two-dimensional sstems. s we saw in Section 6.2, the free-bod diagram associated with a planar sstem tpicall requires onl a single view of the sstem. sstem in which the loads do not all lie in a single plane can be treated as a planar sstem for the purpose of static analsis if the sstem has a plane of smmetr with regard to its geometr and the loads acting on it. plane of smmetr is one that divides the sstem into two sections that are mirror images of each other. None of the forces acting on the sstem has a component perpendicular to the plane of smmetr, and all moments acting on the sstem are about an ais perpendicular to the plane. igure 6.35b illustrates a sstem that has a plane of smmetr and therefore can be treated as a planar sstem. Other eamples in which a plane of smmetr was used to classif a sstem as planar are the Golden Gate ridge in hapter 2, and the ladder person eample in hapter 4 (igures 4.19, 4.20, and 4.22, but not igure 4.21). igure 6.35c illustrates a sstem with geometric smmetr, but because the cable forces acting on it have a component perpendicular to the plane, we are not able to classif the sstem as planar. Plane of smmetr Sstem 10 N Sstem 10 N 10 N Sstem Sstem (c) 10 N igure 6.35 Sstem that can be modeled as planar; sstem with plane of smmetr can be modeled as planar; (c) and (d) sstem that cannot be modeled as planar (d) 10 N

30 244 H 6 RWING REE-OY IGRM If it is not possible to define a single plane in which all forces and moments lie, or there is no plane of smmetr, the sstem is classified as nonplanar. In the sstem of igure 6.35d, for instance, it is not possible to define a single plane that contains the gravit force and supports and, and there is no plane of smmetr. Nonplanar sstems are referred to as three-dimensional sstems. The free-bod diagram associated with a nonplanar sstem tpicall requires an isometric drawing or multiple views. In Section 6.2 we dealt eclusivel with planar sstems and in Section 6.3 with nonplanar sstems. rawing the free-bod diagram for a planar sstem is generall more straightforward because onl eternal forces in the plane and eternal moments about an ais perpendicular to the plane must be considered. In performing analsis in engineering practice ou will not be told whether a phsical situation can be modeled as a planar sstem or must be modeled as a nonplanar sstem the choice will be up to ou. The discussion in this section is intended to give ou some guidelines with which to make such a judgment. EXMPLE 6.5 IENTIYING PLNR N NONPLNR SYSTEMS onsider the description of each sstem and determine whether the sstem can be classified as planar or nonplanar. (No sstem is reall planar, because we live in a three-dimensional world. Even something as thin as a sheet of paper has a third dimension; UT under certain conditions we can model it as planar for the purpose of static analsis) T Goal We are asked to determine, for a number of different cases, whether a sstem can be classified as planar or nonplanar. Given We are given a specified sstem and the loads that act on it. ssume Unless specified otherwise, assume that gravit is considered and acts in the negative direction. raw n additional drawing is not required to determine the classification of each sstem; however, ou ma want to draw a free-bod diagram to help ou better understand the geometr of the sstem and the eternal loads. Situation : The uniform bar in igure 6.36a weighs 60 N and is pulled on b a rope at. The sstem is the arm and the wheel at. 60 N T nswer Planar. The gravit force of 60 N is in the plane. urthermore, the forces at the supports (normal force and tension in rope ) and (collar guide) are also in the plane. ecause it is possible to define a single plane that contains all known forces and moments, gravit force, and forces applied at supports, this sstem can be treated as planar. The free-bod diagram of is shown in igure 6.36b. igure 6.36 Question: If gravit acted in the direction, would we reach the same conclusion? If the pulle at is not in the plane, would we reach the same conclusion?

31 6.4 PLNR N NONPLNR SYSTEMS 245 Situation : The space truss in igure 6.37a is of negligible weight and is supported b rollers at,, and. It supports a vertical 800-N force at. The sstem is the space truss. nswer Nonplanar. It is not possible to define a single plane that contains the points of application of supports (,, ) and the 800-N force. Therefore, this sstem must be treated as a nonplanar sstem. Our answer would be unchanged if we had included gravit forces acting on the space truss. The free-bod diagram of the space truss is shown in igure 6.37b. Situation : The beam in igure 6.38a is pinned to its surroundings at and rests against a rocker at. Ignore gravit. The sstem is the beam. nswer Planar. The 800-N m moment at and the 500-N force are in the plane, as are the points of application of supports at and. The free-bod diagram of the beam is shown in igure 6.38b. Question: If gravit is considered and acts in the negative direction, would we reach the same conclusion? Situation : The beam in igure 6.39a rests on a block at. In addition, there is a pin connection at and a rocker at. The sstem is the beam. 800 N (4, 3, 4) m (6, 0, 0) m (5, 0, 6) m igure N 500 N 800 N m nswer Nonplanar. It is not possible to define a single plane that contains the gravit force and the normal contact force at and the 500-N force (both in the plane). The free-bod diagram of beam is shown in igure 6.39b. Question: If gravit acted in the positive direction, would we reach the same conclusion? How would our conclusion change if we were to ignore gravit? M N M 800 N m N W 500 N igure 6.38 igure 6.39

32 246 H 6 RWING REE-OY IGRM 150 N 30 N 30 N 150 N igure 6.40 igure 6.41 Situation E: The 150-N door in igure 6.40a is supported at and b hinges. The sstem is the door. able all and socket nswer Planar. If we assume that the door is of uniform densit, we place the 150-N gravit force at the door s center. urthermore, because the door is thin relative to its other dimensions, this gravit force and the loads due to the supports at and can be assumed to lie in the plane. The free-bod diagram of the door is shown in igure 6.40b. able Situation : The 150-N door in igure 6.41a is supported at and b hinges. Someone attempts to open the door b appling a force of 30 N to the handle, but because of a high spot in the floor at, the door won t open. The sstem is the door. nswer Nonplanar. It is not possible to define a single plane that contains all of the forces. The free-bod diagram of the door is shown in igure 6.41b. 2 cable igure N Situation G: semicircular plate in igure 6.42a weighs 300 N, which is represented b a point force at the center of gravit. Vertical cables support the plate at and, and a ball-and-socket joint supports the plate at. The sstem is the plate. nswer Planar. The plane is a plane of smmetr for this sstem the portion of the sstem at (a quarter circle and cable force) is the mirror image of the portion of the sstem at (a quarter circle and cable force). onsequentl, it is possible to represent all of the forces as projections onto the plane. The free-bod diagram of the plate is shown in igure 6.42b.

33 6.4 PLNR N NONPLNR SYSTEMS 247 Question: If the 300 N-force acted in the positive direction, would we reach the same conclusion? If it acted in the positive direction, what conclusion should be drawn? Situation H: The same plate as in G is supported b diagonal cables at and, and a ball-and-socket joint at, as shown in igure 6.43a. The sstem is the plate. 300 N nswer Nonplanar. s in G, the plane is a plane of smmetr for this sstem. However, since the cable forces have components perpendicular to the plane of smmetr, we cannot represent this as a planar sstem. The free-bod diagram of the plate is shown in igure 6.43b. Situation I: man weighing 800 N sits at the picnic table halfwa between its two ends (igure 6.44a). His center of gravit (G man ) is noted. The table weighs 200 N, with a center of gravit at G table. ssume that an friction between the legs and the ground can be neglected. The sstem is the table. nswer Planar. The plane is a plane of smmetr for this sstem the portion of the sstem at (half of the picnic table and supports at and ) is a mirror image of the portion of the sstem at (the other half of the picnic table). The free-bod diagram of the table is shown in igure 6.44b. igure N Situation J: child sits down net to the man at the picnic table in I (igure 6.45a). The sstem is again the table. nswer Nonplanar. With the child sitting net to the man, there is no longer an plane of smmetr. The free-bod diagram of the table is shown in igure 6.45b. G man G table Question: If the child sits directl across from the man, could the sstem be modeled as planar? W man W M G table W table 800 N W child W T 200 N, back leg, back leg, front leg 2 2, front leg igure 6.44 igure 6.45

34 248 H 6 RWING REE-OY IGRM EXMPLE 6.6 USING QUESTIONS TO ETERMINE LOS T SUPPORTS 100 N E 200 N m bar is supported at b a frictionless collar guide. t it rests against a rough surface. Known forces act at and E, as shown in igure What loads act at and? Use the general rule about the surroundings preventing translation and/or rotation at each support to answer this question. raw a free-bod diagram of the bar. igure 6.46 Solution The bar (defined as the sstem) can be classified as planar. This means that in considering motion, we need consider onl translations in the plane and rotations about the ais. ' ' * * t : efine the coordinate sstem at (see igure 6.47a). Possible motion nswer Implication Is translation at Yes There is no force acting possible? on the bar in the direction, since it is frictionless. Is translation at No There is a force. possible? Is rotation at No There is a moment possible? about the ais, M., M * + * t : efine the ** coordinate sstem at (see igure 6.47a). M * * Possible motion nswer Implication Is * translation at No, unless the force There is a force *. possible? applied in the * direction eceeds the maimum friction force that can be applied b the rough surface igure 6.47 Is * translation at No, it is not possible in There is a force * possible? the negative direction. in the positive It is possible in the * direction. positive * direction Is rotation at Yes There is no moment possible? about the ais. nswer t :, M t : *, * (in the positive * direction). The free-bod diagram of the bar is as shown. nswer See igure 6.47b.

35 6.4 PLNR N NONPLNR SYSTEMS 249 EXMPLE 6.7 USING QUESTIONS TO ETERMINE LOS T SOLI SUPPORTS crank is supported at b a pin connection, as shown in igure cable (in the plane) is attached to the bracket at. What loads act at and? Use the general rule about the surroundings preventing translation and/or rotation at each support to answer this question. raw a free-bod diagram of the crank. Solution The crank (defined as the sstem) can be classified as planar. This means that in considering motion, we need consider onl translations in the plane and rotations about the ais. able igure 6.48 rank t there is a pin connection: Possible motion nswer Implication Is translation at No There is a force. possible? Is translation at No There is a force. possible? Is rotation at Yes There is no moment possible? about the ais. heck The answers can be confirmed with Table 6.1 for a pin connection. t : * t :, * * t a cable is attached to the sstem: efine the ** coordinate sstem at (see igure 6.49). igure 6.49 Possible motion nswer Implication Is * translation at No, it is not possible in There is a force * possible? the positive * direction. in the negative It is possible in the * direction. negative * direction. Is * translation at Yes There is no force in the possible? positive * direction. Is rotation at Yes There is no moment possible? about the ais. * * * nswer t :, t : * in the negative * direction See igure The free-bod diagram of the crank is as shown. nswer See igure igure 6.50

36 250 H 6 RWING REE-OY IGRM EXMPLE 6.8 USING QUESTIONS TO ETERMINE LOS T SUPPORTS G 20 N 10 N H * * n L-shaped bar is supported at b a hinge and rests against a rough surface at. Known loads act as shown in igure Ignore the weight of the bar. What loads act at and? Use the general rule about the surroundings preventing translation and/or rotation at each support to answer this question. raw a free-bod diagram of the bar. igure 6.51 * Solution The bar (defined as the sstem) is nonplanar. This means that in considering motion, we must consider translations and rotation in all three directions. t : efine the coordinate sstem at (igure 6.51). Possible motion nswer Implication Is translation at No, if we can assume There is force in the possible? that the connection direction,. at and/or prevents motion in the direction. Is translation at No There is force in the possible? direction,. Is translation at No There is force in the possible? direction,. Is rotation about the Yes There is no moment ais possible at? about ais. Is rotation about the No There is moment about ais possible at? the ais, M. Is rotation about the No There is moment about ais possible at? the ais, M. t : efine the *** coordinate sstem at (igure 6.51). Possible motion nswer Implication Is * translation at No, unless the force There is a force *. possible? applied eceeds the maimum friction force that can be applied b the rough surface. Is * translation at No, unless the force There is a force *. possible? applied eceeds the maimum friction force that can be applied b the rough surface.

37 6.4 PLNR N NONPLNR SYSTEMS 251 t : + +, M + M t : * + * + * M M G H 20N 10N G H * igure 6.52 * * Possible motion nswer Implication Is * translation at No, it is not possible in There is a force * possible? the negative direction. in the positive It is possible in the * direction. positive direction. Is rotation about the Yes There is no moment * ais possible at? about the * ais. Is rotation about the Yes There is no moment * ais possible at? about the * ais. Is rotation about the Yes There is no moment * ais possible at? about the * ais. nswer t :,,, M, M t : *, *, * (in the positive * direction) The free-bod diagram of the bar is shown in igure nswer See igure E X E R I S E S onsider the description of each sstem in E6.4.1 and determine whether it can be classified as planar or nonplanar. escribe our reasoning. Unless otherwise stated, ignore the effect of gravit. a. The uniform L-bar is pinned to its surroundings at, and slides along a wall at. vertical force acts at. Gravit acts in the negative direction. The sstem is taken as the L-bar (E6.4.1a). b. The wheelbarrow is loaded, with a center of gravit as shown in E6.4.1b. The sstem is taken as the wheelbarrow. c. The wheelbarrow is loaded as shown in E6.4.1c. The sstem is taken as the wheelbarrow. d. tower 70 m tall is tethered b three cables, as shown in E6.4.1d. The sstem is taken as the tower. e. uniform glass rod having a length L is placed in a smooth hemispherical bowl having a radius r. The sstem is taken as the rod (E.6.4.1e). f. or the situation shown in E6.4.1e, define the bowl as the sstem. W E6.4.1

38 252 H 6 RWING REE-OY IGRM r E (c) (d) (e) E6.4.1 (ont.) onsider the description of each sstem in E6.4.2 and determine whether it can be classified as planar or nonplanar. escribe our reasoning. Unless otherwise stated, ignore the effect of gravit. a. bar is fied to a wall at end. t end, a force acts, as shown in E6.4.2a. The sstem is taken as the bar. b. bar is fied to a wall at end. t end and at, forces act, as shown in E6.4.2b. The sstem is taken as the bar. c. bar is fied to a wall at end. t end a force acts, as shown in E6.4.2c. The sstem is taken as the bar. d. The frame is supported at and. Loads act at,, E, and, as shown in E6.4.2d. The sstem is taken as the frame. e. The bracket in E6.4.2e is tethered as shown with cable. The bracket is taken as the sstem. f. The bracket in E6.4.2f is tethered as shown with cable. The bracket is taken as the sstem. g. The airplane in E6.4.2g weighing 8000 N sits on the tarmac. It has one front wheel and two rear wheels. Its center of gravit is as shown. E (c) (d) G W (e) (f) (g) E6.4.2

39 6.4 PLNR N NONPLNR SYSTEMS cable pulls on the bracket in E6.4.3 with a force of 1.5 kn. t the bracket is attached to the wall with a pin connection, and at there is a pin-in-slot connection. If the sstem is defined as the bracket and is considered to be planar, what loads act on the bracket at and? Use the general rule about prevention of motion to answer this question (Strateg: Review Eamples ) onfirm that our answers are consistent with the information on loads in Table 6.1 or Table 6.2 (whichever is appropriate). Present our answer in words and as a sketch of the bracket that shows these loads acting on it at and. lso comment on whether the sketch ou created is or is not a free-bod diagram. (Strateg: Review Eamples ) onfirm that our answers are consistent with the information on loads in Table 6.1 or Table 6.2 (whichever is appropriate). Present our answer in words and as a sketch of the bracket that shows the loads acting on it at and. lso comment on whether the sketch ou created is or is not a free-bod diagram kn E kn E steel sphere sits in the groove, as shown in E Surfaces and are smooth. If the sstem is defined as the sphere and is considered to be planar, what loads act on the sphere at and? Use the general rule about prevention of motion to answer this question. (Strateg: Review Eamples ) onfirm that our answers are consistent with the information on loads in Table 6.1 or Table 6.2 (whichever is appropriate). Present our answer in words and as a sketch of the sphere that shows the loads acting on it at and. lso comment on whether the sketch ou created is or is not a free-bod diagram ecause of a combination of soil conditions and the tension in the single power cable, the utilit pole in E6.4.6 has developed the indicated 5 lean. The 9-m uniform pole has a mass per unit length of 25 kg/m, and the tension in the power cable is 900 N. If the sstem is defined as the power pole, what loads act on the pole at its base where it is fied into the ground? Use the general rule about prevention of motion to answer this question. (Strateg: Review Eamples ) onfirm that our answer is consistent with the information on loads in Table 6.1 or Table 6.2 (whichever is appropriate). Present our answer in words and as one or more sketches of the pole that show the loads acting on its base. lso comment on whether the sketch ou created is or is not a free-bod diagram E cable pulls on the bracket with a force of 2.5 kn (E6.4.5). t the bracket rests against a smooth surface, and at it is pinned to the wall. If the sstem is defined as the bracket and is considered to be planar, what loads act on the bracket at and? Use the general rule about prevention of motion to answer this question. 30 E6.4.6

40 254 H 6 RWING REE-OY IGRM steel sphere sits in the grooved trough, as shown in E Surfaces,, and are smooth. Gravit acts in the negative direction. If the sstem is defined as the sphere, what loads act on the sphere at,, and? Use the general rule about prevention of motion to answer this question. (Strateg: Review Eamples ) onfirm that our answer is consistent with the information on loads in Table 6.1 or Table 6.2 (whichever is appropriate). Present our answer in words and as one or more sketches of the sphere that show the loads acting on it at,, and. Surface Section Surface Surface 30 E6.4.7 Hand pressing down on table hand igure 6.53 Hand pressing down on a table modeled as a force 6.5 ISTRIUTE ORES Up to this point we have modeled supports as loads acting at a single location on the sstem boundar. In actualit, all supports consist of forces distributed over a finite surface area. or eample, if ou press down on a table with our hand, the force ou appl to the table is distributed over a finite area (igure 6.53a). or man practical applications, we can condense this distributed force into a single point force (igure 6.53b). There are, however, some situations for which we eplicitl consider the loads to be distributed; igure 6.54 shows some eamples. The ke idea we want to get across is that these distributed forces must be included in the sstem s free-bod diagram. The can be represented in the diagram either as distributed forces (igure 6.55b) or as an equivalent point force (igure 6.55c). This equivalent point force is the total force represented b the distributed force and is located so as to create the same moment as the distributed force. or the uniforml distributed Sandbags ar headrest igure 6.54 istributed loads: 60-lb bags of concrete stacked on beam ; a head presses back on a head-rest

41 6.5 ISTRIUTE ORES 255 Sandbags (each weighs 134 N) ω = 402 N/m 1608 N 4 m 2 m (c) igure 6.55 Sandbags sitting on a beam; weight of sandbags represented as distributed load; (c) weight of sandbags represented as an equivalent force force in igure 6.55a we are able to find this location b inspection. In hapter 8 we shall show how to find the location for nonuniforml distributed forces. The important point to remember for our current work is that these distributed forces must be included in the free-bod diagram. their ver nature, fluids acting on a sstem boundar are distributed. Like distributed loads associated with supports, the loads at fluid boundaries are included in a free-bod diagram, either as distributed loads or as an equivalent force. In hapter 8 we discuss in greater detail distributed loads due to fluids acting on the sstem. E X E R I S E S onsider the coat rack in Eercise Redraw the sketch showing the distributed load between the base of the coat rack and the floor onsider a building with a nominall flat roof. The actual roof surface is slightl irregular and can be represented b the profile shown in E fter a night of heav rain, an average rainfall total of 2 in. was recorded. Make a sketch of the distributed force that rain water applies to the nominall flat roof. Indicate the magnitude of the forces with the length of the vectors onsider a cement truck with a tank that is half full in E raw the distributed load applied to the inside of the tank. Indicate the magnitude of the loads with the length of the vectors. ement E in. E Identif three different sstems on which distributed forces act. Make a sketch of each sstem and show what ou think the distributed forces look like; indicate the magnitude of the forces with the length of vectors onsider a person wanting to cross a froen pond. She has a choice of going on foot, wearing snow shoes, or using skis. a. Make three sketches showing the distribution of her bod weight on the froen pond given the three tpes of footwear. b. Which tpe of footwear would ou choose? Wh?

42 256 H 6 RWING REE-OY IGRM hdraulic clinder works b pumping fluid in and out of a piston assembl as shown in E6.5.6a. raw the loads acting on piston assembl shown in E6.5.6b. luid O-ring seal Piston linder Piston rod Piston assembl O-ring seal 100 kg E REE-OY IGRM ETILS We now outline a process for drawing a free-bod diagram of a sstem; this is the RW step in our engineering analsis procedure. 1. efore diving into drawing, take time to stud the phsical situation. onsider what loads are present at boundaries and ask ourself whether ou have ever seen a similar support. Stud actual hardware (if available); pick it up or walk around it to reall get a sense of how the loads act on the sstem. This inspection helps in making modeling assumptions. lassif the sstem as planar or nonplanar. If the sstem can be classified as planar, drawing the free-bod diagram and writing and solving the conditions of equilibrium (covered in the net chapter) all become easier. If ou are unsure, consider the sstem to be nonplanar. lso, consider asking for advice and opinions from others. 2. efine (either b imagining or actuall drawing) a boundar that isolates the sstem from the rest of the world, then draw the sstem that is within the boundar. The drawing should contain enough detail so that distances and locations of loads acting on the sstem can be shown accuratel. Sometimes multiple views of the sstem will be needed, especiall if the sstem is nonplanar. Establish a coordinate sstem. State an assumptions ou make. 3. Identif cross-boundar forces acting on the sstem and draw them at appropriate centers of gravit. 1 Include a variable label and the force magnitude (if known). ontinue to state an assumptions ou make. 4. Identif all known loads acting at the boundar and add these to the drawing, placing each known load at its point (or surface area) of application; identif each load on the drawing with a variable label and magnitude. 5. Identif the loads associated with each support, including those loads that act at discrete points and those that consist of distributed forces. If possible, classif each support as one of the standard supports (Table 6.1 for planar sstems and Table 6.2 for nonplanar sstems) to 1 In hapter 8 we will show how to find the center of gravit of a sstem.

43 6.6 REE-OY IGRM ETILS 257 help in identifing the loads. If this is not possible, consider how the surroundings restrict motion at a particular support (either translation and/or rotation) in order to identif the loads acting on the sstem that restrict this motion. dd all these loads to the drawing. Identif each load on the drawing with a variable label. 6. Identif fluid boundaries. dd loads associated with these boundaries to the drawing, showing them either as distributed or discrete point loads. dd variable labels. You now have a free-bod diagram of a sstem, as well as a list of the assumptions made in creating it. The diagram consists of a depiction of the sstem and the eternal loads acting on the sstem. The loads are represented in the diagram as vectors and with variable labels, and magnitudes (if known) are indicated. free-bod diagram is an idealied model of a real sstem. making assumptions about the behavior of supports, dimensions, and the material, ou are able to simplif the compleit of the real sstem into a model that ou can anale. You might want the model to describe the real situation eactl, but this is generall not an achievable goal, due to limitations such as information, time, and mone. What ou do want, however, is a model that ou can trust and that gives results that closel approimate the real situation. In creating a model, an engineer must decide which loads acting on the sstem are significant. or eample, a hinge on a door is often modeled as having no friction about its ais. Yet for most hinges, grease, dust, and dirt have built up, and there is actuall some friction some resistance to rotation. If friction is large enough the engineer should include it in the model. However, if the friction is small enough that the door can still swing freel, the engineer ma conclude that it is not significant for the problem at hand, and model the hinge loads as shown in igure Often the significance of loads is judged b their relative magnitude or location. or eample, the weight of a sack of groceries is insignificant relative to the weight of an automobile carring them but ver significant if the vehicle is a biccle. Whenever ou are in doubt about the significance of a load, consider it significant. In man of the eamples in this book, we will set the stage b making some of the assumptions regarding significance. In others, though, it will be up to ou to judge the significance of a load based either on our own eperience or on the advice of other engineers. n loads considered insignificant are not included in the free-bod diagram and should be noted in the assumption list. igure 6.56 orces acting at hinges and when hinges are frictionless

44 258 H 6 RWING REE-OY IGRM EXMPLE 6.9 RETING REE-OY IGRM O PLNR SYSTEM The lift force on an airplane wing (which is actuall a distributed load) can be modeled b eight forces as shown in igure The magnitude of each force is given in terms of its position on the wing b R i i 17 2 N, with i 1,2,..., 8 (1) igure m 2 m 3m 4m 1600 N The location 0 is at the root of the wing (location R); this is the point where the wing connects with the fuselage. The weight of the wing W 1600 N can be located at midpoint of the wing s length. reate a free-bod diagram of the wing M 6 R 7 R R R 1600 N igure Goal raw a free-bod diagram of the wing. Given We are given a planar view of the wing and some dimensions. In addition, we are told that we can model the lift as eight forces acting upward at locations i 0.5 m, 1.0 m, 1.5 m,..., 4.0 m. We are also told that the weight of the wing is 1600 N, with a point of application at 2.0 m. ssume Ignoring an slight differences in the leading and trailing edges of the wing, we assume there is a plane of smmetr (the plane in igure 6.57); therefore we can treat the wing as a planar sstem. This means we are ignoring an twist on the wings that would occur from asmmetr. raw ased on the information given in the problem and our assumptions, we isolate the wing at its root. The boundar condition at R can be modeled as a fied boundar condition; therefore, from Table 6.1 we find that there will be a force (represented as and components) and a moment present. We also determine the values of the lift force at each of the eight locations using function (1), obtaining the values shown in Table 6.3. The free-bod diagram is given in igure Table 6.3 Point Loads Representing Lift on a Wing (m) (N)

45 6.6 REE-OY IGRM ETILS 259 EXMPLE 6.10 RETING REE-OY IGRM O PLNR SYSTEM The ladder in igure 6.59 rests against the wall of a building at and on the roof of an adjacent building at. If the ladder has a weight of 100 N and length 3 m, and the surfaces at and are assumed smooth, create a free-bod diagram of the ladder. 40 Goal raw a free-bod diagram of the ladder. θ Given We are given a planar view of the ladder and some spatial information (length of ladder and angle of orientation with respect to the roof and wall). In addition we are told that the roof is at 40 with respect to the horiontal and that the surfaces at and are smooth. ssume irst, we assume that the ladder is uniform. this we mean that the rungs (cross-pieces) are identical to one another, as are the two stringers (sides of ladder which the rungs are attached). Net we assume that the weight of the ladder is significant and that its center of mass can be located at its midpoint (since it is uniform). We also assume that gravit works downward in the vertical direction. inall, we assume that there is an plane of smmetr; therefore we can treat the ladder as a planar sstem. raw ased on the information given and our assumptions, we isolate the ladder from the wall (at ) and from the roof (at ). t each surface there is a normal force present that pushes on the ladder. There is no frictional force because both surfaces are smooth. The resulting freebod diagram is shown in igure Notice that there is a factor of two associated with both normal forces; this factor reflects that there are two stringers. igure * 3 m W θ 100N igure * * 40 EXMPLE 6.11 RETING REE-OY IGRM O PLNR SYSTEM 500-N crane boom is supported b a pin at and a hdraulic clinder at. t, the boom supports a stack of lumber weighing W. reate a free-bod diagram of the boom (igure 6.61). Goal raw a free-bod diagram of the boom. Given We are given an isometric view of the boom. The joint at is a pin connection, and member is a hdraulic clinder. The hdraulic clinder acts like a link; therefore there will be a force acting on the boom at that runs along. The boom is being used to lift a load W. 40 G d 2 d 1 oom ssume We assume that the weight of the boom can be modeled as a force acting at the point marked G in igure We also assume that gravit works in the negative direction. inall, we assume that there is an plane of smmetr; therefore we can treat the boom as a planar sstem. igure 6.61

46 260 H 6 RWING REE-OY IGRM 40 d 1 raw ased on the information given and our assumptions, we isolate the boom. t there is a pin connection, and according to Table 6.1, this means that there is a force present (which we represent in terms of its and components). The resulting free-bod diagram is shown in igure d N W igure 6.62 EXMPLE 6.12 RETING REE-OY IGRM O NONPLNR SYSTEM b The L-shaped bar is supported b a bearing at and rests on a smooth horiontal surface at, with 800 N and b 1.5 m (igure 6.63). Ignore gravit. reate a free-bod diagram of the bar. 3 m igure 6.63 M 2 m Goal raw a free-bod diagram of the bar. Given We are given the dimensions of the bar and that surface is smooth. Joint looks like a journal or thrust bearing. We are told to ignore the weight of the bar. ssume We will assume that the bearing at is a journal bearing. Since the loads involved at,, and do not lie in a single plane, we must treat the bar as a nonplanar sstem. raw ased on the information given and our assumptions, we isolate the bar. t there is a single thrust bearing; according to Table 6.2, this means there are forces and moments present. t the smooth surface at there is a normal force that acts to push up on the bar. The resulting free-bod diagram is shown in igure M igure 6.64 EXMPLE 6.13 Pulle igure 6.65 Shaft Handle RETING REE-OY IGRM O NONPLNR SYSTEM The cable in igure 6.65 is attached at to a stationar surface and is wrapped around the pulle. t is a single thrust bearing, and at, force (10 N i 30 N j 10 N k) acts. Ignore gravit. reate a freebod diagram of the shaft handle pulle sstem. Goal We are to draw a free-bod diagram of the shaft handle pulle sstem.

47 6.6 REE-OY IGRM ETILS 261 Given We are given dimensions of the shaft, pulle, and handle. In addition, we are told the magnitude and direction of the applied force, and that the bearing at is a thrust bearing. ssume Since the loads acting on the shaft handle pulle sstem do not lie in a single plane, we must treat the shaft handle pulle sstem as nonplanar. inall, we assume that the weights of the shaft, handle, and pulle are negligible (because we are not told anthing about their weights). raw ased on the information given and our assumptions, we isolate the shaft handle pulle sstem. t we show the pull of the cable as. t the single bearing at we include a force and moment (Table 6.2). inall, at we show the force. igure 6.66 shows the completed free-bod diagram. M igure 6.66 = 10Ni 30Nj 10Nk M EXMPLE 6.14 RETING REE-OY IGRM O NONPLNR SYSTEM igure 6.67 shows an improved design relative to Eample The bearing at is a thrust bearing, and the bearing at E is a journal bearing. t, force (10 N i 30 N j 10 N k) acts. Ignore gravit. reate a free-bod diagram of the shaft handle pulle sstem. Wh do ou think this design is improved relative to Eample 6.13? Goal raw a free-bod diagram of the shaft handle pulle sstem. Given We are given the dimensions of the shaft, pulle, and handle. In addition, we are told the magnitude and direction of the applied force, and to ignore gravit. ssume We assume that the bearings at and E are properl aligned. This means (according to Table 6.2) that forces are applied to the bar at the location of each bearing. In addition, we are told that bearing is a thrust bearing; therefore it will appl an aial force to the shaft. Since the loads involved do not lie in a single plane, we must treat the shaft handle pulle sstem as nonplanar. raw ased on the information given and our assumptions, we isolate the shaft handle pulle sstem. t we include the pull of the cable. t the thrust bearing at we include radial (, ) forces and an aial force ( ). t the journal bearing at E we include radial ( E, E ) forces. inall, at we show the force (igure 6.68). This design is superior to the design in Eample 6.13 because the bearings appl onl forces to the shaft; this results in better wear of both the shaft and the bearings. In addition, there will be less radial pla of the sstem, generall a desirable characteristic of rotating sstems. It is generall good practice to design sstems with two properl aligned bearings. E igure 6.67 = 10Ni 30Nj 10Nk igure 6.68 E E E

48 262 H 6 RWING REE-OY IGRM E X E R I S E S 6. 6 or each of the eercises in this section, follow the steps outlined in Section 6.6 for presenting our work tape guide assembl is subjected to the loading as shown in E raw the free-bod diagram of the tape guide assembl. T T Reconsider the simple truss described in E efine the simple truss as the sstem and draw its freebod diagram Reconsider the beam described in E efine a. the beam as the sstem and draw its free-bod diagram b. the beam and the 100-N clinder as the sstem and draw its free-bod diagram beam is fied to the wall at and rests against a block at. dditional loads act on the beam, as shown in E efine the beam as the sstem. raw its free-bod diagram. E N m wheel and pulle assembl is subjected to the loading as shown in E raw the free-bod diagram of the assembl. θ P Sharp corner 100 N E n engine is lifted with the pulle sstem shown in E efine a. the pulle at as the sstem and draw its free-bod diagram b. the engine and chain as the sstem and draw its free-bod diagram c. the ring at as the sstem and draw its free-bod diagram Rough surface E Reconsider the cable pulle clinder assembl described in E efine a. the clinder as the sstem and draw its free-bod diagram b. the pulle as the sstem and draw its free-bod diagram Reconsider the belt-tensioning device described in E efine a. the pulle as the sstem and draw its free-bod diagram b. the mass as the sstem and draw its free-bod diagram c. the pulle, L-arm, and mass as the sstem and draw its free-bod diagram 200 kg P E concrete hopper and its contents have a combined mass of 400 kg, with mass center at G. The hopper is being elevated at constant velocit along its vertical guide b cable tension T. The design calls for two sets of guide rollers at, one on each side of the hopper, and two sets at. efine

49 6.6 REE-OY IGRM ETILS 263 a. the hopper and the triangular guides (including the wheels) at and as the sstem and draw its free-bod diagram b. the triangular guide at (including the two wheels) as the sstem and draw its free-bod diagram c. the hopper minus the triangular guides at and as the sstem and draw its free-bod diagram H T ar, and the dimes roll into the right collection column. Pennies, on the other hand, are heav enough (3.06 grams mass each) so that the triangular portion pivots clockwise, and the pennies roll into the left collection column. efine a. the triangular portion in Position 1 (E6.6.10a) as the sstem and draw its free-bod diagram b. the triangular portion in Position 2 as the sstem (E6.6.10b) (the triangle has just started to pivot) and draw its free-bod diagram The throttle-control sector pivots freel at O. n internal torsional spring at O eerts a return moment of magnitude M 2 N m on the sector when in the position shown. efine the sector as the sstem. raw its free-bod diagram. T G O E E portion of a mechanical coin sorter is shown in E6.6.10a. Pennies and dimes roll down the 20 incline, the triangular portion of which pivots freel about a horiontal ais through O. imes are light enough (2.28 grams mass each) so that the triangular portion remains station- M O Sector E The following three cases involve a 2 4 wooden board and a wrench. In each case a force is applied, and hands are used to react to this force, as shown in E Imagine what forces the hands would need to appl to the 2 4, then: a. onsider ase 1. efine the sstem as the 2 4, bolt and wrench. raw its free-bod diagram. b. onsider ase 2. efine the sstem as the 2 4, bolt and wrench. raw its free-bod diagram. c. onsider ase 3. efine the sstem as the 2 4, bolt and wrench. raw its free-bod diagram. d. Repeat a, b, and c if the sstem is defined as just being the 2 4 and the bolt. Pennies Positon 1 O Pennies Position 2 imes imes E onsider the mechanism used to weigh mail in E package placed at causes the weight pointer to rotate through an angle. Neglect the weights of the members ecept for the counterweight at, which has a mass of 4 kg. or a particular package, 20. efine a. the sstem as shown in E6.6.13a and draw its freebod diagram b. the sstem as shown in E6.6.13b and draw its freebod diagram c. the sstem as shown in E6.6.13c and draw its freebod diagram d. the sstem as shown in E6.6.13d and draw its freebod diagram e. the sstem as shown in E6.6.13e and draw its freebod diagram

50 264 H 6 RWING REE-OY IGRM g Wrench 100 N 0.5 m package wrapped in brown paper * * 0.5 m 0.5 m ounterweight 30 ase g 0.5 m Wrench 0.5 m 0.5 m 100 N ase g 0.5 m (c) 8 cm 8 cm (d) 100 N Wrench (e) (c) ase E E6.6.13

51 6.6 REE-OY IGRM ETILS onsider the pair of pliers in E efine a. Member 1 (E6.6.14a) as the sstem and draw its free-bod diagram b. Member 2 (E6.6.14b) as the sstem and draw its free-bod diagram c. Member 3 (E6.6.14c) as the sstem and draw its free-bod diagram d. the entire pair of pliers as the sstem (minus the shaft it is clamping) and draw its free-bod diagram onsider the frame in E efine a. the entire frame as the sstem and draw its freebod diagram b. Member 1 as the sstem and draw its free-bod diagram c. Member 2 as the sstem and draw its free-bod diagram 200 lb/ft Member N 150 N shaft 30 mm E 30 mm 70 mm 30 mm 30 mm Member 1 E (c) Member 2 E Member 3 E Member 1 Member 1 Member 2 E E onsider the frame in E efine a. the entire frame as the sstem and draw its freebod diagram b. Member 1 as the sstem and draw its free-bod diagram c. Member 2 as the sstem and draw its free-bod diagram E onsider the frame in E efine a. the entire frame as the sstem and draw its freebod diagram b. Member 1 as the sstem and draw its free-bod diagram c. Member 2 as the sstem and draw its free-bod diagram onsider the frames in E efine Pulle 2 Pulle 1 Member 1 W E Member lb Member 2 E Member lb ft E α Member 1 α Member 2 E6.6.18

52 266 H 6 RWING REE-OY IGRM a. raw the free-bod diagrams for the entire frame (E6.6.18a), Member 1, Member 2, Pulle 1, and Pulle 2. b. raw the free-bod diagrams for the entire frame (E6.6.18b), Member 1, and Member onsider the eercise frame in E efine the entire frame as the sstem and draw its free-bod diagram Reconsider the utilit pole described in E6.4.6 and draw the free-bod diagram of the pole N force is applied to the handle of the hoist in the direction shown in E There is a thrust bearing at and a journal bearing at. raw a free-bod diagram of the handle shaft pulle assembl. Make sure to record an assumptions N lb E E roof access cover is buried under a pile of snow as shown in E a. raw the free-bod diagram of the cover. b. escribe in words how ou showed the snow load and wh ou chose to show it in this manner. m E Reconsider the welded tubular frame described in E6.3.4 and draw the free-bod diagram of the frame Reconsider the tower crane described in E6.3.2 and draw the free-bod diagram of the crane Reconsider the steel shaft described in E6.3.3 and draw the free-bod diagram of the shaft Three workers are carring a 4-ft b 8-ft panel in the horiontal position shown in E The panel weight is 100 lb. efine the panel as the sstem and draw a free-bod diagram of the panel. E The bent bar is loaded and attached as shown in E raw the free-bod diagram of the bar. E E W E bracket is bolted to the shaft at O. ables load the bracket, as shown in E efine the bracket as the sstem and draw a free-bod diagram.

53 6.7 JUST THE TS kn carpenter is slowl pushing the 90-kg roof truss into place. In the current position shown in E it is oriented 20 from the vertical. The mass center of the smmetric triangular truss is located up one-third of its 2.25-m dimension from its base. efine the truss of the sstem and draw its free-bod diagram. O kn E hanging chair is suspended, as shown in E person weighing 800 N is sitting in the chair (but is not shown). efine a. the ring at as the sstem and draw its free-bod diagram b. the chair (including the knot at E) as the sstem and draw its free-bod diagram c. the eelet fastener at as the sstem and draw its free-bod diagram (2.0, 1.0, 1.0) m (1.0, 0, 2.0) m E ( 0.5, 1.0, 0.5) m m 3.5 m 2.25 m 800 N E E JUST THE TS In this chapter we looked at free-bod diagrams what the are and how to create them. To create a free-bod diagram: 1. Stud the phsical situation. lassif the sstem as planar or nonplanar. planar sstem is one in which all the forces acting on the sstem lie in the same plane and all moments are about an ais perpendicular to that plane or there is a plane of smmetr. Planar sstems are also referred to as two-dimensional sstems. If a sstem is not planar, it is nonplanar and is also referred to as three-dimensional. 2. efine (either b imagining or actuall drawing) a boundar that isolates the sstem from its surroundings, then draw the sstem that is within the boundar. Establish a coordinate sstem. Planar sstems tpicall require a single view drawing, whereas nonplanar sstems ma require multiple views or an isometric drawing. State an assumptions ou make.

54 268 H 6 RWING REE-OY IGRM 3. Identif cross-boundar forces (e.g., gravit) acting on the sstem and draw them at appropriate centers of gravit. Include a variable label and the force magnitude (if known). ontinue to state an assumptions ou make. 4. Identif all known loads acting at the boundar and add these to the drawing, placing each known load at its point (or surface area) or application; identif each load on the drawing with a variable label and magnitude. 5. Identif the loads associated with each support, both those loads that act at discrete points and those that consist of distributed forces. The loads associated with various planar supports (e.g., normal contact, links, cable) are presented in Table 6.1, and those with various nonplanar supports (e.g., hinges, journal bearings) are presented in Table 6.2. Irrespective of whether a sstem is planar or nonplanar, if a particular support is not described in Table 6.1 (planar sstems) or Table 6.2 (nonplanar sstems), the general rule that describes a boundar s restriction of motion can be used to identif the loads at the support. This rule states: If a support prevents the translation of the sstem in a given direction, then a force acts on the sstem at the location of the support in the opposite direction. urthermore, if rotation is prevented, a moment opposite the rotation acts on the sstem at the location of the support. 6. Identif fluid boundaries. dd the loads at these boundaries to the drawing, showing them either as distributed or discrete point loads. dd variable labels. You now have a free-bod diagram of a sstem, as well as a list of the assumptions made in creating it. The diagram consists of a depiction of the sstem and the eternal loads acting on the sstem. The loads are represented in the diagram as vectors and with variable labels, and magnitudes (if known) are indicated. In the net chapter we consider how to use a free-bod diagram in conjunction with Newton s first law to consider equilibrium of the sstem.

55 SYSTEM NLYSIS (S) EXERISES 269 S Y S T E M N L Y S I S ( S ) E X E R I S E S (up position) S6.1 heck on the esign of a hair To increase the seating capacit during basketball games, collapsible and portable floors on rollers have been installed, as shown in igure S The chairs that go with this portable flooring sstem are placed onto the floors but are not connected to the floor. igure S6.1.2 shows the dimensions of one of these chairs. Situation: The basketball game of Wolfpack against UN hapel Hill is underwa. s Sierra, an engineering student taking Statics, is cheering her team, she notices a woman sitting on the front edge of one of the chairs described above. ecause of the small sie of the hinge that Hinge bracket 90 slot in hinge bracket Weld between and chair frame Stop-pin welded to hinge bracket 5 cm onnector Pin pin at 5 cm between connector pin and chair frame (welded to ) igure S6.1.3 asic design of hinge for seat (down position) 3 cm Hinge bracket connected to seat 3 cm between connector pin and stop-pin igure S6.1.1 oliseum 40 cm Mobile floor and chairs in the Renolds holds the chair s seat, Sierra becomes concerned for the safet of the woman. Imagine that ou are in Sierra s place and re-create what goes through her mind as she has a sudden flashback to moments and the concept of free-bod diagrams covered in her Statics class. ssuming that the mass of the woman is 61 kg, what is the maimum moment that the slotted hinge has to bear? igure S6.1.3 ma be helpful. s indicated in igure S6.1.3, the entire moment created b the woman has to be held b the connector pins and stop-pins. onsider the chair with the dimensions in igures S raw free-bod diagrams of the: (1) seat with hinge bracket and (2) hinge bracket with 90 slot. The dimensions in igure S6.1.4 will be useful. Remember that the 90 slot is milled into bracket while the stop-pin is welded to bracket at a distance of 3 cm from, the center of rotation at the connector pin. 80 cm 90 range of motion Seat with hinge bracket Seat Hinge bracket Hinge 33 cm 33 cm 3 cm igure S cm 25 cm 26 cm 10 cm imensions of the chair 3 cm 26 cm 2 cm 3 cm igure S6.1.4 imensions associated with seat with hinge bracket ; hinge bracket with 90 slot

56 270 H 6 RWING REE-OY IGRM (c) ased on the free-bod diagram of the seat with hinge bracket created in, write epressions for the equivalent load (consisting of an epression for the equivalent force and an epression for the equivalent moment) acting at a moment center at. If the magnitude of the epression for equivalent moment is ero, what force must the stop-pin appl to the seat plate? (d) igure S6.1.5 provides a plot that shows the maimum allowable force that the stop-pin can safel hold. What sie pin is needed to ensure that the woman sitting on the chair is safe? (Remember: Each chair has two hinges and stop-pins.) (e) ssume that the maimum mass of a person for which each chair should be designed is 100 kg. What pin sie would ou recommend? (f) inall, it is ver likel that in the process of sitting down, a person might actuall fall into the chair. Since ou did not cover the dnamic effect et, let s assume ou should triple the static force to account llowable orce on stop-pin (N) 10,000 8,000 6,000 4,000 2, Stop-pin diameter (mm) igure S6.1.5 Shear strength versus pin diameter for the person s deceleration. What would our final recommendation be to the seat manufacturer regarding the pin sie? (Hint: You ma have to etend the plot b identifing the function that underlies the curve, which depends on the cross-sectional area of the pin and a fied force per cm 2.) S6.2 ollow the Path of the Gravitational orce s ou are leaving the coliseum after the game, ou recognie still another mechanical beaut in a corner a large trash cart read for action. Here is a picture of it and its dimensions (igure S6.2.1). We can safel assume that the cart would be able to carr a total of 150 kg in in in in. 3.0 in. G 24.4 in. 6.0 in in in. 3.5 in. Handle 37.0 in in. Part I: Lifting a load Situation 1: raw a free-bod diagram for the situation in which the janitor is just starting to dump a full cart b moving the handle on the back of the cart upward. The center of gravit for the full load distributed is shown in igure S If the equivalent moment about a moment center at (the contact point between the front wheels and the ground) is igure S6.2.1 Trash cart 6.5 in. 6.0 in in. 7.5 in. igure S6.2.2 imensions of trash cart load distributions ero, what is the magnitude of the force that the janitor must appl to the bin when the rear wheels just lift off the ground? State an assumptions ou make. Situation 2: Instead of the load shown in igure S6.2.1, the cart is loaded with several heav concrete pieces with a total mass of 150 kg that are placed close to the handle at position ; see igure S6.2.3a. raw a free-bod diagram for the situation in which the janitor is just starting to dump the cart b moving the handle on the back of the cart upward. If the equivalent moment about a moment center at is ero, what is the magnitude of the force that the janitor must appl to the bin when the rear wheels just lift off the ground? State an assumptions ou make.

57 SYSTEM NLYSIS (S) EXERISES in. 3.0 in in. 9.5 in in in in in. 6.0 in. 6.0 in in. 7.5 in in. 7.5 in. igure S6.2.3 The cart is carring 150 kg mass at Position ; Position (c) Situation 3: If the heav concrete pieces with a total mass of 150 kg are placed at (igure S6.2.3b), what force must the janitor appl to just lift the rear wheels off the ground? Should the janitor worr more about hurting his or her back in Situation 1, 2, or 3, and wh? (d) Will the magnitude of the force that the janitor is required to appl to the cart to move it from Position 1 to Position 2 (see igure S6.2.4) decrease, increase, or remain the same as the cart goes from Position 1 to Position 2? Include the rationale for our answer (no calculations are required). Part II: How the load is transferred to the ground You notice that the cart has large front wheels and a sturd main ale that connects the wheels to one another. The main ale is attached to a trash bin as shown in igure S ecause the ale is attached to each of the front wheels with a bearing, the wheels rotate and the ale does not. Let s consider how the weight of the trash contained in the bin is transferred to the ground. To do this we will create a series of free-bod diagrams. onsider the following cross-sectional view of the trash bin in igure S6.2.6 with various components labeled. raw a free-bod diagram of the sstem defined b oundar 1 (the cart and the trash it is holding). raw a free-bod diagram of the sstem defined b oundar 2 (the trash). (c) raw a free-bod diagram of the main ale and wheel assembl. (d) Now we are read to separate the main ale from the wheels b pushing the shaft out of the bearing wheel hub. igure S6.2.7 shows details of the ale wheel Position 1 oundar 1 Position 2 oundar 2 Trash Wheel bearing igure S6.2.4 rom Position 1 to Position 2 igure S6.2.6 Main ale supporting bin Swiveling rear wheels ross section through cart igure S6.2.5 art suspension sstem igure S6.2.7 le and wheels