Ch5 Equilibrium

6
Summary of Important Equations First condition of equilibrium ~F~0 (5.1) First condition of equilibrium ~ Fx ~ 0 ~ Fy ~ 0 Torque T = r.l..F = rF.L = rFsin (J (5.4) (5.5) (5.21) Questions for Chapter 5 Second condition of equilibrium 1: T = 0 l: WiYi Yel: = -----w- Center of mass l: miX; 1: miX; x ~---~--- em l:mj M 1; miYI 1: miYi Yem = 1: rnj =----;;;- (5.35) (5.32) (5.25) Second condition of equilibrium 1: T c .. = 1: Tee .. (5.34) (5.24) Center of gravity l: WjXj Xcg = ----w- (5.31) 1. Why can a body moving at constant 7. Where would you expect the center 11. Is it possible for the center of gravity velocity be considered as a body in of gravity of a sphere to be located? of a booy to lie outside of the body? equilibrium? A cylinder? (Hint: consider a doughnut.) 2. Why cannot an accelerated body be t 8. When lifting heavy objects why is it tI2. Why does an obese person have morc considered as in equilibrium? said that you should bend your knees trouble with lower back problems 3. Why can a point outside the body in and lift with your legs instead of your than a thin person? equilibrium be considered as an axis back? Explain. 13. Describe how a lever works in terms to compute torques? 9. A short box and a tall box arc sitting of the concept of torque. 4. What is the difference between the on the floor of a truck. If the truck tt4. Describe how you could determine center of mass of a body and its makes a sudden stop, which box is the center of gravity of an irregular center of gravity? more likely to tumble over? Why? body such as a plate, experimentally. 5. A ladder is resting against a wall and tJO. A person is sitting at the end of a tIS. Engineers often talk about the a person climbs up the ladder. Is the row boat that is at rest in the middle moment of a force acting on a body. ladder more likely to slip out at the of the lake. If the person gets up and Is there any difference between the bottom as the person climbs closer to walks toward the front of the boat, concept of a torque acting on a body the top of the ladder? Explain. what will happen to the boat? and the moment of a force acting on 6. When flying an airplane a pilot Explain in terms of the center of a body? frequently changes from the fuel mass of the system. tank in the right wing to the fuel tank in the left wing. Why does he do this? Problems for Chapter 5 5. 1 The First Condition of Equilibrium 1. In a laboratory experiment on a force table, three forces are in equilibrium. One force of 0.300 N acts at an angle of 40.0°. A second force of 0.800 N acls al an angle of 120 0 What is the magnitude and direction of the force that causes equilibrium? 2. Two ropes each 10.0 ft long are attached to the ceiling at two points located 15.0 ft apart. The ropes are tied together in a knot at their lower end and a load of 70.0 lb is hung on the knot. What is the tension in each rope? Char.ner5 Equilibrium 3. What force must be applied parallel to the plane to make the block move up the frictionless plane at constant speed? 4. Two ropes are attached to the ceiling as shown, making angles of 40.0° and 20.0°. A weight of 100 N is hung from the knot. What is the tension in each rope? F 149

Transcript of Ch5 Equilibrium

Page 1: Ch5 Equilibrium

Summary of Important Equations

First condition of equilibrium~ F ~ 0 (5.1)

First condition of equilibrium~ Fx ~ 0~ Fy ~ 0

TorqueT = r.l..F = rF.L

= rFsin (J

(5.4)(5.5)

(5.21)

Questions for Chapter 5

Second conditionof equilibrium1: T = 0

l: WiYiYel: = -----w-Center of mass

l: miX; 1: miX;x ~---~---em l:mj M

1; miYI 1: miYiYem = 1: rnj =----;;;- (5.35)

(5.32)

(5.25)

Second conditionof equilibrium1: Tc .. = 1: Tee ..

(5.34)(5.24)

Center of gravityl: WjXj

Xcg = ----w- (5.31)

1. Why can a body moving at constant 7. Where would you expect the center 11. Is it possible for the center of gravityvelocity be considered as a body in of gravity of a sphere to be located? of a booy to lie outside of the body?equilibrium? A cylinder? (Hint: consider a doughnut.)

2. Why cannot an accelerated body be t 8. When lifting heavy objects why is it tI2. Why does an obese person have morcconsidered as in equilibrium? said that you should bend your knees trouble with lower back problems

3. Why can a point outside the body in and lift with your legs instead of your than a thin person?equilibrium be considered as an axis back? Explain. 13. Describe how a lever works in termsto compute torques? 9. A short box and a tall box arc sitting of the concept of torque.

4. What is the difference between the on the floor of a truck. If the truck tt4. Describe how you could determinecenter of mass of a body and its makes a sudden stop, which box is the center of gravity of an irregularcenter of gravity? more likely to tumble over? Why? body such as a plate, experimentally.

5. A ladder is resting against a wall and tJO. A person is sitting at the end of a tIS. Engineers often talk about thea person climbs up the ladder. Is the row boat that is at rest in the middle moment of a force acting on a body.ladder more likely to slip out at the of the lake. If the person gets up and Is there any difference between thebottom as the person climbs closer to walks toward the front of the boat, concept of a torque acting on a bodythe top of the ladder? Explain. what will happen to the boat? and the moment of a force acting on

6. When flying an airplane a pilot Explain in terms of the center of a body?frequently changes from the fuel mass of the system.tank in the right wing to the fueltank in the left wing. Why does he dothis?

Problems for Chapter 5

5. 1 The First Conditionof Equilibrium

1. In a laboratory experiment on a forcetable, three forces are in equilibrium.One force of 0.300 N acts at anangle of 40.0°. A second force of0.800 N acls al an angle of 1200

What is the magnitude and directionof the force that causes equilibrium?

2. Two ropes each 10.0 ft long areattached to the ceiling at two pointslocated 15.0 ft apart. The ropes aretied together in a knot at their lowerend and a load of 70.0 lb is hung onthe knot. What is the tension in eachrope?

Char.ner5 Equilibrium

3. What force must be applied parallelto the plane to make the block moveup the frictionless plane at constantspeed?

4. Two ropes are attached to the ceilingas shown, making angles of 40.0°and 20.0°. A weight of 100 N ishung from the knot. What is thetension in each rope?

F

149

Page 2: Ch5 Equilibrium

5. Find the force F. paratiel to thefrictionless plane, that will allow thesystem to move at constant speed.

m2 = 20 kg

6. A weightless rope is stretchedhorizontally between two poles 25.0ft apart. Spiderman, who weighs 160lb, balances himself at the center ofthe rope, and the rope is observed tosag 0.500 ft at the center. Find thetension in each part of the rope.

7. A weightless rope is stretchedhorizontally between two poles 25.0ft apart. Spiderman, who weighs 160lb, balances himself 5.00 ft from oneend, and the rope is observed to sag0.300 ft there. What is the tension ineach part of the rope?

8. A force of 15.0 N is applied to a15.0-N block on a rough inclinedplane that makes an angle of 52.0°with the horizontal. The force isparallel to the plane. The blockmoves up the plane at constantvelocity. Find the coefficient ofkinetic friction between the block andthe plane.

9. With what force must a 5.00-Neraser be pressed against ablackboard for it to be in staticequilibrium? The coefficient of staticfriction between the board and theeraser is 0.250.

10. A traffic light, weighing 150 lb ishung from the center of a cable ofnegligible weight that is stretchedhorizontally between two poles thatare 60.0 ft apart. The cable isobserved to sag 2.00 ft. What is thetension in the cable?

11. A traffic light that weighs 600 N ishung from the cable as shown. Whatis the tension in each cable? Assumethe cable to be massless.

30°40°

150 /

12. Your car is stuck in a snow drift. Youattach one end of a 50.0-ft rope tothe front of the car and attach theother end to a nearby tree, as shownin the figure. If you can exert a forceof 150 lb on the center of the rope,thereby displacing it 3.00 ft to theside, what will be the force exertedon the car?

Tree Car

5.2 The Concept of Torque15. A force of 10.0 lb is applied to a door

knob perpendicular to a 30.0-in. door.What torque is produced to open thedoor?

16. A horizontal force of 10.0 lb isapplied at an angle of 40.0° to thedoor knob of a 30.0-in. door. Whattorque is produced to open the door?

17. A horizontal force of 50.0 N isapplied at an angle of 28.5° to a doorknob of a 75.0-cm door. What torqueis produced to open the door?

18. A door knob is placed in the centerof a 30.0-in. door. If a force of 10.0lb is exerted perpendicular to thedoor at the knob, what torque isproduced to open the door?

19. Compute the net torque acting on thepulley in the diagram if the radius ofthe pulley is 0.250 m and thetensions are T1 = 50.0 Nand T2 =30.0N.

r

F = 150 Ib

t13. What force is indicated on the scalein part a and part b of the diagram ifm1 = m2 = 20.0 kg?

(a)

(b)

t14. Find the tension in each cord of thefigure, if the block weighs 100 N.

T2 = 30 N

r

T1 = 50 N

20. Find the torque produced by thebicycle pedal in the diagram if theforce F = 25.0 lb, the radius of thecrank r = 7.00 inches, and angle()= 37.0°.

5.5 Examples of Rigid Bodiesin Equilibrium

Parallel Forces21. Two men are carrying a 25.0-ft

telephone pole that weighs 200 lb. Ifthe center of gravity of the pole is10.0 ft from the right end, and themen lift the pole at the ends, howmuch weight must each mansupport?

Mechanics

Page 3: Ch5 Equilibrium

22. Two men are carrying a 9.00-mtelephone pole that has a mass of 115kg. If the center of gravity of thepole is 3.00 m from the right end,and the men lift the pole at the ends,how much weight must each mansupport?

23. A uniform board that is 5.00 m longand weighs 450 N is supported bytwo wooden horses, 0.500 m fromeach end. If a 800-N person standson the board 2.00 m from the rightend, what force will be exerted oneach wooden horse?

24. A 300-N boy and a 250-N girl sit atopposite ends of a 4.00-m seesaw.Where should another 250-N girl sitin order to balance the seesaw?

25. A uniform beam 10.0 ft long andweighing 15.0 lb carries a load of20.0 lb at one end and 45.0 lb at theother end. It is held horizontal, whileresting on a wooden horse 4.00 ftfrom the heavier load. What torquemust be applied to keep it at rest inthis position?

26. A uniform beam 3.50 m long andweighing 90.0 N carries a load of110 N at one end and 225 N at theother end. It is held horizontal, whileresting on a wooden horse 1.50 mfrom the heavier load. What torquemust be applied to keep it at rest inthis position?

27. A uniform pole 5.00 m long andweighing 100 N is to be carried at itsends by a man and his son. Whereshould a 250-N load be hung on thepole, such that the father will carrytwice the load of his son?

28. A meter stick is hung from twoscales that are located at the 20.0-and 70.0-cm marks of the meterstick. Weights of 2.00 N are placedat the 10.0- and 40.0-cm marks,while a weight of 1.00 N is placed atthe 90.0-cm mark. The weight of theuniform meter stick is 1.50 N.Determine the scale readings at Aand B in the diagram.

Fa

Chapter ~ Equilibrium

Center of Gravity of a Body29. A tapered pole 10.0 ft long weighs

25.0 lb. The pole balances at its mid-point when a 5.00-lb weight hangsfrom the slimmer end. Where is thecenter of gravity of the pole?

t30. A loaded wheelbarrow that weighs75.0 lb has its center of gravity 2.00ft from the front wheel axis. If thedistance from the wheel axis to theend of the handles is 6.00 ft, howmuch of the weight of thewheelbarrow is supported by eacharm?

t31. Find the center of gravity of thecarpenters square shown in thediagram.

1.29"

I2.0,

t37. A I.OO-kgcircular metal plate ofradius 0.500 m has attached to it asmaller circular plate of the samematerial of 0.100 m radius, as shownin the diagram. Find the center ofmass of the combination with respectto the center of the large plate.

t38. This is the same problem as 37except the smaller circle of materialis removed from the larger plate.Where is the center of mass now?

Crane Boom Problems39. A horizontal uniform boom that

weighs 200 N and is 5.00 m longsupports a load WL of 1000 N, asshown in the figure. Find all theforces acting on the boom.

40. A horizontal, uniform boom 4.00 mlong that weighs 200 N supports aload WL of 1000 N. A guy wire thathelps to support the boom, isattached 1.00 m in from the end ofthe boom. Find all the forces actingon the boom.

151~

~----24.0" "

32. The front and rea'r axles of a 3Jboo-lbcar are 8.00 ft apart. If the ce ter ofgravity of the car is located 3.od ftbehind the front axle, find the loadsupported by the front and rearwheels of the car.

33. The front and rear axles of a IIIO-kgcar are 2.50 m apart. If the center ofgravity of the car is located 1.15 mbehind the front axle, find the loadsupported by the front and rearwheels of the car.

34. A very bright but lonesome childdecides to make a seesaw for one.The child has a large plank, and awooden horse to act as a fulcrum.Where should the child place thefulcrum, such that the plank willbalance, when the child is sitting onthe end? The child weighs 60.0 lband the plank weighs 40.0 lb and is10.0 ft long. (Hint: find the center ofgravity of the system.)

Center of Mass35. Four masses of 20.0, 40.0, 60.0, and

80.0 g are located at the respectivedistances of 10.0, 20.0, 30.0, and40.0 cm from an origin. Find thecenter of mass of the system.

36. Three masses of 15.0, 45.0, and25.0 g are located on the x-axis at10.0,25.0, and 45.0 cm. Two massesof 25.0 and 33.0 g are located on they-axis at 35.0 and 50.0 cm,respectively. Find the center of massof the system.

Page 4: Ch5 Equilibrium

Diagram for problem 40.

41. A horizontal, uniform boom 4.50 mlong that weighs 250 N supports a

1.001:) ~ N load WL. A guy wire thathelps to support the boom is attached1.0 m in from the end of the boom,as in the diagram for problem 40. Ifthe maximum tension that the cable

~

can withstand is 3500 N, how far outon the boom can a 95.0-kg repairman

~ walk without the cable breaking?42. A uniform beam 4.00 m long that

weighs 200 N is supported, as shownin the figure. The boom lifts a loadWL of 1000 N. Find all the forcesacting on t7~.

./

t43. A uniform beam 4.00 m long thatweighs 200 N is supported, as shownin the figure. The boom lifts a loadWL of 1000 N. Find all the forcesacting on the boom.

152

44. An 80.0-lb sign is hung on a uniformsteel pole that weighs 25.0 lb, asshown in the figure. Find all theforces acting on the boom.

3 ft .L' --,I

Applications to the Health Sciences51. A weight lifter is lifting a dumbbell

as in the example shown in figure5.22 only now the forearm makes anangle of 30.0° with the horizontal.Using the same data as for thatproblem find the force FM exerted bythe biceps muscle and the reactionforce at the elbow joint FJ• Assumethat the force FM remainsperpendicular to the arm.

52. Consider the weight lifter in theexample shown in figure 5.23.Determine the forces FM and FR ifthe angle (J = 00.0°.

t53. The weight of the upper body of theperson in the accompanying diagram·acts downward about 8.00 em infront of the fifth lumbar vertebra.This weight produces a torque aboutthe fifth lumbar vertebra. Tocounterbalance this torque themuscles in the lower back exert aforce FM that produces a countertorque. These muscles exert theirforce about 5.00 em behind the fifthlumbar vertebra. If the person weighs180 lb find the force exerted by thelower back muscles FM and thereaction force FR that the sacrumexerts upward on the fifth lumbarvertebra. The weight of the upperportion of the body is about 65% ofthe total body weight.

5th Lumbarvertebra

Mechanics

Ladder Problems45. A uniform ladder 6.00 m long

weighing 120 N leans against africtionless wall. The base of theladder is 1.00 m away from the wall.Find all the forces acting on theladder.

46. A uniform ladder 6.00 m longweighing 120 N leans against africtionless wall. A girl weighing 400N climbs three-fourths of the way uP.the ladder. If the base of the laddermakes an angle of 75.0° with theground, find all the forces acting onthe ladder. Compute all torquesabout the base of the ladder.

47. Repeat problem 46, but compute alltorques about the top of the ladder. 'Is there any difference in the results,of the problem?

48. A uniform ladder 15.0 ft longweighing 25.0 lb leans against africtionless wall. If the base of theladder makes an angle of 40.0° withthe ground, what is the minimumcoefficient of friction between theladder and the ground such that theladder will not slip out?

t49. A uniform ladder 20.0 ft longweighing 35.0 lb leans against africtionless wall. The base of theladder makes an angle of 60.0~ withthe ground. If the coefficient of.friction between the ladder and theground is 0.300, how high can a160-lb man climb the ladder beforethe ladder starts to slip? '

50. A uniform ladder 5.50 m long with amass of 12.5 kg leans against africtionless wall. The base of the,ladder makes an angle of 48.0° withthe ground. If the coefficient offriction between the ladder and theground is 0.300, how high can a82.3-kg man climb the ladder beforethe ladder starts to slip?

Page 5: Ch5 Equilibrium

t54. Consider the same situation as inproblem 53 except that the person isoverweight. The center of gravitywith the additional weight is nowlocated 15.0 cm in front of the fifthlumbar vertebra instead of theprevious 8.00 cm. Hence a greatertorque will be exerted by thisadditional weight. The distance ofthe lower back muscles is onlyslightly greater at 6.00 cm. If theperson weighs 240 Ib find the forceFR on the fifth lumbar vertebra andthe force FM exerted by the lowerback muscles.

5th Lumbarvertebra

t55. A 668-N person stands evenly on theballs of both feet. The Achillestendon, which is located at the backof the ankle, provides a tension TA tohelp balance the weight of the bodyas seen in the diagram. The distancefrom the ball of the foot to theAchilles tendon is approximately 18.0cm. The tibia leg bone pushes downon the foot with a force FT. Thedis'tance from the tibia to the ball ofthe foot is about 14.0 cm. Theground exerts a reaction force FNupward on the ball of the foot that isequal to half of the body weight.Draw a free body diagram of theforces acting and determine the forceexerted by the Achilles tendon andthe tibia.

Fibula\

Tibia

Achilles tendon(attaches here)

Chapter 5 Equilibrium

Additional Problemst56. If W weighs 100 N, find (a) the

tension in ropes 1, 2, and 3 and(b) the tension in ropes 4,5, and 6.The angle 8 = 52.0° and the angle<I> = 33.0°.

w

t57. Block A rests on a table and isconnected to another block B by arope that is also connected to a wall.If MA = 15.0 kg and µs = 0.200,what must be the value O~MB tostart the system into motion?

A 1----

58. In the pulley system shown, whatforce F is necessary to keep thesystem in equilibrium?

t59. A sling is used to support a leg asshown in the diagram. The leg iselevated at an angle of 20.0°. Thebed exerts a reaction force R on thethigh as shown. The weight of thethigh, leg, and ankle are given by WT,

WL, and wA, respectively, and thelocations of these weights are asshown. The sling is located 27.0 in.from the point 0 in the diagram. Afree body diagram is shown in part bof the diagram. Find the weight wthat is necessary to put the leg intoequilibrium.

W

(a)

wA=6.8Ib

wL = 19.21b_...1 _

WT = 431b

(b)

t60. Find the tensions Tlo T, and T2 in thefigure if WI = 500 Nand Wz = 300N. The angle 8 = 35.0° and theangle <I> = 25.0°.

t61. The steering wheel of an auto has adiameter of 18.0 in. The axle that itis connected to has a diameter of2.00 in. If a force of 25.0 lb isexerted on the rim of the wheel,(a) what is the torque exerted on thesteering wheel, (b) what is the torqueexerted on the axle, and (c) whatforce is exerted on the rim of theaxle?

153

Page 6: Ch5 Equilibrium

62. One type of simple machine is calleda wheel and axle. A wheel of radius35.0 cm is connected to an axle of2.00 cm radius. A force of Fin =10.0 N is applied tangentially to thewheel. What force Fout is exerted onthe axle? The ratio of the outputforce Fout to the input force Fin iscalled the ideal mechanicaladvantage OMA) of the system. Findthe IMA of this system.

t63. A box 1.00 m on a side rests on afloor next to a small piece of woodthat is fixed to the floor. The boxweighs 500 N. At what height hshould a force of 400 N be applied soas to just tip the box?

t64. A 200-N door, 0.760 m wide and2.00 m long, is hung by two hinges.The top hinge is located 0.230 mdown from the top, while the bottomhinge is located 0.330 m up from thebottom. Assume that the center ofgravity of the door is at itsgeometrical center. Find thehorizontal force exerted by eachhinge on the door.

t6S. A uniform ladder 6.00 m longweighing 100 N leans against africtionless wall. If the coefficient offriction between the ladder and theground is 0.400, what is the smallestangle () that the ladder can makewith the ground before the ladderstarts to slip?

t66. If an 800-N man wants to climb adistance of 5.00 m up the ladder ofproblem 65, what angle () should theladder make with the ground suchthat the ladder will not slip?

154

t67. A uniform ladder 20.0 ft longweighing 30.0 Ib leans against arough wall, that is, a wall wherethere is a frictional force between thetop of the ladder and the wall. Thecoefficient of static friction is 0.400.If the base of the ladder makes anangle () of 40.0° with the groundwhen the ladder begins to slip downthe wall, find all the forces acting onthe ladder. (Hint: With a rough wallthere will be a vertical force J. actingupward at the top of the ladder. Ingeneral, this force is unknown but wedo know that it must be less thanµsFN. At the moment the ladderstarts to slip, this frictional force isknown and is given by the equationof static friction, namely,J. = µsFN= µsF. Although there are now fourunknowns, there are also fourequations to solve for them.)

t68. A 225-lb person stands three-quarters of the way up a stepladder.The step side weighs 20.0 Ib, is 6.00ft long, and is uniform. The rear sideweighs 10.0 lb, is also uniform, and isalso 6.00 ft long. A hinge connectsthe front and back of the ladder atthe top. A weightless tie rod, 1.50 ftin length, is connected 2.00 ft fromthe top of the ladder. Find the forcesexerted by the floor on the ladder andthe tension in the tie rod.

Interactive Tutorialsr;;!69. Two ropes are attached to the

ceiling, making angles () = 20.0° and1> = 40.0°, suspending a massm = 50.0 kg. Calculate the tensionsT1 and T2 in each rope.

r;;!70. A uniform beam of length L = 10.0m and mass m = 5.00 kg is held upat each end by a force FA (at 0.00 m)and force FB (at 10.0 m). If a weightW = 400 N is placed at the positionx = 8.00 m, calculate forces FA andFB•

r;;!71. The crane boom. A uniform boom ofweight WB = 250 N and length I =8.00 m is connected to the mast by ahinge pin at the point A in figure5.20. A load WL = 1200 N issupported at the other end. A cable isconnected at the end of the boommaking an angle () = 55.0°, asshown in the diagram. Find thetension T in the cable and thevertical Vand horizontal H forcesthat the hinge pin exerts on theboom. -

r;;!72. A uniform ladder of weight WI =100 N and length L = 20.0 m leansagainst a frictionless wall at a baseangle () = 60.0°. A person weighingwp = 150 N climbs the ladder adistance d = 6.00 m from the baseof the ladder. Calculate thehorizontal H and vertical V forcesacting on the ladder, and the force Fexerted by the wall on the top of theladder.

Mechanics