Problems on Force System

Problems on

Coplanar concurrent, non-concurrent and parallel

Force Systems

Assignment by: Md. Kamrul Ahsan Lecturer, Dept. of Civil Engineering, Khulna University of Engineering & Technology, Khulna

Problems Source: http://www.mathalino.com

Solves Collected and Composed by: Tariqul Islam (1301047),

Dept. of Civil Engineering, Khulna University of Engineering & Technology, Khulna

Problem 01 The magnitude of vertical force F shown in Fig. P-016 is 8000 N. Resolve F into components parallel to the bars AB and AC.

Solution

By Sine Law:

° °

4256.71 answer

° °

10778.37 answer

Problem 02 If the force F shown in Fig. P-017 is resolved into components parallel to the bars AB and BC, the magnitude of the component parallel to bar BC is 4 kN. What are the magnitudes of F and its component parallel to AB?

Solution

.

.

56.31°

.

.

20.56°

90° 90° 56.31°

33.69°

90° 90° 20.56°

69.44°

180° 180° 33.69° 69.44°

76.87°

By Sine law

. °

. °

7.02 answer

. °

. °

6.75 answer

Problem 03 A vertical force P at A and another vertical force F at B in Fig. P-251 produce a resultant of 100 lb down at D and a counterclockwise couple C of 200 lb·ft. Find the magnitude and direction of forces P and F.

Solution

7

3 7 100 200

300 answer

∑

100

300 100

200 answer

Problem 04 A parallel force system acts on the lever shown in Fig. P-236. Determine the magnitude and position of the resultant.

Solution

∑ 30 60 20 40 110 downward

∑

2 30 5 60 7 20 11 40

660 . clockwise

110 660

6 to the right of A

Thus, R = 110 lb downward at 6 ft to the right of A. answer

Problem 05 The beam AB in Fig. P-238 supports a load which varies an intensity of 220 N/m to 890 N/m. Calculate the magnitude and position of the resultant load.

Solution

6 220 1320

6 670 2010

1320 2010

3330

3 4

3330 3 1320 4 2010

3330 12000

3.6

Thus, R = 3330 N downward at 3.6 m to the left of A. answer

Problem 06 The cable and boom shown in Fig. P-308 support a load of 600 lb. Determine the tensile force T in the cable and the compressive for C in the boom.

Solution

∑ 0

45° 30° 1.2247

∑ 0

30° 45° 600

30° 1.2247 45° 600

1.366 600

439.24 answer

1.2247 439.24

537.94 answer

Another Solution (By Rotation of Axes)

∑ 0 75° 600 45° 439.23 (okay!)

∑ 0

75° 600 45° 439.23 75° 600 45° 537.94 (okay!)

Another Solution (By Force Polygon)

° ° °

439.23 (okay!)

537.94 (okay!)

Problem 07 A cylinder weighing 400 lb is held against a smooth incline by means of the weightless rod AB in Fig. P-309. Determine the forces P and N exerted on the cylinder by the rod and the incline.

Solution

∑ 0

25° 55° 0.9038

∑ 0

25° 55° 400

0.9038 25° 55° 400

0.9556 400

418.60 answer

0.9038 418.60

378.34 answer

Another Solution (By Rotation of Axes)

∑ 0

30° 400 55°

378.35 (ok!)

∑ 0

30° 400 55°

378.35 30° 400 55°

418.60 (ok!)

Another Solution (By Force Polygon)

° ° °

378.35 (ok!)

418.60 (ok!)

Problem 08 Determine the magnitude of P and F necessary to keep the concurrent force system in Fig. P-312 in equilibrium.

Solution

∑ 0

60° 200 45° 300 30°

317.16 1.7320

∑ 0

60° 200 45° 30°

317.16 1.7320 60° 200 45° 30°

274.67 1.5 141.42 0.5

133.25 answer

317.16 1.7320 133.25

86.37 answer

Problem 09 The five forces shown in Fig. P-314 are in equilibrium. Compute the values of P and F.

Solution

∑ 0 30° 40 15° 30 30° 20 60°

0.5 6.3165 12.63 answer

∑ 0 20 60° 40 15° 30 30° 30° 10 10.35 25.98 13.63 0.4660 5.31 answer

Problem 10 The system of knotted cords shown in Fig. P-317 support the indicated weights. Compute the tensile force in each cord.

Solution

From the knot where 400-lb load is hanging ∑ 0

75° 30° 0.5176

∑ 0

75° 30° 400 0.5176 75° 30° 400

400 answer

0.5176 400 207.06 answer

From the knot where 300-lb load is hanging

∑ 0

45° 300 30° 45° 300 400 30° 914.16 answer

∑ 0

45° 30° 914.16 45° 400 30° 846.41 answer

Problem 11 Cords are loop around a small spacer separating two cylinders each weighing 400 lb and pass, as shown in Fig. P-319 over a frictionless pulleys to weights of 200 lb and 400 lb . Determine the angle θ and the normal pressure N between the cylinders and the smooth horizontal surface.

Solution

∑ 0 400 200

0.5 60° answer

∑ 0 N 400sinθ 800 N 400sin60° 800 N 453.59lb answer

Problem 12 Determine the amount and direction of the smallest force P required to start the wheel in Fig. P-325 over the block. What is the reaction at the block?

Solution

cosβ.

β 41.41°

30° β 71.41°

ϕ 18.45° α

θ 90° α

. °

P. °

. °

. ° . °

. °

2000sin71.41° cos 18.59° α 0

cos 18.59° α 0

18.59° α 90°

α 71.41° answer

P. °

. ° . °

P 1895.65lb answer

ϕ 18.59° 71.41° 90°

θ 90° 71.41° 18.59°

. ° °

R 637.59lb answer

Problem 13 Forces P and F acting along the bars shown in Fig. P-327 maintain equilibrium of pin A. Determine the values of P and F.

Solution

∑ 0

√30

√50 ... ... ... (1)

∑ 0

√ 18

√22.5

Substitute F of Equation (1)

√ √

50 22.5

√ 27.5

14.76 answer

From Equation (1)

√14.76 50

39 answer

Problem 14 Two weightless bars pinned together as shown in Fig. P-328 support a load of 35 kN. Determine the forces P and F acting respectively along bars AB and AC that maintain equilibrium of pin A.

Solution

∑ 0

√ √

0.7104

∑ 0

√ √35

√0.7104

√35

0.3883 35

90.14 answer

0.7104 90.14

64.03 answer

Problem 15 Determine the reactions R1 and R2 of the beam in Fig. P-333 loaded with a concentrated load of 1600 lb and a load varying from zero to an intensity of 400 lb per ft.

Solution

∑ 0

12 4 12 400

800 ∑ 0

12 8 12 400

1600 ∑ 0 16 12 1600 12 16 13 1600 12 800

1900 answer ∑ 0 16 3 1500 4 16 16 3 1600 4 800 16 1600

2100 answer

Problem 16 Determine the reactions for the beam loaded as shown in Fig. P-334.

Solution

∑ 0

7.5 6 12 4.5 3 6 1 3 15

23.4 answer

∑ 0

7.5 1.5 12 3 3 6 6.5 3 15

29.1 answer

Problem 17 The upper beam in Fig. P-337 is supported at D and a roller at C which separates the upper and lower beams. Determine the values of the reactions at A, B, C, and D. Neglect the weight of the beams.

Solution

∑ 0

10 4 60 6 190

90 answer

∑ 0

10 14 60 4 190

160 answer

Problem 18 The two 12-ft beams shown in Fig. 3-16 are to be moved horizontally with respect to each other and load P shifted to a new position on CD so that all three reactions are equal. How far apart will R2 and R3 then be? How far will P be from D?

Solution

From FBD of beam CD

∑ 0

0.5 960

640

0.5 640 320 answer

∑ 0

12 960

12 320 960

4

Thus, P is 8 ft to the left of D. answer

From the figure above, Rc is at the midspan of AB to produce equal reactions R1 and R2. Thus, R2 and R3 are 6 ft apart. answer

From FBD of beam AB

0.5 640 320 answer 0.5 640 320 answer

Problem 19 For the system of pulleys shown in Fig. P-340, determine the ratio of W to P to maintain equilibrium. Neglect axle friction and the weights of the pulleys.

Solution

From the lowermost pulley

∑ 0

9

9 answer

Problem 20 A boom AB is supported in a horizontal position by a hinge A and a cable which runs from C over a small pulley at D as shown in Fig. P-346. Compute the tension T in the cable and the horizontal and vertical components of the reaction at A. Neglect the size of the pulley at D.

Solution

∑ 0

4√

2 200 6 100

279.51 answer ∑ 0

√200 100

√279.51 300

50 answer

∑ 0

√

√279.51

125 answer

Problem 21 The beam shown in Fig. P-351 is supported by a hinge at A and a roller on a 1 to 2 slope at B. Determine the resultant reactions at A and B.

Solution

∑ 0

4√

3 40

33.54

∑ 0

4 1 40

10

∑ 0

√ √33.54

15

√15 10

18.03

33.69°

Thus, 18.03 up to the right at 33.69° from horizontal. answer

Another Solution

From Equilibrium of Concurrent Force System, three coplanar forces in equilibrium are concurrent.

tanθ

33.69° okay

63.43°

90° 56.31°

90° 26.57°

97.12°

. ° . ° . °

18.03 okay

33.54 okay

Problem 22 The uniform rod in Fig. P-357 weighs 420 lb and has its center of gravity at G. Determine the tension in the cable and the reactions at the smooth surfaces at A and B.

Solution

Distance AB AB √8 8 8√2 ∑ 0

45°

∑ 0

2 6 420 8√2

2 2520 8√2

1260 4√2

45° 1260 4√2

4.9497 1260

254.56 answer

254.56 45°

180 answer

∑ 0

45° 420

254.56 45° 420

240 answer

Problem 23 A 4-m bar of negligible weight rests in a horizontal position on the smooth planes shown in Fig. P-359. Compute the distance x at which load T = 10 kN should be placed from point B to keep the bar horizontal.

Solution

From the Force Polygon

° °

21.96

From the Free Body Diagram

∑ 0

4 30° 20 3 10

4 21.96 30° 20 3 10

10 16.072

1.61 answer