CAPITULO II MARCO TEÓRICO
II. 11 Aporte teórico Patricia Benner
From the geometry, we calculate the distances:
16 in.
2 12 in.
2 20 in.
AC
20 in.
2 21 in.
2 29 in.
BC
Then, from the Free Body Diagram of point C:
16 21
Knowing that D 25 ,q determine the tension (a) in cable AC, (b) in rope BC.
SOLUTION
Free-Body Diagram Force Triangle
Law of Sines:
5 kN sin115 sin 5 sin 60
AC BC
T T
q q q
(a) 5 kN
sin115 5.23 kN sin 60
TAC q
q TAC 5.23 kNW
(b) 5 kN
sin 5 0.503 kN sin 60
TBC q
q TBC 0.503 kNW
Knowing that D 50q and that boom AC exerts on pin C a force directed long line AC, determine (a) the magnitude of that force, (b) the tension in cable BC.
SOLUTION
Free-Body Diagram Force Triangle
Two cables are tied together at C and are loaded as shown. Knowing that D 30 ,q determine the tension (a) in cable AC, (b) in cable BC.
SOLUTION
Free-Body Diagram Force Triangle
Law of Sines:
2943 N sin 60 sin 55 sin 65
AC BC
T T
q q q
(a) 2943 N
sin 60 2812.19 N sin 65
TAC q
q TAC 2.81 kNW
(b) 2943 N
sin 55 2659.98 N sin 65
TBC q
q TBC 2.66 kNW
A chairlift has been stopped in the position shown. Knowing that each chair weighs 300 N and that the skier in chair E weighs 890 N, determine that weight of the skier in chair F.
SOLUTION
Free-Body Diagram Point B
Force Triangle
Free-Body Diagram Point C
Force Triangle
In the free-body diagram of point B, the geometry gives:
1 9.9
Thus, in the force triangle, by the Law of Sines:
1190 N
In the free-body diagram of point C (with W the sum of weights of chair and skier) the geometry gives:
11.32
tan 10.39
CD 7.2
T q
Hence, in the force triangle, by the Law of Sines:
7468.6 N
Finally, the skier weight 1608.5 N300 N 1308.5 N
skier weight 1309 NW
A chairlift has been stopped in the position shown. Knowing that each chair weighs 300 N and that the skier in chair F weighs 800 N, determine the weight of the skier in chair E.
SOLUTION
Free-Body Diagram Point F
Force Triangle
Free-Body Diagram Point E
Force Triangle
In the free-body diagram of point F, the geometry gives:
1 12
Thus, in the force triangle, by the Law of Sines:
1100 N
In the free-body diagram of point E (with W the sum of weights of chair and skier) the geometry gives:
1 9.9
tan 30.51
AE 16.8
T q
Hence, in the force triangle, by the Law of Sines:
5107.5 N
skier weight 514 NW
Four wooden members are joined with metal plate connectors and are in equilibrium under the action of the four fences shown. Knowing that FA 510 lb and FB 480 lb, determine the magnitudes of the other two forces.
SOLUTION Free-Body Diagram
Resolving the forces into x and y components:
0: 510 lb sin15 480 lb cos15 0
x C
F F
6 q q
or FC 332 lbW
0: 510 lb cos15 480 lb sin15 0
y D
F F
6 q q
or FD 368 lbW
Four wooden members are joined with metal plate connectors and are in equilibrium under the action of the four fences shown. Knowing that FA 420 lb and FC 540 lb, determine the magnitudes of the other two forces.
SOLUTION
Resolving the forces into x and y components:
0: cos15 540 lb 420 lb cos15 0 or 671.6 lb
x B B
F F F
6 q q
672 lb
FB W
0: 420 lb cos15 671.6 lb sin15 0
y D
F F
6 q q
or FD 232 lbW
Two forces P and Q are applied as shown to an aircraft connection.
Knowing that the connection is in equilibrium and the P 400 lb and Q 520 lb, determine the magnitudes of the forces exerted on the rods A and B.
SOLUTION
Free-Body Diagram Resolving the forces into x and y directions:
A B 0
In the y-direction (one unknown force)
Two forces P and Q are applied as shown to an aircraft connection.
Knowing that the connection is in equilibrium and that the magnitudes of the forces exerted on rods A and B are FA 600 lb and FB 320 lb, determine the magnitudes of P and Q.
SOLUTION
Free-Body Diagram Resolving the forces into x and y directions:
A B 0
In the x-direction (one unknown force)
Two cables tied together at C are loaded as shown. Knowing that W 840 N, determine the tension (a) in cable AC, (b) in cable BC.
SOLUTION
Free-Body Diagram From geometry:
The sides of the triangle with hypotenuse CB are in the ratio 8:15:17.
The sides of the triangle with hypotenuse CA are in the ratio 3:4:5.
Thus:
Solving Equations (1) and (2) simultaneously:
(a) TCA 750 NW
(b) TCB 1190 NW
Two cables tied together at C are loaded as shown. Determine the range of values of W for which the tension will not exceed 1050 N in either cable.
SOLUTION
Free-Body Diagram From geometry:
The sides of the triangle with hypotenuse CB are in the ratio 8:15:17.
The sides of the triangle with hypotenuse CA are in the ratio 3:4:5.
Thus: Then, from Equations (1) and (2)
680 N 17
The cabin of an aerial tramway is suspended from a set of wheels that can
Note: In Problems 2.55 and 2.56 the cabin is considered as a particle. If considered as a rigid body (Chapter 4) it would be found that its center of gravity should be located to the left of the centerline for the line CD to be vertical.
The cabin of an aerial tramway is suspended from a set of wheels that can roll freely on the support cable ACB and is being pulled at a constant speed by cable DE. Knowing that D 42q and E 32q, that the tension in cable DE is 20 kN, and assuming the tension in cable DF to be negligible, determine (a) the combined weight of the cabin, its support system, and its passengers, (b) the tension in the support cable ACB.
SOLUTION Free-Body Diagram
First, consider the sum of forces in the x-direction because there is only one unknown force:
0.1049TACB 14.863 kN
(b) TACB 141.7 kNW
A block of weight W is suspended from a 500-mm long cord and two springs of which the unstretched lengths are 450 mm. Knowing that the constants of the springs are kAB 1500 N/m and kAD 500 N/m, determine (a) the tension in the cord, (b) the weight of the block.
SOLUTION
Free-Body Diagram At A First note from geometry:
The sides of the triangle with hypotenuse AD are in the ratio 8:15:17.
The sides of the triangle with hypotenuse AB are in the ratio 3:4:5.
The sides of the triangle with hypotenuse AC are in the ratio 7:24:25.
Then:
(b) and
3 24 8
0: 150 N 66.18 N 115 N 0
5 25 17
Fy W
6
or W 208 NW
A load of weight 400 N is suspended from a spring and two cords which are attached to blocks of weights 3W and W as shown. Knowing that the constant of the spring is 800 N/m, determine (a) the value of W, (b) the unstretched length of the spring.
SOLUTION
Free-Body Diagram At A
First note from geometry:
The sides of the triangle with hypotenuse AD are in the ratio 12:35:37.
The sides of the triangle with hypotenuse AC are in the ratio 3:4:5.
The sides of the triangle with hypotenuse AB are also in the ratio 12:35:37.
(b) Have spring force
s AB o
F k L L Where
AB AB AB o
F k L L
and
0.360 m
2 1.050 m
2 1.110 m
LAB
So:
0
281.74 N 800 N/m 1.110L m
or L0 758 mmW
For the cables and loading of Problem 2.46, determine (a) the value of D for which the tension in cable BC is as small as possible, (b) the corresponding value of the tension.
SOLUTION
The smallest TBC is when TBC is perpendicular to the direction of TAC
Free-Body Diagram At C Force Triangle
(a) D 55.0q W
(b) TBC
2943 N sin 55
q
2410.8 N
2.41 kN
TBC W
Knowing that portions AC and BC of cable ACB must be equal, determine the shortest length of cable which can be used to support the load shown if the tension in the cable is not to exceed 725 N.
SOLUTION
Two cables tied together at C are loaded as shown. Knowing that the maximum allowable tension in each cable is 200 lb, determine (a) the magnitude of the largest force P which may be applied at C, (b) the corresponding value of D.
SOLUTION
Free-Body Diagram: C Force Triangle
Force triangle is isoceles with
2E 180q 85q E 47.5q
(a) P 2 200 lb cos 47.5
q 270 lb
Since P !0, the solution is correct. P 270 lbW
(b) D 180q 55q 47.5q 77.5q D 77.5q W
Two cables tied together at C are loaded as shown. Knowing that the maximum allowable tension is 300 lb in cable AC and 150 lb in cable BC, determine (a) the magnitude of the largest force P which may be applied at C, (b) the corresponding value of D.
SOLUTION
Free-Body Diagram: C Force Triangle
(a) Law of Cosines:
For the structure and loading of Problem 2.45, determine (a) the value of D for which the tension in cable BC is as small as possible, (b) the corresponding value of the tension.
SOLUTION
TBC must be perpendicular to FAC to be as small as possible.
Free-Body Diagram: C Force Triangle is
a right triangle
(a) We observe: D 55q D 55q W
(b) TBC
400 lb sin 60
q
or TBC 346.4 lb TBC 346 lbW
Boom AB is supported by cable BC and a hinge at A. Knowing that the boom exerts on pin B a force directed along the boom and that the tension in rope BD is 70 lb, determine (a) the value of D for which the tension in cable BC is as small as possible, (b) the corresponding value of the tension.
SOLUTION
Free-Body Diagram: B (a) Have: TBD FAB TBC 0
where magnitude and direction of TBD are known, and the direction of FAB is known.
Then, in a force triangle:
By observation, TBC is minimum when D 90.0q W (b) Have TBC
70 lb sin 180
q 70q 30q
68.93 lb
68.9 lb
TBC W
Collar A shown in Figure P2.65 and P2.66 can slide on a frictionless vertical rod and is attached as shown to a spring. The constant of the spring is 660 N/m, and the spring is unstretched when h 300 mm.
Knowing that the system is in equilibrium when h 400 mm, determine the weight of the collar.
SOLUTION
Free-Body Diagram: Collar A
Have: Fs k L
cAB LAB
The 40-N collar A can slide on a frictionless vertical rod and is attached as shown to a spring. The spring is unstretched when h 300 mm.
Knowing that the constant of the spring is 560 N/m, determine the value of h for which the system is in equilibrium.
SOLUTION
Free-Body Diagram: Collar A
2 2
A 280-kg crate is supported by several rope-and-pulley arrangements as shown. Determine for each arrangement the tension in the rope. (Hint:
The tension in the rope is the same on each side of a simple pulley. This can be proved by the methods of Chapter 4.)
SOLUTION
Free-Body Diagram of pulley (a)
Solve parts b and d of Problem 2.67 assuming that the free end of the rope is attached to the crate.
Problem 2.67: A 280-kg crate is supported by several rope-and-pulley arrangements as shown. Determine for each arrangement the tension in the rope. (Hint: The tension in the rope is the same on each side of a simple pulley. This can be proved by the methods of Chapter 4.)
SOLUTION
Free-Body Diagram of pulley and crate
A 350-lb load is supported by the rope-and-pulley arrangement shown.
Knowing that E 25q, determine the magnitude and direction of the force P which should 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 Chapter 4.)
SOLUTION
Free-Body Diagram: Pulley A
0: 2 sin 25 cos 0
A 350-lb load is supported by the rope-and-pulley arrangement shown.
Knowing that D 35 ,q determine (a) the angle E, (b) the magnitude of the force P which should 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 Chapter 4.)
SOLUTION
Free-Body Diagram: Pulley A
0: 2 sin cos 25 0
Fx P E P
6 q
Hence:
(a) 1
sin cos 25
E 2 q or E 24.2q W
(b) 6Fy 0: 2 cosP E Psin 35q 350 lb 0 Hence:
2 cos 24.2P q Psin 35q 350 lb 0
or P 145.97 lb P 146.0 lbW
A load Q is applied to the pulley C, which can roll on the cable ACB. The pulley is held in the position shown by a second cable CAD, which passes over the pulley A and supports a load P. Knowing that P 800 N, determine (a) the tension in cable ACB, (b) the magnitude of load Q.
SOLUTION
Free-Body Diagram: Pulley C
(a) 6Fx 0: TACB
cos 30q cos 50q
800 N cos 50
q 0
Hence TACB 2303.5 N
2.30 kN
TACB W
(b) 6Fy 0: TACB
sin 30q sin 50q
800 N sin 50
q Q 0
2303.5 N sin 30
q sin 50q
800 N sin 50
q Q 0
or Q 3529.2 N Q 3.53 kNW
A 2000-N load Q is applied to the pulley C, which can roll on the cable ACB. The pulley is held in the position shown by a second cable CAD, which passes over the pulley A and supports a load P. Determine (a) the tension in the cable ACB, (b) the magnitude of load P.
SOLUTION
Free-Body Diagram: Pulley C
(a) Substitute Equation (1) into Equation (2):
0.3473 1306 N 453.57 N P
454 N
P W
Determine (a) the x, y, and z components of the 200-lb force, (b) the anglesTx,Ty, and Tz that the force forms with the coordinate axes.
SOLUTION
(a) Fx
200 lb cos 30 cos 25
q q 156.98 lb
157.0 lb
Fx W
200 lb sin 30
100.0 lb
Fy q
100.0 lb
Fy W
200 lb cos 30 sin 25
73.1996 lb
Fz q q
73.2 lb Fz W
(b) 156.98
cosTx 200 or Tx 38.3q W
100.0
cosTy 200 or Ty 60.0q W
73.1996
cosTz 200 or Tz 111.5q W
Determine (a) the x, y, and z components of the 420-lb force, (b) the
To stabilize a tree partially uprooted in a storm, cables AB and AC are attached to the upper trunk of the tree and then are fastened to steel rods anchored in the ground. Knowing that the tension in cable AB is 4.2 kN, determine (a) the components of the force exerted by this cable on the tree, (b) the angles Tx,Ty, and Tz that the force forms with axes at A which are parallel to the coordinate axes.
SOLUTION
(a) Fx
4.2 kN sin 50 cos 40
q q 2.4647 kN
2.46 kN
Fx W
Fy
4.2 kN cos 50
q 2.6997 kN
2.70 kN
Fy W
Fz
4.2 kN sin 50 sin 40
q q 2.0681 kN
2.07 kN
Fz W
(b) 2.4647
cosTx 4.2
x 54.1
T q W
2.7 cosTy 4.2
130.0 Ty q W
2.0681 cosTz 4.0
z 60.5
T q W
To stabilize a tree partially uprooted in a storm, cables AB and AC are attached to the upper trunk of the tree and then are fastened to steel rods anchored in the ground. Knowing that the tension in cable AC is 3.6 kN, determine (a) the components of the force exerted by this cable on the tree, (b) the angles Tx,Ty, and Tz that the force forms with axes at A which are parallel to the coordinate axes.
SOLUTION
(a) Fx
3.6 kN cos 45 sin 25
q q 1.0758 kN
1.076 kN
Fx W
3.6 kN sin 45
2.546 kN
Fy q
2.55 kN
Fy W
3.6 kN cos 45 cos 25
2.3071 kN
Fz q q
2.31 kN
Fz W
(b) 1.0758
cosTx 3.6
2.546 cosTy 3.6
135.0 Ty q W
2.3071 cosTz 3.6
z 50.1
T q W
A horizontal circular plate is suspended as shown from three wires which
A horizontal circular plate is suspended as shown from three wires which
A horizontal circular plate is suspended as shown from three wires which are attached to a support at D and form 30q angles with the vertical.
Knowing that the tension in wire CD is 120 lb, determine (a) the components of the force exerted by this wire on the plate, (b) the angles Tx,Ty, and Tz that the force forms with the coordinate axes.
A horizontal circular plate is suspended as shown from three wires which are attached to a support at D and form 30q angles with the vertical.
Knowing that the x component of the forces exerted by wire CD on the plate is –40 lb, determine (a) the tension in wire CD, (b) the angles Tx,Ty, andTz that the force exerted at C forms with the coordinate axes.
SOLUTION
Determine the magnitude and direction of the force
Determine the magnitude and direction of the force
A force acts at the origin of a coordinate system in a direction defined by the angles Tx 64.5q and Tz 55.9q. Knowing that the y component of the force is –200 N, determine (a) the angle Ty, (b) the other components and the magnitude of the force.
SOLUTION
(a) We have
cosTx
2
cosTy
2
cosTz
2 1
cosTy
2 1
cosTy
2
cosTz
2
Since Fy we must have cos0 Ty 0
Thus, taking the negative square root, from above, we have:
2
2
A force acts at the origin of a coordinate system in a direction defined by the angles Tx 75.4q and Ty 132.6q. Knowing that the z component of the force is –60 N, determine (a) the angle Tz, (b) the other components and the magnitude of the force.
SOLUTION
(a) We have
cosTx
2
cosTy
2
cosTz
2 1
cosTy
2 1
cosTy
2
cosTz
2
Since Fz we must have cos0 Tz 0
Thus, taking the negative square root, from above, we have:
2
2
A force F of magnitude 400 N acts at the origin of a coordinate system.
A force F of magnitude 600 lb acts at the origin of a coordinate system.
A transmission tower is held by three guy wires anchored by bolts at B, C, and D. If the tension in wire AB is 2100 N, determine the components of the force exerted by the wire on the bolt at B.
SOLUTION
A transmission tower is held by three guy wires anchored by bolts at B, C, and D. If the tension in wire AD is 1260 N, determine the components of the force exerted by the wire on the bolt at D.
SOLUTION
A rectangular plate is supported by three cables as shown. Knowing that the tension in cable AB is 204 lb, determine the components of the force exerted on the plate at B.
SOLUTION
A rectangular plate is supported by three cables as shown. Knowing that the tension in cable AD is 195 lb, determine the components of the force exerted on the plate at D.
SOLUTION
A steel rod is bent into a semicircular ring of radius 0.96 m and is supported in part by cables BD and BE which are attached to the ring at B. Knowing that the tension in cable BD is 220 N, determine the components of this force exerted by the cable on the support at D.
SOLUTION
A steel rod is bent into a semicircular ring of radius 0.96 m and is supported in part by cables BD and BE which are attached to the ring at B. Knowing that the tension in cable BE is 250 N, determine the components of this force exerted by the cable on the support at E.
SOLUTION
Find the magnitude and direction of the resultant of the two forces shown
Find the magnitude and direction of the resultant of the two forces shown knowing that P 600 N and Q 400 N.
SOLUTION
Using the results from 2.93:
600 lb
>
0.2241 0.50 0.8365@
Knowing that the tension is 850 N in cable AB and 1020 N in cable AC, determine the magnitude and direction of the resultant of the forces exerted at A by the two cables.
SOLUTION
Assuming that in Problem 2.95 the tension is 1020 N in cable AB and 850 N in cable AC, determine the magnitude and direction of the resultant of the forces exerted at A by the two cables.
SOLUTION
For the semicircular ring of Problem 2.91, determine the magnitude and direction of the resultant of the forces exerted by the cables at B knowing that the tensions in cables BD and BE are 220 N and 250 N, respectively.
SOLUTION
For the solutions to Problems 2.91 and 2.92, we have
120 N
140 N
120 N
To stabilize a tree partially uprooted in a storm, cables AB and AC are attached to the upper trunk of the tree and then are fastened to steel rods anchored in the ground. Knowing that the tension in AB is 920 lb and that the resultant of the forces exerted at A by cables AB and AC lies in the yz plane, determine (a) the tension in AC, (b) the magnitude and direction of the resultant of the two forces.
SOLUTION