tut1 chapters 2to4

44 Statics of Particles 2.61 Two cables tied together at C are loaded as shown. Knowing that the maximum allowable tension in each cable is 800 N, deter-mine (a) the magnitude of the largest force P that can be applied at C, (b) the corresponding value of a.

Fig. P2.61 and P2.62

35ºA B

C

P

50º

a

50 lb

x

C

B

A

P

20 in.


A

P

B

!

"

160 kg


T

TT T T

(a) (b) (c) (d) (e)

Fig. P2.67

2.62 Two cables tied together at C are loaded as shown. Knowing that the maximum allowable tension is 1200 N in cable AC and 600 N in cable BC, determine (a) the magnitude of the largest force P that can be applied at C, (b) the corresponding value of a.

2.63 Collar A is connected as shown to a 50-lb load and can slide on a frictionless horizontal rod. Determine the magnitude of the force P required to maintain the equilibrium of the collar when (a) x 5 4.5 in., (b) x 5 15 in.

2.64 Collar A is connected as shown to a 50-lb load and can slide on a frictionless horizontal rod. Determine the distance x for which the collar is in equilibrium when P 5 48 lb.

2.65 A 160-kg load is supported by the rope-and-pulley arrangement shown. Knowing that b 5 20°, determine the magnitude and direc-tion of the force P that must be exerted on the free end of the rope to maintain equilibrium. (Hint: The tension in the rope is the same on each side of a simple pulley. This can be proved by the methods of Chap. 4.)

2.66 A 160-kg load is supported by the rope-and-pulley arrangement shown. Knowing that a 5 40°, determine (a) the angle b, (b) the magnitude of the force P that must be exerted on the free end of the rope to maintain equilibrium. (See the hint for Prob. 2.65.)

2.67 A 600-lb crate is supported by several rope-and-pulley arrange-ments as shown. Determine for each arrangement the tension in the rope. (See the hint for Prob. 2.65.)

b ee2 9 4 0 0 _ch0 2 _0 1 4 - 0 7 1 .ind d Page 4 4 1 1 / 2 8 / 0 8 9 :2 0 :5 7 PM user- s1 7 3b ee2 9 4 0 0 _ch0 2 _0 1 4 - 0 7 1 .ind d Page 4 4 1 1 / 2 8 / 0 8 9 :2 0 :5 7 PM user- s1 7 3 / Volum es / 2 0 4 / MHDQ0 7 6 / work%0 / ind d %0/ Volum es / 2 0 4 / MHDQ0 7 6 / work%0 / ind d %0

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Mahesh Tirumkudulu

The tutorial problems have been highlighted in yellow.

67

REVIEW PROBLEMS

2.127 The direction of the 75-lb forces may vary, but the angle between the forces is always 50°. Determine the value of a for which the resultant of the forces acting at A is directed horizontally to the left.

2.128 A stake is being pulled out of the ground by means of two ropes as shown. Knowing the magnitude and direction of the force exerted on one rope, determine the magnitude and direction of the force P that should be exerted on the other rope if the resul-tant of these two forces is to be a 40-lb vertical force.

2.129 Member BD exerts on member ABC a force P directed along line BD. Knowing that P must have a 240-lb vertical component, determine (a) the magnitude of the force P, (b) its horizontal component.

2.130 Two cables are tied together at C and loaded as shown. Determine the tension (a) in cable AC, (b) in cable BC.

Fig. P2.127

240 lb

75 lb

75 lb

50°

30°A

!

Fig. P2.128

30 lb P

!25°

40°Q

D

A B C

Fig. P2.129

8.5 ft5 ft

12 ft 7.5 ft

396 lb

A B

C 9 ft

Fig. P2.130

2.131 Two cables are tied together at C and loaded as shown. Knowing that P 5 360 N, determine the tension (a) in cable AC, (b) in cable BC.

2.132 Two cables are tied together at C and loaded as shown. Determine the range of values of P for which both cables remain taut. Fig. P2.131 and P2.132

A B

P

Q = 480 N

C

34

600 mm

250 mm

b ee2 9 4 0 0 _ch0 2 _0 1 4 - 0 7 1 .ind d Page 6 7 1 1 / 2 8 / 0 8 9 :2 1 :4 5 PM user- s1 7 3b ee2 9 4 0 0 _ch0 2 _0 1 4 - 0 7 1 .ind d Page 6 7 1 1 / 2 8 / 0 8 9 :2 1 :4 5 PM user- s1 7 3 / Volum es / 2 0 4 / MHDQ0 7 6 / work%0 / ind d %0/ Volum es / 2 0 4 / MHDQ0 7 6 / work%0 / ind d %0

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90 Rigid Bodies: Equivalent Systems of Forces 3.9 A winch puller AB is used to straighten a fence post. Knowing that the tension in cable BC is 1040 N and length d is 1.90 m, deter-mine the moment about D of the force exerted by the cable at C by resolving that force into horizontal and vertical components applied ( a) at point C, ( b) at point E.

3.10 It is known that a force with a moment of 960 N ? m about D is required to straighten the fence post CD . If d 5 2.80 m, determine the tension that must be developed in the cable of winch puller AB to create the required moment about point D.

A

B

C

D

E

d

0.875 m

0.2 m

Fig. P3.9 , P3.10, and P3.11

3.11 It is known that a force with a moment of 960 N ? m about D is required to straighten the fence post CD . If the capacity of winch puller AB is 2400 N, determine the minimum value of distance d to create the specified moment about point D.

3.12 and 3.13 The tailgate of a car is supported by the hydraulic lift BC . If the lift exerts a 125-lb force directed along its centerline on the ball and socket at B, determine the moment of the force about A.

Fig. P3.12

A

BC

15.3 in.

12.0 in.

12.0 in.

2.33 in.

17.2 in.

4.38 in.

7.62 in.

20.5 in.

A

BC

Fig. P3.13

bee29400_ch03_072-155.indd Page 90 11/28/08 9:36:51 PM user-s172bee29400_ch03_072-155.indd Page 90 11/28/08 9:36:51 PM user-s172 /Volumes/204/MHDQ076/work%0/indd%0/Volumes/204/MHDQ076/work%0/indd%0

Mahesh Tirumkudulu

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Mahesh Tirumkudulu

105Problems 3.49 To lift a heavy crate, a man uses a block and tackle attached to the bottom of an I-beam at hook B . Knowing that the moments about the y and the z axes of the force exerted at B by portion AB of the rope are, respectively, 120 N ? m and –460 N ? m, determine the distance a .

3.50 To lift a heavy crate, a man uses a block and tackle attached to the bottom of an I-beam at hook B . Knowing that the man applies a 195-N force to end A of the rope and that the moment of that force about the y axis is 132 N ? m, determine the distance a .

3.51 A small boat hangs from two davits, one of which is shown in the figure. It is known that the moment about the z axis of the resultant force R A exerted on the davit at A must not exceed 279 lb ? ft in absolute value. Determine the largest allowable ten-sion in line ABAD when x 5 6 ft.

3.52 For the davit of Prob. 3.51, determine the largest allowable dis-tance x when the tension in line ABAD is 60 lb.

3.53 To loosen a frozen valve, a force F of magnitude 70 lb is applied to the handle of the valve. Knowing that u 5 25°, M x 5 261 lb ? ft, and M z 5 243 lb ? ft, determine f and d .

3 ft

x

y

z

A

C

D7.75 ft

x

B

Fig. P3.51

x

y

d

z

B

A

q

F

4 in.

11 in.

f


x

y

z

A

BC

D

O

a

1.6 m2.2 m

4.8 m


3.54 When a force F is applied to the handle of the valve shown, its moments about the x and z axes are, respectively, M x 5 277 lb ? ft and M z 5 281 lb ? ft. For d 5 27 in., determine the moment M y of F about the y axis.


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Mahesh Tirumkudulu

3.100 The handpiece for a miniature industrial grinder weighs 0.6 lb, and its center of gravity is located on the y axis. The head of the handpiece is offset in the xz plane in such a way that line BC forms an angle of 25° with the x direction. Show that the weight of the handpiece and the two couples M 1 and M 2 can be replaced with a single equivalent force. Further, assuming that M 1 5 0.68 lb · in. and M 2 5 0.65 lb · in., determine ( a ) the magnitude and the direction of the equivalent force, ( b ) the point where its line of action intersects the xz plane.

3.17 REDUCTION OF A SYSTEM OF FORCES TO ONE FORCE AND ONE COUPLE

Consider a system of forces F 1 , F 2 , F 3 , . . . , acting on a rigid body at the points A 1 , A 2 , A 3 , . . . , defined by the position vectors r 1 , r 2 , r 3 , etc . ( Fig. 3.41 a ). As seen in the preceding section, F 1 can be moved from A 1 to a given point O if a couple of moment M 1 equal to the moment r 1 3 F 1 of F 1 about O is added to the original system of forces. Repeating this procedure with F 2 , F 3 , . . . , we obtain the

3.99 A 46-lb force F and a 2120-lb · in. couple M are applied to corner A of the block shown. Replace the given force-couple system with an equivalent force-couple system at corner H .

x

y

z

A

B

C

M1

M2

25°

Fig. P3.100

M

F

3 in.

x

y

zA

B

C

D

E

F

H

J

14 in.

18 in.

25 in.

45 in.

Fig. P3.99

(c)

R

MOR

O

(b)

F1

F2

M1

M2

M3

=O

F3

(a)

F1

F2

F3r2r3

A2

A3

=O

r1

A1

Fig. 3.41

3.17 Reduction of a System of Forces to One Force and One Couple 123


Mahesh Tirumkudulu

139Problems 3.109 A couple of magnitude M 5 54 lb ? in. and the three forces shown are applied to an angle bracket. ( a ) Find the resultant of this sys-tem of forces. ( b ) Locate the points where the line of action of the resultant intersects line AB and line BC .

3.110 A couple M and the three forces shown are applied to an angle bracket. Find the moment of the couple if the line of action of the resultant of the force system is to pass through ( a ) point A , ( b ) point B , ( c ) point C .

3.111 Four forces act on a 700 3 375-mm plate as shown. ( a ) Find the resultant of these forces. ( b ) Locate the two points where the line of action of the resultant intersects the edge of the plate.

3.112 Solve Prob. 3.111, assuming that the 760-N force is directed to the right.

3.113 A truss supports the loading shown. Determine the equivalent force acting on the truss and the point of intersection of its line of action with a line drawn through points A and G .

A B

C

10 lb 30 lb

60°12 in.

45 lb

M 8 in.

Fig. P3.109 and P3.110

A B

D EC

500 N

600 N

760 N

340 N

500 mm200 mm

375 mm

Fig. P3.111

CA

B D F

EG

240 lb 160 lb 300 lb

40°

180 lb

70°

x

y4 ft

8 ft 8 ft

8 ft 8 ft 8 ft

6 ft

Fig. P3.113

2 in.

1 in.

210 lb

150 lb25°

25°

4 in.

120 lb 160 lb

A B

C D

EF

r = 2 in. r = 1 in.12

6 in.6 in.

Fig. P3.114

3.114 Pulleys A and B are mounted on bracket CDEF . The tension on each side of the two belts is as shown. Replace the four forces with a single equivalent force, and determine where its line of action intersects the bottom edge of the bracket.

C

A B

DF

E

G H

P

200 N240 mm

120 N

70°15°

50 mm

50 mm

50 mm

80 N42 N•m

40 N•m 180 mm

640 mm

520 mm

Fig. P3.115

3.115 A machine component is subjected to the forces and couples shown. The component is to be held in place by a single rivet that can resist a force but not a couple. For P 5 0, determine the location of the rivet hole if it is to be located ( a ) on line FG , ( b ) on line GH .

3.116 Solve Prob. 3.115, assuming that P 5 60 N.


Mahesh Tirumkudulu

153Review Problems 3.156 Four ropes are attached to a crate and exert the forces shown. If the forces are to be replaced with a single equivalent force applied at a point on line AB , determine ( a ) the equivalent force and the dis-tance from A to the point of application of the force when a 5 30°, ( b ) the value of a so that the single equivalent force is applied at point B .

3.157 A blade held in a brace is used to tighten a screw at A . ( a ) Determine the forces exerted at B and C , knowing that these forces are equiva-lent to a force-couple system at A consisting of R 5 2(30 N) i 1 R y j 1 Rz k and M R A 5 2(12 N ? m) i . ( b ) Find the corresponding values of R y and R z . ( c ) What is the orientation of the slot in the head of the screw for which the blade is least likely to slip when the brace is in the position shown?

A B

D C

46 in.

66 in.

10 in.

25°

65°

36 in.

100 lb

160 lb

400 lb

90 lb

a

Fig. P3.156

x

B

Czk

Cxi

Cyj

–B k

200 mm

200 mm

150 mm

y

z

A

C

Fig. P3.157

3.158 A concrete foundation mat in the shape of a regular hexagon of side 12 ft supports four column loads as shown. Determine the magnitudes of the additional loads that must be applied at B and F if the resultant of all six loads is to pass through the center of the mat.

A

BC

D

EF

O

15 kips

20 kips

10 kips30 kips

y

x

z

12 ft

Fig. P3.158


Mahesh Tirumkudulu

174 Equilibrium of Rigid Bodies

B

D

30°

500 N

C

200 mm

250 mm

250 mm

A

Fig. P4.28

4.27 A rod AB, hinged at A and attached at B to cable BD, supports the loads shown. Knowing that d 5 150 mm, determine (a) the tension in cable BD, (b) the reaction at A.

4.28 A lever AB is hinged at C and attached to a control cable at A. If the lever is subjected to a 500-N horizontal force at B, determine (a) the tension in the cable, (b) the reaction at C.

4.23 and 4.24 For each of the plates and loadings shown, determine the reactions at A and B.

40 lb 40 lb

50 lb 50 lb

AB

(a)

30°

10 in.

AB

(b)

20 in.

10 in.

4 in. 4 in.

20 in.

Fig. P4.23

40 lb 40 lb

50 lb 50 lb

AB

(a)

30º

20 in.

10 in.

AB

(b)

20 in.

10 in.

4 in. 4 in.

Fig. P4.24

4.25 Determine the reactions at A and B when (a) a 5 0, (b) a 5 90°, (c) a 5 30°.

4.26 A rod AB, hinged at A and attached at B to cable BD, supports the loads shown. Knowing that d 5 200 mm, determine (a) the tension in cable BD, (b) the reaction at A.

10 in. 10 in.

12 in.

a

A

B

750 lb ! in.

Fig. P4.25

90 N

100 mm

100 mm100 mm100 mm

A

B

d

D

90 N



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Mahesh Tirumkudulu

177Problems 4.39 Two slots have been cut in plate DEF, and the plate has been placed so that the slots fit two fixed, frictionless pins A and B. Knowing that P 5 15 lb, determine (a) the force each pin exerts on the plate, (b) the reaction at F.

P A

B

D E

F

4 in. 4 in. 7 in. 2 in.

30º

30 lb

3 in.

Fig. P4.39

4.40 For the plate of Prob. 4.39 the reaction at F must be directed downward, and its maximum allowable value is 20 lb. Neglecting friction at the pins, determine the required range of values of P.

4.41 Bar AD is attached at A and C to collars that can move freely on the rods shown. If the cord BE is vertical (a 5 0), determine the tension in the cord and the reactions at A and C.

A B

E

CD

30°

80 Na

0.2 m 0.2 m

30°

0.2 m

Fig. P4.41

4.42 Solve Prob. 4.41 if the cord BE is parallel to the rods (a 5 30°).

4.43 An 8-kg mass can be supported in the three different ways shown. Knowing that the pulleys have a 100-mm radius, determine the reaction at A in each case.

B

A A A

B B

8 kg 8 kg 8 kg

(a) (b) (c)

1.6 m 1.6 m 1.6 m

Fig. P4.43


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180 Equilibrium of Rigid Bodies 4.55 A collar B of weight W can move freely along the vertical rod shown. The constant of the spring is k, and the spring is unstretched when u 5 0. (a) Derive an equation in u, W, k, and l that must be satisfied when the collar is in equilibrium. (b) Knowing that W 5 300 N, l 5 500 mm, and k 5 800 N/m, determine the value of u corre-sponding to equilibrium.

4.56 A vertical load P is applied at end B of rod BC. The constant of the spring is k, and the spring is unstretched when u 5 90°. (a) Neglecting the weight of the rod, express the angle u corre-sponding to equilibrium in terms of P, k, and l. (b) Determine the value of u corresponding to equilibrium when P 5 1

4 kl.

4.57 Solve Sample Prob. 4.5, assuming that the spring is unstretched when u 5 90°.

4.58 A slender rod AB, of weight W, is attached to blocks A and B that move freely in the guides shown. The constant of the spring is k, and the spring is unstretched when u 5 0. (a) Neglecting the weight of the blocks, derive an equation in W, k, l, and u that must be satisfied when the rod is in equilibrium. (b) Determine the value of u when W 5 75 lb, l 5 30 in., and k 5 3 lb/in.

4.59 Eight identical 500 3 750-mm rectangular plates, each of mass m 5 40 kg, are held in a vertical plane as shown. All connections consist of frictionless pins, rollers, or short links. In each case, determine whether (a) the plate is completely, partially, or improp-erly constrained, (b) the reactions are statically determinate or indeterminate, (c) the equilibrium of the plate is maintained in the position shown. Also, wherever possible, compute the reactions.

A

B

q

l

Fig. P4.55A

B

C

P

q

l

l

Fig. P4.56

A

BW

q

l

Fig. P4.58

Fig. P4.59

A B

CD

1 2 3 4

5 6 7 8


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198 Equilibrium of Rigid Bodies 4.94 Two transmission belts pass over sheaves welded to an axle supported by bearings at B and D. The sheave at A has a radius of 2.5 in., and the sheave at C has a radius of 2 in. Knowing that the system rotates at a constant rate, determine (a) the tension T, (b) the reactions at B and D. Assume that the bearing at D does not exert any axial thrust and neglect the weights of the sheaves and axle.

T

720 N

y

80 mm 120 mm

120 mm

200 mm

AE

B

C

D

x

z

40 mm

Fig. P4.95

4.95 A 200-mm lever and a 240-mm-diameter pulley are welded to the axle BE that is supported by bearings at C and D. If a 720-N verti-cal load is applied at A when the lever is horizontal, determine (a) the tension in the cord, (b) the reactions at C and D. Assume that the bearing at D does not exert any axial thrust.

30 lb

T

24 lb

18 lb

y8 in.

6 in.

A

BC

D

x

z

6 in.

Fig. P4.94

4.96 Solve Prob. 4.95, assuming that the axle has been rotated clockwise in its bearings by 30° and that the 720-N load remains vertical.


Mahesh Tirumkudulu

213

REVIEW PROBLEMS

4.142 A hand truck is used to move two kegs, each of mass 40 kg. Neglecting the mass of the hand truck, determine (a) the vertical force P that should be applied to the handle to maintain equilib-rium when a 5 35°, (b) the corresponding reaction at each of the two wheels.

A

P

B

G1

G280 mm

350 mm

300 mm

500 mm

a

Fig. P4.142

800 mm

200 mm300 N

200 mm300 N

a

AB

C

Fig. P4.143

A

B

D

12 in.

20°75 lb

C10 in.

15 in.

Fig. P4.144 A B C

150 mm

200 mm80 mm 80 mm

120 N

D

Fig. P4.145

4.143 Determine the reactions at A and C when (a) a 5 0, (b) a 5 30°.

4.144 A lever AB is hinged at C and attached to a control cable at A. If the lever is subjected to a 75-lb vertical force at B, determine (a) the tension in the cable, (b) the reaction at C.

4.145 Neglecting friction and the radius of the pulley, determine (a) the tension in cable ADB, (b) the reaction at C.


Mahesh Tirumkudulu

215Review Problems 4.152 The rigid L-shaped member ABF is supported by a ball-and-socket joint at A and by three cables. For the loading shown, determine the tension in each cable and the reaction at A.

AA

B B

C C

P

A

B

C

P

P

A

B

C

P

45°

45°

(a) (b)

(c) (d)

a = 30°

30°

aa

a

aa

a

aa

a

aa

a

Fig. P4.153

x

y

z

A

B

C D

E F

G

J

H

24 lb

24 lb

9 in.

16 in.

16 in.

8 in.

12 in.

16 in.

8 in.

8 in.

8 in.

O

Fig. P4.152

4.153 A force P is applied to a bent rod ABC, which may be supported in four different ways as shown. In each case, if possible, determine the reactions at the supports.


Mahesh Tirumkudulu

tut1 chapters 2to4

Documents

largest force p

angle b

qda b c

vertical force

ttt t ta b c d efig

cable bc

ind d page

cable ac