PROBLEM 2.1 1

PROBLEM 2.1

Two forces are applied to an eye bolt fastened to a beam. Determine

graphically the magnitude and direction of their resultant using (a) the

parallelogram law, (b) the triangle rule.

SOLUTION

(a)

(b)

We measure: 8.4 kNR =

19α = °

8.4 kN=R 19°

1

PROBLEM 2.2

The cable stays AB and AD help support pole AC. Knowing that the

tension is 500 N in AB and 160 N in AD, determine graphically the

magnitude and direction of the resultant of the forces exerted by the stays

at A using (a) the parallelogram law, (b) the triangle rule.

SOLUTION

We measure: 51.3 , 59α β= ° = °

(a)

(b)

We measure: 575 N, 67α= = °R

575 N=R 67°

2

PROBLEM 2.3

Two forces P and Q are applied as shown at point A of a hook support.

Knowing that P = 15 lb and Q = 25 lb, determine graphically the

magnitude and direction of their resultant using (a) the parallelogram law,

(b) the triangle rule.

SOLUTION

(a)

(b)

We measure: 37 lb, 76α= = °R

37 lb=R 76°

3

PROBLEM 2.4

Two forces P and Q are applied as shown at point A of a hook support.

Knowing that P = 45 lb and Q = 15 lb, determine graphically the

magnitude and direction of their resultant using (a) the parallelogram law,

(b) the triangle rule.

SOLUTION

(a)

(b)

We measure: 61.5 lb, 86.5α= = °R

61.5 lb=R 86.5°

4

PROBLEM 2.5

Two control rods are attached at A to lever AB. Using trigonometry and

knowing that the force in the left-hand rod is F1 = 120 N, determine

(a) the required force F2 in the right-hand rod if the resultant R of the

forces exerted by the rods on the lever is to be vertical, (b) the

corresponding magnitude of R.

SOLUTION

Graphically, by the triangle law

We measure: 2 108 NF ≅

77 NR ≅

By trigonometry: Law of Sines

2 120

sin sin 38 sin

F R

α β= =

°

90 28 62 , 180 62 38 80α β= ° − ° = ° = ° − ° − ° = °

Then:

2 120 N

sin 62 sin 38 sin80

F R= =

° ° °

or (a) 2 107.6 NF =

(b) 75.0 NR =

5

PROBLEM 2.6

Two control rods are attached at A to lever AB. Using trigonometry and

knowing that the force in the right-hand rod is F2 = 80 N, determine

(a) the required force F1 in the left-hand rod if the resultant R of the

forces exerted by the rods on the lever is to be vertical, (b) the

corresponding magnitude of R.

SOLUTION

Using the Law of Sines

1 80

sin sin 38 sin

F R

α β= =

°

90 10 80 , 180 80 38 62α β= ° − ° = ° = ° − ° − ° = °

Then:

1 80 N

sin80 sin 38 sin 62

F R= =

° ° °

or (a) 1 89.2 NF =

(b) 55.8 NR =

6

PROBLEM 2.7

The 50-lb force is to be resolved into components along lines -a a′ and

- .b b′ (a) Using trigonometry, determine the angle α knowing that the

component along -a a′ is 35 lb. (b) What is the corresponding value of

the component along - ?b b′

SOLUTION

Using the triangle rule and the Law of Sines

(a) sin sin 40

35 lb 50 lb

β °=

sin 0.44995β =

26.74β = °

Then: 40 180α β+ + ° = °

113.3α = °

(b) Using the Law of Sines:

50 lb

sin sin 40

bbF

α′ =

°

71.5 lbbbF ′ =

7

PROBLEM 2.8

The 50-lb force is to be resolved into components along lines -a a′ and

- .b b′ (a) Using trigonometry, determine the angle α knowing that the

component along -b b′ is 30 lb. (b) What is the corresponding value of

the component along - ?a a′

SOLUTION

Using the triangle rule and the Law of Sines

(a) sin sin 40

30 lb 50 lb

α °=

sin 0.3857α =

22.7α = °

(b) 40 180α β+ + ° = °

117.31β = °

50 lb

sin sin 40

aaF

β′ =

°

sin

50 lbsin 40

β′

= ° aaF

69.1 lbaaF ′ =

8

PROBLEM 2.9

To steady a sign as it is being lowered, two cables are attached to the sign

at A. Using trigonometry and knowing that α = 25°, determine (a) the

required magnitude of the force P if the resultant R of the two forces

applied at A is to be vertical, (b) the corresponding magnitude of R.

SOLUTION

Using the triangle rule and the Law of Sines

Have: ( )180 35 25α = ° − ° + °

120= °

Then: 360 N

sin 35 sin120 sin 25

P R= =

° ° °

or (a) 489 NP =

(b) 738 NR =

9

PROBLEM 2.10

To steady a sign as it is being lowered, two cables are attached to the sign

at A. Using trigonometry and knowing that the magnitude of P is 300 N,

determine (a) the required angle α if the resultant R of the two forces

applied at A is to be vertical, (b) the corresponding magnitude of R.

SOLUTION

Using the triangle rule and the Law of Sines

(a) Have: 360 N 300 N

sin sin 35α=

°

sin 0.68829α =

43.5α = °

(b) ( )180 35 43.5β = − ° + °

101.5= °

Then: 300 N

sin101.5 sin 35

R=

° °

or 513 NR =

10