Chapter 3: Equilibrium (6th Edition)

SECTION A: EQUILIBRIUM IN TWO DIMENSIONS

3/1 Determine the force P required to maintain the 200-kg engine in the position for which θ = 30°. The diameter of the pulley at B is negligible.

3/2 The mass center G of the 1400-kg rear-engine car is located as shown in the figure. Determine the normal force under each tire when the car is in equilibrium. State any assumptions.

3/3 A carpenter carries a 6-kg uniform board as shown. What downward force does he feel on his shoulder at A?

3/4 In the side view of a 70-kg television resting on a 24-kg cabinet, the mass centers are labeled G1 and G2. Determine the force reactions at A and B. (Note that the mass center of most televisions is located well forward because of the heavy nature of the front portion of picture tubes.)

3/5 The roller stand is used to support portions of long boards as they are being cut on a table saw. If the

board exerts a 25-N downward force on the roller C, determine the vertical reactions at A and D. Note that the connection at B is rigid, and that the feet A and D are fairly lengthy horizontal tubes with a nonslip coating.

3/6 The 450-kg uniform I-beam supports the load shown. Determine the reactions at the supports. 

3/7 Calculate the force and moment reactions at the bolted base O of the overhead traffic-signal assembly. Each traffic signal has a mass of 36 kg, while the masses of members OC and AC are 50 kg and 55 kg, respectively.

3/8 The 20-kg homogeneous smooth sphere rests on the two inclines as shown. Determine the contact forces at A and B.

3/9 A 54-kg crate resets on the 27-kg pickup tailgate. Calculate the tension T in each of the two restraining cables, one of which is shown. The centers of gravity are at G1 and G2. The crate is located midway between the two cables.

3/10 A portable electric generator has a mass of 160 kg with mass center at G. Determine the upward force F necessary to reduce the normal force at A to one half its nominal (F = 0) value.

3/11 With what force magnitude T must the person pull on the cable in order to cause the scale A to read 2000 N? The weights of the pulleys and cables are negligible. State any assumptions.

3/12 The device shown is designed to aid in the removal of pull-tab tops from cans. If the user exerts a 40-N force at A, determine the tension T in the portion BC of the pull tab.

3/13 A woodcutter wishes to cause the tree trunk to fall uphill, even though the trunk is leaning downhill. With the aid of the winch W, what tension T in the cable will be required? The 600-kg trunk has a center of gravity at G. The felling notch at O is sufficiently large so that the resisting moment there is negligible.

3/14 To facilitate shifting the position of a lifting hook when it is not under load, the sliding hanger shown is used. The projections at A and B engage the flanges of a box beam when a load is supported, and the hook projects through a horizontal slot in the beam. Compute the forces at A and B when the hook supports a 300-kg mass.

3/15 Three cables are joined at the junction ring C. Determine the tensions in cables AC and BC caused by the weight of the 30-kg cylinder.

3/16 A 700-N axial force is required to remove the pulley from its shaft. What force F must be exerted on the handle of each of the two prybars? Friction at the contact points B and E is sufficient to prevent slipping; friction at the pulley contact points C and F is negligible.

3/17 The uniform beam has a mass of 50 kg per meter of length. Compute the reactions at the support O. The force loads shown lie in a vertical plane.

3/18 A pipe P is being bent by the pipe bender as shown. If the hydraulic cylinder applies a force of magnitude F = 24 kN to the pipe at C, determine the magnitude of the roller reactions at A and B.

3/19 The uniform 15-m pole has a mass of 150 kg and is supported by its smooth ends against the vertical walls and by the tension T in the vertical cable. Compute the reactions at A and B.

3/20 Determine the reactions at A and E if P = 500 N. What is the maximum value which P may have for static equilibrium? Neglect the weight of the structure compared with the applied loads.

3/21 While digging a small hole prior to planting a tree, a homeowner encounters rocks. If he exerts a horizontal 225-N force on the prybar as shown, what is the horizontal force exerted on rock C? Note that a small ledge on rock C supports a vertical force reaction there. Neglect friction at B. Complete solutions (a) including and (b) excluding the weight of the 18-kg prybar.

3/22 Determine the force P required to begin rolling the uniform cylinder of mass m over the obstruction of height h.

3/23 A 35-N axial force at B is required to open the spring loaded plunger of the water nozzle. Determine the required force F applied to the handle at A and the magnitude of the pin reaction at O. Note that the plunger passes through a vertically-elongated hole in the handle at B, so that negligible vertical force is transmitted there.

3/24 A person holds a 30-kg suitcase by its handle as indicated in the figure. Determine the tension in each of the four identical links AB.

3/25 A block placed under the head of the claw hammer as shown greatly facilitates the extraction of the nail. If a 200-N pull on the handle is required to pull the nail, calculate the tension T in the nail and the magnitude A of the force exerted by the hammer head on the block. The contacting surfaces at A are
sufficiently rough to prevent slipping.

3/26 The indicated location of the center of mass of the 1600-kg pickup truck is for the unladen condition. If a load whose center of mass is x = 400 mm behind the rear axle is added to the truck, determine the mass mL for which the normal forces under the front and rear wheels are equal.

3/27 The wall-mounted 2.5-kg light fixture has its mass center at G. Determine the reactions at A and B and also calculate the moment supported by the adjustment thumbscrew at C. (Note that the lightweight
frame ABC has about 250 mm of horizontal tubing, directed into and out of the paper, at both A and B.)

3/28 To test the validity of aerodynamic assumptions made in the design of the aircraft, its model is being tested in a wind tunnel. The support bracket is connected to a force and moment balance, which is zeroed when there is no airflow. Under test conditions, the lift L, drag D, and pitching moment MG act as shown. The force balance records the lift, drag, and a moment MP. Determine MG in terms of L, D, and MP.

3/29 The chain binder is used to secure loads of logs, lumber, pipe, and the like. If the tension T1 is 2 kN when θ = 30°, determine the force P required on the lever and the corresponding tension T2 for this position. Assume that the surface under A is perfectly smooth.

3/30 The device shown is designed to apply pressure when bonding laminate to each side of a countertop near an edge. If a 120-N force is applied to the handle, determine the force which each roller exerts on its corresponding surface.

3/31 The two light pulleys are fastened together and form an integral unit. They are prevented from turning about their bearing at O by a cable wound securely around the smaller pulley and fastened to point A. Calculate the magnitude R of the force supported by the bearing O for the applied 2-kN load.

3/32 In a procedure to evaluate the strength of the triceps muscle, a person pushes down on a load cell with the palm of his hand as indicated in the figure. If the load-cell reading is 160 N, determine the vertical tensile force F generated by the triceps muscle. The mass of the lower arm is 1.5 kg with mass center at G. State any assumptions.

3/33 A person is performing slow arm curls with a 10-kg weight as indicated in the figure. The brachialis muscle group (consisting of the biceps and brachialis muscles) is the major factor in this exercise. Determine the magnitude F of the brachialis-muscle-group force and the magnitude E of the elbow joint reaction at point E for the forearm position shown in the figure. Take the dimensions shown to locate the effective points of application of the two muscle groups; these points are 200 mm directly above E and 50 mm directly to the right of E. Include the effect of the 1.5-kg forearm mass with mass center at point G. State any assumptions.

3/34 A woman is holding a 3.6-kg sphere in her hand with the entire arm held horizontally as shown in the figure. A tensile force in the deltoid muscle prevents the arm from rotating about the shoulder joint O; this force acts at the 21° angle shown. Determine the force exerted by the deltoid muscle on the upper arm at A and the x- and y-components of the force reaction at the shoulder joint O. The mass of the upper arm is mU =1.9 kg, the mass of the lower arm is mL = 1.1 kg, and the mass of the hand is mH = 0.4 kg; all the corresponding weights act at the locations shown in the figure.

3/35 With his weight W equally distributed on both feet, a man begins to slowly rise from a squatting position as indicated in the figure. Determine the tensile force F in the patellar tendon and the magnitude of the force reaction at point O, which is the contact area between the tibia and the femur. Note that the line of action of the patellar tendon force is along its midline. Neglect the weight of the lower leg.

3/36 The elements of an on-off mechanism for a table lamp are shown in the figure. The electrical switch S requires a 4-N force in order to depress it. What corresponding force F must be exerted on the handle at A?

3/37 The uniform 18-kg bar OA is held in the position shown by the smooth pin at O and the cable AB. Determine the tension T in the cable and the magnitude and direction of the external pin reaction at O.

3/38 A person attempts to move a 20-kg shop vacuum by pulling on the hose as indicated. What force F will cause the unit to tip clockwise if wheel A is against an obstruction?

3/39 The exercise machine is designed with a lightweight cart which is mounted on small rollers so that it is free to move along the inclined ramp. Two cables are attached to the cart—one for each hand. If the hands are together so that the cables are parallel and if each cable lies essentially in a vertical plane, determine the force P which each hand must exert on its cable in order to maintain an equilibrium position. The mass of the person is 70 kg, the ramp angle θ is 15°, and the angle is 18
. In addition, calculate the force R which the ramp exerts on the cart

3/40 The device shown is used to test automobile-engine valve springs. The torque wrench is directly connected to arm OB. The specification for the automotive intake-valve spring is that 370 N of force should reduce its length from 50 mm (unstressed length) to 42 mm. What is the corresponding reading M on the torque wrench, and what force F exerted on the torque-wrench handle is required to produce this reading? Neglect the small effects of changes in the angular position of arm OB.

3/41 During an engine test on the ground, a propeller thrust T = 3000 N is generated on the 1800-kg airplane with mass center at G. The main wheels at B are locked and do not skid; the small tail wheel at A has no brake. Compute the percent change n in the normal forces at A and B as compared with their
“engine-off” values.

3/42 A rocker arm with rollers at A and B is shown in the position when the valve is open and the valve
spring is fully compressed. In this position, the spring force is 900 N. Determine the force which the rocker arm exerts on the camshaft C. Also calculate the magnitude of the force supported by the rocker-arm shaft O.

3/43 The hook wrench or pin spanner is used to turn shafts and collars. If a moment of is required to turn the 200-mm-diameter collar about its center O under the action of the applied force P, determine the contact force R on the smooth surface at A. Engagement of the pin at B may be considered to occur at the periphery of the collar.

3/44 The dolly shown is useful in the handling of large drums. Determine the force F necessary to hold a drum in the position shown. You may neglect the weight of the dolly in comparison with that of the 250-kg drum, whose center of mass is at G. There is sufficient friction to prevent slipping at the contact point P.

3/45 In sailing at a constant speed with the wind, the sailboat is driven by a 4-kN force against its mainsail and a 1.6-kN force against its staysail as shown. The total resistance due to fluid friction through the water is the force R. Determine the resultant of the lateral forces perpendicular to motion applied to the hull by the water.

3/46 Estimate the force F required to lift the rear tires of the race car off the ground. You may assume that part of CD of the dolly jack is horizontal. The mass of the car and the driver combined is 700 kg with mass center at G. The driver applies the brakes during the jacking. State any additional assumptions.

3/47 A portion of the shifter mechanism for a manual car transmission is shown in the figure. For the 8-N force exerted on the shift knob, determine the corresponding force P exerted by the shift link BC on the transmission (not shown). Neglect friction in the ball-and-socket joint at O, in the joint at B, and in the slip tube near support D. Note that a soft rubber bushing at D allows the slip tube to self-align with link BC.

3/48 The small sailboat may be tipped at its moorings as shown to effect repairs below the waterline. One attached rope is passed under the keel and secured to the dock. The other rope is attached to the mast and is used to tip the boat. The boat shown has a displacement (which equals the total mass) of 5000 kg with mass center at G. The metacenter M is the point on the centerline of the boat through which the vertical resultant of the buoyant forces passes, and . Calculate the tension T required to hold the boat in the position shown.

3/49 A torque (moment) of is required to turn the bolt about its axis. Determine P and the forces between the smooth hardened jaws of the wrench and the corners A and B of the hexagonal head. Assume that the wrench fits easily on the bolt so that contact is made at corners A and B only.

3/50 Determine the moment M which must be applied to the shaft in order to hold the homogeneous hemisphere in an arbitrary angular position as measured by the angle θ. The radii of gear A, gear B, and the hemisphere are rA, rB, and r, respectively. Assume the friction in all bearings to be negligible.

3/54 The member OBC and sheave at C together have a mass of 500 kg, with a combined center of mass at G. Calculate the magnitude of the force supported by the pin connection at O when the 3-kN load is applied. The collar at A can provide support in the horizontal direction only.

3/55 It is desired that a person be able to begin closing the van hatch from the open position shown with a 40-N vertical force P. As a design exercise, determine the necessary force in each of the two hydraulic struts AB. The mass center of the 40-kg door is 37.5 mm directly below point A. Treat the problem as two-dimensional.

3/60 Certain elements of an in-refrigerator ice-cube maker are shown in the figure. (A “cube” has the form of a cylindrical segment!) Once the cube freezes and a small heater (not shown) forms a thin film of water between the cube and supporting surface, a motor rotates the ejector arm OA to remove the cube. If there are eight cubes and eight arms, determine the required torque M as a function of θ. The mass of eight cubes is 0.25 kg, and the center-of-mass distance . Neglect friction, and assume that the resultant of the distributed normal force acting on the cube passes through point O.