104 Mechanics, Module 1

MODULE 11. Two equal forces are acting at a point with an angle 60º between them. If the resultant is equal to

20√3N. Find the magnitude of each force. Ans = 20 N2. The resultant of two forces when they act at 60º is 14 N . If the same forces are acting at right angle

their resultant is √136 N. Determine the magnitude of the two forces. Ans : P = 6 or 10 N , Q = 10 or 6 N3. Two forces are acing at point O as shown in fig.Determine the resultant in magnitude and direction. Q=100 N

P = 50 N 30º 15° Ans : R = 145.46 N , θ = 20.10º with force P4. The resultant of two concurrent force is 1500 N and the angle between the forces is 90°.The resultant

makes an angle 36° with one of the force. Find the magnitude of each force. Ans : P = 1213.86 N , Q = 881.26 N5. The sum of two concurrent force is 270 N and their resultant is 180 N. The angle between the force P

and the resultant is 90º. Find the magnitude of each force and angle between them. Ans : P = 75 N , Q = 195 N

6. The greatest and least resultant of two forces acting on a particle are 35 KN and 5 KN resp. If 25 KN is the resultant for the given system of force F1 and F2.Prove that the force are at rt.angle.

Ans : F1 =20 KN , F2 = 15 KN θ = 90° 7. If two forces F1 = 20 KN , F2 = 15 KN and on a particle. Find the resultant by Parallelogram law and

triangle law. Ans R = 28.813 KN θ = 29.3 º

6. The greatest and least resultant of two forces acting on a particle are 35 KN and 5 KN respectively. If 35 KN is the resultant of given system of forces. Prove that the forces are at right angle. Ans : F1=20 KN, F2=15 KN θ = 90 º

8. Two forces F and 2F act on a particle. If the resultant is increased by 12 KN and second force is doubled , the direction of the resultant remains unchanged. Find the value of F. Ans F=12 KN

9. The greatest and least resultant of two forces are resp 17 KN and 3 KN. Determine the angle between the forces when there resultant is √149 KN. Ans : θ = 90 º

10 The resultant of two forces F1, F2 is (2k+1) F12+ F22 when the included

angle is θ and it becomes(2k-1) F12+ F22 when the included angle is( 90 – θ).

Show that the included angle between the forces is given by tan θ = k-1 k+111 The forces acting at an angle of 120 º.The bigger force is of 40 N and the resultant is perpendicular to

the smaller one. Find the smaller force. Ans P = 20 N

Prepared by Manju P M ,Civil Department SNGCE

15

2070º

1

12.Find the magnitude of forces such that if they act at right angle their resultant is √10 KN but if they act at 60 º the resultant is √13 KN Ans P=3 KN , Q=1 KN

13.Two forces of magnitude 240N and 200 N acting at a point O as shown in fig. If the angle between the force is 60° . Determine the magnitude of the resultant force .Also determine the angle β and γ as shown in fig.

240 N α β γ 200 N

Ans : β = 33° , γ = 26.966º14.Two forces P and Q are acting at a point as shown in fig.The resultant force is 400 N and angle β and γ

are 35° and 25º resp. Find the force P and Q.

R = 400 N P α β γ Q

Ans : P =264.93 N, Q = 195. 19 N15 .Two forces of magnitude 240N and 200 N acting at a point O as shown in fig. If the resultant is equal

to 400 N . Determine the the angle α , β and γ as shown in fig.

R = 400 N

240 N α β γ 200 N

Ans :α = 49.27º , β = 33° , γ = 26.966º16. A transmission cable AB exerts a force of 3.2 KN on the pole in the direction of the tangent to the cable

at A as shown in fig. The pole is kept vertical by means of a guy rope AC. Determine the tension in the rope and the force exerted on the pole. Ans :R = 7.547 KN , T = 7.12 N

A 20° 25°

B

R C R

15° Prepared by Manju P M ,Civil Department SNGCE

2

17.Find the resultant of a set of coplanar forces acting on a lamina as shown in fig .Each square has a side of a 10 m. Y 20 KN

X O 15 KN 5 KN

Ans :R = 6.48 KN , θ = 43.167°

18.The forces 2P ,3P ,4P are directed along the sides of an equilateral triangle ABC as shown in fig.Determine the resultant considering C as origin.

A

3P 4P

B 2P C Ans : R = √3P , θ = -30°19. Replace the force system by a single force and a couple acting at origin O Y (200,300) 200 KNm 3 4 5 KN

(400,0) X O 5 KN 60° 6 KN (0,-300)

Ans : R =10.57 KN , θ = 217.96° , Mo = 23.1 KNm


3

20. Determine a single force couple system and its distance from A which is equivalent to force system of fig 3KN

6 KN 400 mm A 200mm

3 KN

400mm 200mm21. Resolve the given forces into its X and Y component

65°

45 N

Ans : X = -45 sin 65° , Y = -45 cos 65°

70°

45 N

Ans : X = +45 cos 70° , Y = -45 sin 70°

155 KN

20°

Ans : Y = -155 cos 20° , X = +155 sin 20°


4

60 N

50°

Ans : X = +60 cos 50° , Y = +60 sin 50°22. A weight of 1000 N is supported by two chains as shown in fig. Determine the tension in each chain.

A B

30° 60º

C E

Ans : T1 = 500 N , T2 = 866 N23.A weight of 900 N is supported by two chain of lengths 4m and 3m .Determine the tension in the string

A 5m B

4m 3m C

Ans : T1 = 539.959 N , T2 = 720 N

24.An electric light fixture weighing 15 N hangs from a point C ,by two strings AC ,BC .AC inclined at an angle 60° to the horizontal and BC at 45° to the vertical as shown in fig. Using lamis theorem determine the forces in the string AC and BC O A Ans : T1 = 539.959 N , T2 = 720 N 60°

B 45°

C E


5

1000 N

900 N

15 N

25. Determine the resultant in magnitude and direction.

24 º

3 º 10 º

9 º

228N Ans R=253.619 KN, θ = 11.4 º

26. The guy wire of an electric pole shown in fig makes an angle 30 º to the pole and

is subjected to 20KN force . Find the horizontal and vertical component of force. Ans Y=-17.321 KN, X= -10KN

27. Determine the resultant in magnitude and direction.

200N 120N 4 1

3 2

60°

40° Ans R=160.159 KN, θ = 24.6 º

50 KN 100 KN28. Determine the resultant of given system of forces. 105 KN 15 KN

75 KN 15º X 40º 35º 45 KN 60 KN

Ans : R = 52.421 KN , θ = 39.66ºPrepared by Manju P M ,Civil Department SNGCE

30º20KN

6

252N

156N

104N

29. If five forces act on a particle as shown in fig and the algebric sum of the hrizonal component of all the forces is – 324.904 KN , calculate the magnitude of P and the resultant of all the forces. . .

P Y 282.843 KN 15 KN 165 KN 3 40º 4 X O 75 KN Ans : R = 339.946 KN , θ = 17.11º

30. Three forces act on a particle as shown in fig.Determine the value of F such that the resultant of the three forces are horizontal. Find the magnitude and direction of fourth force which when acting along with the three forces will keep O in equilibrium.

30 KN Y F 18 KN 40º 10º 30º X O Ans : R = 53.67 KN, Equilibrant = 53.67 KN31. Five forces 4 , √ 3 , 5 , √ 3 and 3 KN resp act at one of the angular poinr of a regular hexagon

towards other five angular points . Find the magnitude and direction of resultant force. Ans : R =11.52 KN , θ = 55.7°

32. A wheel has five equally spaced radial spokes and all are in tension. If the tension in three consecutive spokes are 1000 KN ,1400 KN and 1200 KN resp. Find the tension in other two spokes considering the wheel in equilibrium.

Ans : F1 = 1123.7 KN , F2 = 1447.2 KN 33. Find the moment of given forces about O 50 KN

2m O 3m 6m 75 N O 20 KN

3m 65° 3m 6m O 20° O


7

15KN 5m 1m 40 N 3°

3m

OAns 1) Mo = +100 KN m , 2) Mo = -300 KNm ,3) Mo = -422.861 KNm ,

4) Mo =+ 29.02 KNm , 5) Mo = -253.42 KNm34.Determine the moment of the coplanar force system. 6 KN 30° 5 KN

20 KN

45° 10 KN 6 KNAns :- 0.35 KNm

35. Determine the moment of a couple wrt E and F 60 N E A 200mm 200mm B

100 cm F

100 cm D C

60 NMe = -24 KNm Mf = -24 KNm

36.. A system of load acting on a beam is shown in fig. Determine the resultant of load. . 20 KN 30KN 20KN

1.5m 1.5m 3.0m 60º 2m

A C D E BAns R=68.0592 KN, θ = 6.73 º , d = 3.290m X intercept =3.326m


8

37.Determine the resultant and position of resultant from A.

100N 150N 25N 200N

0.9m 1.2m 0.75m

A B C DAns R=125 N, x = 3.06m

38. A system of parallel forces are acting on a rigid bar as shown in fig. Reduce the system to 1) Single force and a couple at A 2) Single force 3) Single force and a couple at B.

32.5 N 150 N 67.5 N 10 N

1.0m 1.0m 1.5m A C D B

60 NAns 2) 0.833m A E 60 N

1)

Ma = 49.48 Nm (clockwise)

60 N 3)

Mb = 160 Nm (anticlockwise)

39. Fig shows two vertical forces and a couple of moment 2000Nm acting on a horizontal rod which is fixed at end A.1) Determine the resultant.2) Determine the equivalent system through A


9

4000N 2000Nm 2500N 1m 0.8m

A C D B

1.5mAns R=1500 N , Ma = 250 Nm40. Replace the given system of force to 1) an equivalent system and a couple at A 2) an equivalent

force couple system at B 3) a single force.500KN 200KN 300 KN 100KN

2m 2m 2m A C D BAns 3) R=500 N ( ) x = 0.4m ,1) R = 500 N N ( ) M a = -200Nm2) R = 500 N ( ) , M b = +3200 Nm41.Find the magnitude direction and position of the resultant force.

1000N 1500N 1000N 500N

4m 60° 4m 45° 4m 30°

Ans R=3760N, θ = 61.48 º X =4.25m42. Fig shows the coplanar system of force acting on a flat plate. Determine the resultant and X and Y

intercept of the resultant. 1500 N 1805 N

60 º 33.67º

D 2M 2 E

3m

O 4m A 63.43 º

2240 N Ans R=2114.4N, θ = 53.7 º X intercept =0.97m right of O, Y intercept = 1.32 m below O


1

2m

3m

43. Replace the force couple system a O by a single force applied at point located on line 1) AB 2) CD. In each case find the distance from centre O and the point of application of force.

C 30 KN 250 KN

A B Φ = 0.6m

30KN D Ans : R = 250 N ,M o = 18 KN- m , X intercept = 0.112 m ,Y intercept = 0.094 m

250 KN

18 18 KN –m

250 KN

OP = 112mm

250 KN

OQ = 94mm

1944. A bracket is subjected to a force as shown in fig Determine 1) An equivalent force couple system at A

2) An equivalent force couple system at B.


1

O

O

P

O

Q Q

B 0.3m

0.5m C

40º

0.8m

36KN D 1m

Ans : R = 36KN ,θ = 40º,M a = -16.008 KN- m, M b = -9.066 KN-m ,, X intercept = 0.112 m ,Y intercept = 0.094 m

B 16.008 KN -m

C

36KN D B 9.066 KN-m

C

45. Determine the resultant of three forces acting on the dam shown in fig and locate its intersection with base AB. For a good design this intersection should occur within the middle third of base. Comment whether it is a good design or not.


1

A

O

A

O

40º

A

40º

O

246 KN 62 KN

103.5 KN 1.8m 2.2

30º 2.1m 5.7 m.

46 Draw the FBD

A Ans : F

C B Rc

W W (String can pull along its length)

47 A circular roller of weight 100N and radius 10 cm hangs by a tie rod AB = 20cm and rest against a smooth vertical wall at C as shown in fig. Determine the 1) Force in the tie rod 2) the reaction at point C

A 20cm Ans : Rc = 57.73 N , F = 115.47 N

C

OC = 10cm48 A ball of weight 120 N rest in a right angled groove as shown in fig. The sides of the groove are

inclined at an angle of 30º and 60º to the horizontal. If all he surfaces are smooth then determine the reaction at A and C.

Ans : Rc = 103.92 N , Ra = 60 N


1

o

200N

A C 10cm C 100N 30º 60º θ D A 49 A circular roller of radius 5 cm and of weight 100N rest on a smooth horizontal surface and is held in

position by an inclined bar AB of length 10 cm as shown in fig. A horizontal force of 200 N IS acting at B .Find the tension in the bar AB and vertical reaction at C(fig shown above)

Ans F = 230.94 N Rc = 215. 47 N

50 A rope 9m long is connected at A and B two points on the same level 8m apart. A load of 300N is suspended from a point C on the rope , 3m from A . What load connected from point D on the rope is necessary to keep portion CD parallel to AB.

8m A B

3m 2m C 4m D 300 N W Ans : T ac =619.58 N , Tcd = 542.11 N , Tdb = 788.56 N W =572.66 N.

51 A fine light string ABCDE whose extremities A is fixed has weight W1 and W2 attached to it at end B and C . It passes around a smooth peg at D carrying a weight 40 Kg –f at the free end E as shown in fig. If in the position of equilibrium , BC is horizontal and AB and CD makes 150º and 120° with BC . Find the tension in the portion AB ,BC, CD of the string and magnitude of W1 and W2.

D

A

E 150º 120° G - 40kgf B C

W1 W2Ans : Tcd = T de = 40 kgf , Tbc = 20 kgf , W2 = 34.64kgf , Tab = 23.1 kgf ,W1 = 11.55kgf.


1

B

52 Two cylinder A and B rest in a box as shown in fig. A has a dia of 300mm and weighs 1200N. B has a dia of 200mm and weighs 360 N . The box is 450 mm wide at bottom. Assume all surface to be smooth. Find the reaction at supporting surface.

3000N

1500 N 1500N

D G E

F D E 450mm53 Two smooth cylinder each 300 mm in dia and each weighing 1500 N rest on a smooth horizontal

surface. They are connected at their centre by a rope PQ = 500mm long. They support a third cylinder of same dia weighing 3000 N as shown in fig above. Find the tension T in the rope AB and reaction at ground surface.

Ans Vd =Ve = 300 N , T= 2261.334N , R = 2713.60 N54 Two identical roller each of weight 500N are supported by an inclined plane and a vertical wall as

shown in fig. Assuming smooth surface . Find the reactions induced in the support A ,B and C . Ans : Rc = 577.4N , Rb =721.68 N , Ra =433 N

D 7KN

45° C A 5KN B 30° 30°

55 A roller of weight 10 KN rest on a smoth horizontal floor and is connected to the bar AC as shown in fig above. Determine the force in the bar AC and the reaction from the floor , if the roller is subjected to a horizontal force of 7 KN Ans : T ac = 0.058 KN , R = 4.978 KN

56 A roller of radius r = 400mm and weight 3000 N is to pulled over a curb of height 200mm by a horizontal force P applied to the end of a string wound round the circumference of the roller. Find the magnitude of P required to start the roller move over the curb. If in the problem P is applied horizontally at the centre of the roller. What would be the magnitude of this force. Also determine the least force and its line of action at the roller center for turning the roller over the rectangular block.


1

AB

P B

B C l 200mm

A aAns P =1732.0 N,P least = 2598N, Ra = 0 Rb = 3464.2 N P at centre = 5196 N

57 A uniform rod of length l and weight W remains in equilibrium in a vertical plane lodged between a smooth vertical wall and a peg C as shown in fig. Ignoring friction find inclination θ between wall and the rod.

Ans : θ = sin -1 3 √2a/l

58. In problem no:56 if the wheel is 600mm dia , weighing 5 KN rest against a rigid rectangular block of 150mm height.Find the least pull through the centre of wheel required just to turn the wheel over the corner A of the block.Also find the reaction of the block..

Ans : P = 4.33 KN, R = 2.5 KN

59. A 4m long joist weighing 100N is raised by a man by pulling the rope connected to it as shown in fig. Find tension T in the rope and reaction at A. B

Ans : Ra = 136.6 N θ =60° C B

30° T

45º A A D

60 .The above fig shows a cylinder of radius 0.2m supporting a rod AB 0.60m long weighing 100 N .The end A of the rod is hinged to the horizontal surface AD. The cylinder is also attached to the hinge by a string of length 0.20 m . Find the tension in the string.

Ans : T = 25√ 3 N , Rc = 25√ 3 N


1

61. A uniform rod AB is in equilibrium with one end resting against a smooth vertical wall , while other end is attached to a string AC as shown in fig. If AB = 2l and BC=l. Find the length of the string.

Ans AC = l√7 B

C

B T D 4m

60 ° 30° A A 2m62. A man weighing 750N stands at the middle of uniform ladder weighing 250N as shown in fig.

Assuming that the floor and the wall are absolutely smooth and slipping is prevented by the connecting string DE .Determine the tension in the string and reaction at the floor and wall

Ans: T = 405.827 N, Ha = 351.457 N Va = 1202.914 N

63 .An L shaped body ABC is hinged at A with a force F acting at end C . Determine the angle θ which the force should make with the horizontal to keep the edge AB of the body vertical.

25cm F θ B C A

15º

40cm

B 45º

A Ans θ = 57.99º

64. A roller of weight 500N rest on a smooth inclined plane and is kept free from rolling down by a string as shown in fig. Find the tension in the string and rection at B.

Ans :T = 732.06 N , Rb = 896.592 N

65 .Determine the axial force S1 and S2 induced in the bar AC and BC due to the action of horizontal load applied at C and to the foundation at A and B.

Ans : S1 = 780.87 N , S2 = - 640.3 N


1

A

C

C 1000 N

2.5m

A B

3m 2m66. A ball of weight W rest on a smooth horizontal plane and is attached to its centre by a string AB and

AC which passes over a smooth pulley at B and C and carry loads P and Q resp as shown in fig. If the string AB is horizontal. Find the angle αthat the string AC makes with the horizontal when ball is in equilibrium. Find also force excerted by the ball on the plane.

Ans : R = W - √(Q2 – P2) C

W

67. A slender prismatic bar AB of weight Q and length 2l rest on a very small frictionless roller at D and against a smooth vertical wall at A( refer fig of problem 57). Find the angle α that the bar must make with horizontal in the condition of equilibrium.

Ans cos α = 3√ a/l

68. A spherical ball of weight W rest in a v- shaped trough whose sides are inclined at angle α ,β to the horizontal. Find the reaction on each side of trough, if the second ball of equal weight and size are placed on the sides of inclination α , so as to rest above the first ball. Find the reaction of lower ball on the sides of inclination β.


1

B

A α

P

Q

Ans :2W sin α / (sin( α +β))

β α

69. A 600 N cylinder is supported by he frame BCD as shown in fig. The frame is hinged at D. Determine the reaction ot A ,B , C and D. Ans : Rb = 600 N, Ra = 200 N , Rd = 632.456 N ,Rc = 200 N

300 mm D

P

60º

600 mm 15°

C A 60º 45º

B70 Two cylinder of weight A of weight 4000N and B of weight 2000 N rest on a smooth incline as

shown in fig. They are connected by a bar of negligible weight hinged to each other at its geometrical centre by a smooth pins. Find the force P to be applied to the ball B such that it will hold the system in given position. Ans : C = 4898.98 N ,P = 1071.8169 N

71 Find the resultant of the force system shown in fig acting on a lamina of equilateral triangle. 80 N B 80 N

100 N Ans : 91.9 N , α = 35.86º G E

60º A 100 mm 30º C F 60º

120 N H72 . Two cylinder P and Q resting in a channel as shown in fig. The cylinder P has a dia of 100mm and

weighs 200 N whereas the cylinder Q has dia of 180mm dia and weighs 500 N. Find the reactions at E , F ,H. Ans :

73. A bar AB 15m long is hinged to a vertical wall at A making an angle 60º to the vertical and carries load 200 KN hanging vertically from B. The bar is supported by a horizontal rope CD where the point D is at 10m from point A. Find the tension in the rope and reaction of hinge at A

Ans :Ra = 558.7 N , T = 521 N


W

1

150 mm

A

B

B F B C D

60º A D θ A

74. A cylinder of radius R and weight W is supported horizontally against a wall by a bar AB, hinged a A as shown in fig.Find the angle θ that the bar must make with the wall sothat the force F applied horizontally at the free end of the bar is minimum. Neglect friction and weight of bar AB.

Ans :θ = 60°


2

200 KN

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104 Mechanics, Module 1