Express each force as a Cartesian vector and determine the resultant force. The pipe is supported at its ends by a cord AB. The chandelier is supported by three chains which are concurrent at point O. If the force in each chain has a magnitude of 60 lb, express each force as a Cartesian vector and determine the magnitude and coordinate direction angles of the resultant force.
Express this force as a Cartesian vector. If the resultant force at O has a magnitude of lb and is directed along the negative z axis, determine the force in each chain. Determine the magnitude and coordinate direction angles of the resultant force. The window is held open by chain AB. Determine the length of the chain, and express the lb force acting at A along the chain as a Cartesian vector and determine its coordinate direction angles. If the magnitude of the resultant force is N and acts along the axis of the strut, directed from point A towards O, determine the magnitudes of the three forces acting on the strut.
In two dimensions, these problems can readily be solved by trigonometry since the geometry is easy to visualize. In three dimensions, however, this is often difficult, and consequently vector methods should be employed for the solution.
The dot product is often referred to as the scalar product of vectors since the result is a scalar and not a vector. Laws of Operation. The proof of the distributive law is left as an exercise see Prob.
Cartesian Vector Formulation. Equation 2—12 must be used to find the dot product for any two Cartesian unit vectors. Note that the result will be either a positive or negative scalar. The dot product has two important applications in A mechanics. The angle u between the tails of vectors A and B in Fig.
The components of a vector parallel and perpendicular to a line. The component of vector A parallel to or collinear with the line aa in Fig. This component is sometimes referred to as the projection of A onto the line, since a right angle is formed in the construction. Notice that if this result is positive, then Aa has a directional sense which is the same as ua, whereas if Aa is a negative scalar, then Aa has the opposite sense of direction to ua.
There are two possible ways of obtaining A. The graphical representation of the projections is shown in Fig. NOTE: These projections are not equal to the magnitudes of the components of force F along the u and v axes found from the parallelogram law.
They will only be equal if the u and v axes are perpendicular to one another. Determine the magnitude of the components of this force parallel and perpendicular to member AB. The perpendicular component, Fig. Determine the angle u between F and the pipe segment BA and the projection of F along this segment. Then we will determine the angle u between the tails of these two vectors.
Components of F. The component of F along BA is shown in Fig. We must first formulate the unit vector along BA and force F as Cartesian vectors. Determine the angle u between the force and the line AO. Find the magnitude of the projected component of the force along the pipe AO. Determine the angle u between the force and the line AB. Determine the components of the force acting parallel and perpendicular to the axis of the pole. Determine the angle u between the force and the line OA.
D F2— Determine the component of projection of the force along the line OA. Determine the magnitude of the projected component of force FAB acting along the z axis. Determine the angle u between the edges of the sheet-metal bracket. Determine the magnitude of the projected component of force FAC acting along the z axis. Determine the angle u between the sides of the triangular plate. Determine the length of side BC of the triangular plate. Solve the problem by finding the magnitude of rBC; then check the result by first finding u, rAB, and rAC and then use the cosine law.
Determine the projection of the force F along the pole. Determine the angle u between the y axis of the pole and the wire AB. Determine the angle u between cables AB and AC.
Determine the angle f between cable AC and strut AO. Two cables exert forces on the pipe. Determine the magnitude of the projected component of F1 along the line of action of F2. Determine the angle u between the two cables attached to the pipe.
Determine the magnitudes of the components of F acting along and perpendicular to segment BC of the pipe assembly. Determine the magnitude of the projected component of F along AC.
Express this component as a Cartesian vector. The cables each exert a force of N on the post. Determine the angle u between the two cables attached to the post. Determine the angle u between the pipe segments BA and BC. Determine the components of F that act along rod AC and perpendicular to it.
Point B is located at the midpoint of the rod. Point B is located 3 m along the rod from end C. Determine the magnitude of the components F1 and F2 which act along the axis of AB and perpendicular to it.
A 2 A vector has a magnitude and direction, where the arrowhead represents the sense of the vector. Multiplication or division of a vector by a scalar will change only the magnitude of the vector. If the scalar is negative, the sense of the vector will change so that it acts in the opposite sense. If vectors are collinear, the resultant is simply the algebraic or scalar addition. The components form the sides of the parallelogram and the resultant is the diagonal. Resultant F1 FR b F2 To find the components of a force along any two axes, extend lines from the head of the force, parallel to the axes, to form the components.
To obtain the components of the resultant, show how the forces add by tip-to-tail using the triangle rule, and then use the law of cosines and the law of sines to calculate their values. The x, y, z components of u represent cos a, cos b, cos g. To find the resultant of a concurrent force system, express each force as a Cartesian vector and add the i, j, k components of all the forces in the system.
The easiest way to formulate the components of a position vector is to determine the distance and direction that one must travel along the x, y, and z directions—going from the tail to the head of the vector. The force can then be expressed as a Cartesian vector.
If A and B are expressed in Cartesian vector form, then the dot product is the sum of the products of their x, y, and z components.
The dot product can be used to determine the angle between A and B. The dot product is also used to determine the projected component of a vector A onto an axis aa defined by its unit vector ua. Determine the length of the conneting rod AB by first formulating a Cartesian position vector from A to B and then determining its magnitude.
Determine the x and y components of each force acting on the gusset plate of the bridge truss. Show that the resultant force is zero.
Express F1 and F2 as Cartesian vectors. Determine the x and y components of F1 and F2. The cable attached to the tractor at B exerts a force of lb on the framework. Specify its direction measured counterclockwise from the positive x axis.
The hinged plate is supported by the cord AB. What is the length of the cord? In this chapter we will study equilibrium for a particle and show how these ideas can be used to calculate the forces in cables used to hold suspended loads.
Not only is Eq. Since the force system satisfies Eq. Consequently, the particle indeed moves with constant velocity or remains at rest. A drawing that shows the particle with all the forces that act on it is called a free-body diagram FBD. Before presenting a formal procedure as to how to draw a free-body diagram, we will first consider two types of connections often encountered in particle equilibrium problems. If a linearly elastic spring or cord of undeformed length l0 is used to support a particle, the length of the spring will change in direct proportion to the force F acting on it, Fig.
For example, if the spring in Fig. Cables and Pulleys. Unless otherwise stated throughout this book, except in Sec. In Chapter 5, it will be shown that the tension force developed in a continuous cable which passes over a frictionless pulley must have a constant magnitude to keep the cable in equilibrium.
Hence, for any angle u, shown in Fig. To construct a free-body diagram, the following three steps are necessary. Draw Outlined Shape. Show All Forces. Indicate on this sketch all the forces that act on the particle. These forces can be active forces, which tend to set the particle in motion, or they can be reactive forces which are the result of the constraints or supports that tend to prevent motion. Identify Each Force.
The forces that are known should be labeled with their proper magnitudes and directions. Letters are used to represent the magnitudes and directions of forces that are unknown. By drawing a free-body diagram of the bucket we can understand why this is so. This diagram shows that there are only two forces acting on the bucket, namely, its weight W and the force T of the cable.
The 5-kg plate is suspended by two straps A and B. To find the force in each strap we should consider the free-body diagram of the plate. As noted, the three forces acting on it form a concurrent force system. Draw a free-body diagram of the sphere, the cord CE, and the knot at C.
By inspection, there are only two forces acting on the sphere, namely, its weight, 6 kg 9. The free-body diagram is shown in Fig. Cord CE. When the cord CE is isolated from its surroundings, its free-body diagram shows only two forces acting on it, namely, the force of the sphere and the force of the knot, Fig.
Notice that FCE shown here is equal but opposite to that shown in Fig. The knot at C is subjected to three forces, Fig. As required, the free-body diagram shows all these forces labeled with their magnitudes and directions. It is important to recognize that the weight of the sphere does not directly act on the knot. Instead, the cord CE subjects the knot to this force. For equilibrium, these forces must sum to produce a zero force resultant, i.
When applying each of the two equations of equilibrium, we must account for the sense of direction of any component by using an algebraic sign which corresponds to the arrowhead direction of the component along the x or y axis. It is important to note that if a force has an unknown magnitude, then the arrowhead sense of the force on the free-body diagram can be assumed.
Then if the solution yields a negative scalar, this indicates that the sense of the force is opposite to that which was assumed.
For example, consider the free-body diagram of the particle subjected to the two forces shown in Fig. Here it is assumed that the unknown force F acts to the right to maintain equilibrium. Here the negative sign indicates that F must act to the left to hold the particle in equilibrium, Fig. Free-Body Diagram. Equations of Equilibrium. The ring will not move, or will move with constant velocity, provided the summation of these forces along the x and along the y axis equals zero.
If one of the three forces is known, the magnitudes of the other two forces can be obtained from the two equations of equilibrium. Its free-body diagram is shown in Fig. The magnitudes of TA and TC are unknown, but their directions are known. Substituting this into Eq. NOTE: The accuracy of these results, of course, depends on the accuracy of the data, i. For most engineering work involving a problem such as this, the data as measured to three significant figures would be sufficient.
Each rope can withstand a maximum force of 10 kN before it breaks. If AB always remains horizontal, determine the smallest angle u to which the crate can be suspended before one of the ropes breaks. We will study the equilibrium of ring A. There are three forces acting on it, Fig. The magnitude of FD is equal to the weight of the crate, i. FB cos From the problem geometry, it is then possible to calculate the required length of AC. The crate has a weight of lb. Determine the force in each supporting cable.
The block has a mass of 5 kg and rests on the smooth plane. Determine the unstretched length of the spring. The beam has a weight of lb. Determine the shortest cable ABC that can be used to lift it if the maximum force the cable can sustain is lb.
If the mass of cylinder C is 40 kg, determine the mass of cylinder A in order to hold the assembly in the position shown. Neglect the size of the pulley. Also, find the angle u. The members of a truss are pin connected at joint O. Determine the magnitudes of F1 and F2 for equilibrium. Cords AB and AC can each sustain a maximum tension of lb.
If the drum has a weight of lb, determine the smallest angle u at which they can be attached to the drum. Determine the magnitude of F1 and its angle u for equilibrium.
The lift sling is used to hoist a container having a mass of kg. Determine the force in each of the cables AB and AC as a function of u. If the maximum tension allowed in each cable is 5 kN, determine the shortest lengths of cables AB and AC that can be used for the lift.
The center of gravity of the container is located at G. F 3—5. The members of a truss are connected to the gusset plate.
If the forces are concurrent at point O, determine the magnitudes of F and T for equilibrium. The gusset plate is subjected to the forces of four members.
Determine the force in member B and its proper orientation u for equilibrium. The forces are concurrent at point O. The device shown is used to straighten the frames of wrecked autos. Determine the tension of each segment of the chain, i.
Determine the tension developed in wires CA and CB required for equilibrium of the kg cylinder. If cable CB is subjected to a tension that is twice that of cable CA, determine the angle u for equilibrium of the kg cylinder. Also, what are the tensions in wires CA and CB? Two electrically charged pith balls, each having a mass of 0.
Determine the maximum weight of the flowerpot that can be supported without exceeding a cable tension of 50 lb in either cable AB or AC. The concrete pipe elbow has a weight of lb and the center of gravity is located at point G.
Blocks D and F weigh 5 lb each and block E weighs 8 lb. Determine the sag s for equilibrium. Neglect the size of the pulleys. Determine the angle u for equilibrium.
Determine the force in cables BC and BD when the spring is held in the position shown. Determine the stretch in each spring for equilibrium of the 2-kg block. The springs are shown in the equilibrium position. The unstretched length of spring AB is 3 m. If the block is held in the equilibrium position shown, determine the mass of the block at D.
If the spring has an unstretched length of 2 ft, determine the angle u for equilibrium. Cord AB is 2 ft long. Neglect the size of the pulleys at B and D. If the unstretched length of the spring is 12 in. A 4-kg sphere rests on the smooth parabolic surface. Determine the normal force it exerts on the surface and the mass mB of block B needed to hold it in the equilibrium position shown.
If the spring has an unstretched length of 8 in. If the bucket weighs 50 lb, determine the tension developed in each of the wires. Determine the tension developed in each cord required for equilibrium of the kg lamp. Determine the maximum mass of the lamp that the cord system can support so that no single cord develops a tension exceeding N. Determine the maximum weight of the bucket that the wire system can support so that no single wire develops a tension exceeding lb.
Determine the tension developed in each wire which is needed to support the lb flowerpot. If the tension developed in each of the wires is not allowed to exceed 40 lb, determine the maximum weight of the flowerpot that can be safely supported. The single elastic cord ABC is used to support the lb load.
Determine the position x and the tension in the cord that is required for equilibrium. The lb uniform tank is suspended by means of a 6-ft-long cable, which is attached to the sides of the tank and passes over the small pulley located at O.
If the cable can be attached at either points A and B, or C and D, determine which attachment produces the least amount of tension in the cable.
What is this tension? Cable ABC has a length of 5 m. Determine the position x and the tension developed in ABC required for equilibrium of the kg sack. Neglect the size of the pulley at B. The sling BAC is used to lift the lb load with constant velocity. The load has a mass of 15 kg and is lifted by the pulley system shown.
Determine the force F in the cord as a function of the angle u. The cord is fixed to a pin at A and passes over two small pulleys. The ball D has a mass of 20 kg. The concrete wall panel is hoisted into position using the two cables AB and AC of equal length. Establish appropriate dimensions and use an equilibrium analysis to show that the longer the cables the less the force in each cable.
The hoisting cables BA and BC each have a length of 20 ft. If the maximum tension that can be supported by each cable is lb, determine the maximum distance AC between them in order to lift the uniform lb truss with constant velocity.
Chain AB is 1-m long and chain AC is 1. If the distance BC is 1. W Free-Body Diagram. The joint at A is subjected to the force from the support as well as forces from each of the three chains. If the tire and any load on it have a weight W, then the force at the support will be W , and the three scalar equations of equilibrium can be applied to the free-body diagram of the joint in order to determine the chain forces, FB, FC, and FD. Cable AD lies in the x—y plane and cable AC lies in the x—z plane.
The connection at A is chosen for the equilibrium analysis since the cable forces are concurrent at this point. By inspection, each force can easily be resolved into its x, y, z components, and therefore the three scalar equations of equilibrium can be used. NOTE: Since the results for all the cable forces are positive, each cable is in tension; that is, it pulls on point A as expected, Fig.
Determine its smallest vertical distance s from the ceiling if the force developed in any cord is not allowed to exceed 50 N. Due to symmetry, Fig. Also, the angle between each cord and the z axis is g. Equation of Equilibrium. First we will express each force in Cartesian vector form. Thus, solving Eq. Each force on the free-body diagram is first expressed in Cartesian vector form. The force in each of the cords can be determined by investigating the equilibrium of point A.
FD is then determined from Eq. Finally, substituting the results into Eq. Determine the magnitude of forces F1, F2, F3, so that the particle is held in equilibrium.
Determine the tension in these wires. Determine the magnitude and direction of the force P required to keep the concurrent force system in equilibrium. Determine the magnitudes of F1, F2, and F3 for equilibrium of the particle. If the bucket and its contents have a total weight of 20 lb, determine the force in the supporting cables DA, DB, and DC. Determine the stretch in each of the two springs required to hold the kg crate in the equilibrium position shown. Determine the maximum weight of the lamp that can be supported in the position shown.
The force in the pole acts along the axis of the pole. If each one of the ropes will break when it is subjected to a tensile force of N, determine the maximum uplift force F the balloon can have before one of the ropes breaks. The kg pot is supported from A by the three cables. Determine the force acting in each cable for equilibrium. What is the force in each cable for this case?
The flower pot has a mass of 50 kg. Determine the tension developed in cables AB and AC and the force developed along strut AD for equilibrium of the lb crate. If the tension developed in each of the cables cannot exceed lb, determine the largest weight of the crate that can be supported.
Also, what is the force developed along strut AD? Determine the tension developed in each cable for equilibrium of the lb crate. Determine the maximum weight of the crate that can be suspended from cables AB, AC, and AD so that the tension developed in any one of the cables does not exceed lb. Determine the force in each cable needed to support the lb platform. The lb cylinder is supported by three chains as shown. Determine the force in each chain for equilibrium.
If cable AD is tightened by a turnbuckle and develops a tension of lb, determine the tension developed in cables AB and AC and the force developed along the antenna tower AE at point A.
If the tension developed in either cable AB or AC can not exceeded lb, determine the maximum tension that can be developed in cable AD when it is tightened by the turnbuckle.
Also, what is the force developed along the antenna tower at point A? If each wire can sustain a maximum tension of lb before it fails, determine the greatest weight of the chandelier the wires will support in the position shown. The lb chandelier is supported by three wires as shown. Determine the force in each wire for equilibrium.
The thin ring can be adjusted vertically between three equally long cables from which the kg chandelier is suspended. If the ring remains in the horizontal plane and the tension in each cable is not allowed to exceed 1 kN, determine the smallest allowable distance z required for equilibrium.
The lb ball is suspended from the horizontal ring using three springs each having an unstretched length of 1. Determine the vertical distance h from the ring to point A for equilibrium. This requires that all the forces acting on the particle form a zero resultant force. This diagram is an outlined shape of the particle that shows all the forces listed with their known or unknown magnitudes and directions.
F4 3 F3 Two Dimensions The two scalar equations of force equilibrium can be applied with reference to an established x, y coordinate system. If the problem involves a linearly elastic spring, then the stretch or compression s of the spring can be related to the force applied to it.
This requires first expressing each force on the freebody diagram as a Cartesian vector. When the forces are summed and set equal to zero, then the i, j, and k components are also zero. The pipe is held in place by the vise. If the bolt exerts a force of 50 lb on the pipe in the direction shown, determine the forces FA and FB that the smooth contacts at A and B exert on the pipe.
Romeo tries to reach Juliet by climbing with constant velocity up a rope which is knotted at point A. Any of the three segments of the rope can sustain a maximum force of 2 kN before it breaks. Determine if Romeo, who has a mass of 65 kg, can climb the rope, and if so, can he along with Juliet, who has a mass of 60 kg, climb down with constant velocity? When y is zero, the springs sustain a force of 60 lb. When y is zero, the springs are each stretched 1.
The man attempts to pull the log at C by using the three ropes. Determine the direction u in which he should pull on his rope with a force of 80 lb, so that he exerts a maximum force on the log. What is the force on the log for this case? Also, determine the direction in which he should pull in order to maximize the force in the rope attached to B.
What is this maximum force? Determine the maximum weight of the engine that can be supported without exceeding a tension of lb in chain AB and lb in chain AC. Determine the force in each cable needed to support the lb load. The ring of negligible size is subjected to a vertical force of lb.
Determine the required length l of cord AC such that the tension acting in AC is lb. Also, what is the force acting in cord AB? The joint of a space frame is subjected to four member forces. Member OA lies in the x - y plane and member OB lies in the y - z plane.
Determine the forces acting in each of the members required for equilibrium of the joint. This effect is called a moment, and in this chapter we will study how to determine the moment of a system of forces and calculate their resultants. This tendency to rotate is sometimes called a torque, but most often it is called the moment of a force or simply the moment. For example, consider a wrench used to unscrew the bolt in Fig. If a force is applied to the handle of the wrench it will tend to turn the bolt about point O or the z axis.
The magnitude of the moment is directly proportional to the magnitude of F and the perpendicular distance or moment arm d. The larger the force or the longer the moment arm, the greater the moment or turning effect. If F is applied along the wrench, Fig. As a result, the moment of F about O is also zero and no turning can occur.
The moment MO about point O, or about an axis passing through O and perpendicular to the plane, is a vector quantity since it has a specified magnitude and direction. Units of moment magnitude consist of force times distance, e. MO O Direction. The direction of MO is defined by its moment axis, which is perpendicular to the plane that contains the force F and its moment arm d. The right-hand rule is used to establish the sense of direction of MO.
According to this rule, the natural curl of the fingers of the right hand, as they are drawn towards the palm, represent the rotation, or if no movement is possible, there is a tendency for rotation caused by the moment. As this action is performed, the thumb of the right hand will give the directional sense of MO, Fig. Notice that the moment vector is represented three-dimensionally by a curl around an arrow.
In two dimensions this vector is represented only by the curl as in Fig. Since in this case the moment will tend to cause a counterclockwise rotation, the moment vector is actually directed out of the page.
As a convention, we will generally consider positive moments as counterclockwise since they are directed along the positive z axis out of the page. Clockwise moments will be negative. Doing this, the directional sense of each moment can be represented by a plus or minus sign. Using this sign convention, the resultant moment in Fig. Also illustrated is the tendency of rotation of the member as caused by the force. Furthermore, the orbit of the force about O is shown as a colored curl.
Thus, Fig. For this calculation, note how the moment-arm distances for the N and N forces are established from the extended dashed lines of action of each of these forces. The actual rotation would occur if the support at B were removed. The ability to remove the nail will require the moment of FH about point O to be larger than the moment of the force FN about O that is needed to pull the nail out. Before doing this, however, it is first necessary to expand our knowledge of vector algebra and introduce the cross-product method of vector multiplication.
Vector C has a direction that is perpendicular to the plane containing A and B such that C is specified by the right-hand rule; i. The terms of Eq. It is important to note that proper order of the cross products must be maintained, since they are not commutative.
Equation 4—3 may be used to find the cross product of any pair of Cartesian unit vectors. In a similar manner, Fig. A simple scheme shown in Fig. There are four elements in each minor, for example, A11 A21 A12 A22 By definition, this determinant notation represents the terms A11A22 - A12A21 , which is simply the product of the two elements intersected by the arrow slanting downward to the right A11A22 minus the product of the two elements intersected by the arrow slanting downward to the left A12A We will now show that indeed the moment MO, when determined by this cross product, has the proper magnitude and direction.
To establish this angle, r must be treated as a sliding vector so that u can be constructed properly, Fig. The direction and sense of MO in Eq.
The cross product operation is often used in three dimensions since the perpendicular distance or moment arm from point O to the line of action of the force is not needed.
BoMz Wongkongkaew. Periha Ozulu. Mahnoor Athar. Syed Zaidi. Muhamed Bardisy. Karim Walid. Syed Ali Zaidi. Official Site. Shnear Husen. Abuddin Mahdiyar. NesarAhmad Sultani. Yaser Fayez. Hamed Jameli. Show More. Views Total views. Actions Shares. No notes for slide. Total views 19, On Slideshare 0. From embeds 0. Number of embeds These new problems relate to applications in many different fields of engineering.
Also, a significant increase in algebraic type problems has been added, so that a generalized solution can be obtained. Additional Fundamental Problems. These problem sets serve as extended example problems since their solutions are given in the back of the book. Additional problems have been added, especially in the areas of frames and machines, and in friction. Expanded Solutions. Some of the fundamental problems now have more detailed solutions, including some artwork, for better clarification.
Also, some of the more difficult problems have additional hints along with its answer when given in the back of the book. Updated Photos.
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