104 Mechanics, Module 1

1 MODULE 1 1. 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 N 2. 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 N 3. 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 P 4. 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 N 5. 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

15 70º 20 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 angle is θ and it becomes(2k-1)

F12+ F22

when the included

when the included angle is( 90 – θ).

Show that the included angle between the forces is given by tan θ

= k-1 k+1

11 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

2 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 β

α γ

200 N

Ans : β = 33° , γ = 26.966º 14.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 β

α γ

Q

Ans : P =264.93 N, Q = 195. 19 N 15 .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 β

α γ

200 N

Ans :α = 49.27º , β = 33° , γ = 26.966º 16. 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

3 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

Prepared by Manju P M ,Civil Department SNGCE

4 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

200mm

21. 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°

Prepared by Manju P M ,Civil Department SNGCE

5

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 1000 N

Ans : T1 = 500 N , T2 = 866 N 23.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 900 N

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

720 N 60° B 45° C E 15 N

Ans : T1 = 539.959 N , T2 =

6 Prepared by Manju P M ,Civil Department SNGCE 25. Determine the resultant in magnitude and direction

.

156N 24 º

104N

252N 3º

10

º

9º

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

30º

20KN

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 KN

28. Determine the resultant of given system of forces.

105 KN 15 KN 75 KN

15º 40º 35º

45 KN Ans : R = 52.421 KN , θ = 39.66º

60 KN

X

7 Prepared by Manju P M ,Civil Department SNGCE 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 KN 31. 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 3m

6m O

20°

O

65°

8 Prepared by Manju P M ,Civil Department SNGCE 15KN

5m 1m 40 N 3° 3m O

Ans 1) Mo = +100 KN m , 2) Mo = -300 KNm ,3) Mo = -422.861 KNm , 4) Mo =+ 29.02 KNm , 5) Mo = -253.42 KNm 34.Determine the moment of the coplanar force system.

6 KN 30°

5 KN

20 KN 45° 10 KN

6 KN

Ans :- 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 N

Me = -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

1.5m A

30KN

1.5m C

3.0m D

.

20KN

60º E

Ans R=68.0592 KN, θ = 6.73 º , d = 3.290m X intercept =3.326m

2m B

9 Prepared by Manju P M ,Civil Department SNGCE 37.Determine the resultant and position of resultant from A.

100N

150N 0.9m

25N

200N

1.2m

A

0.75m

B

C

D

Ans 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

1.0m A

67.5 N

1.0m C

10 N

1.5m D

B

60 N Ans 2) 0.833m A 60 N

E

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

1 Prepared by Manju P M ,Civil Department SNGCE 4000N

2000Nm

1m

2500N

0.8m

A

C

D

B

1.5m Ans R=1500 N , Ma = 250 Nm 40. 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 2m

300 KN

100KN

2m

2m

A C D Ans 3) R=500 N ( ) x = 0.4m ,1) R = 500 N N ( 2) R = 500 N ( ) , M b = +3200 Nm

B ) M a = -200Nm

41.Find the magnitude direction and position of the resultant force.

1000N

1500N 4m

1000N 4m

60°

45°

500N 4m

30°

Ans R=3760N, θ = 61.48 º X =4.25m 42. 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

2m

3m

O

E

3m

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 Prepared by Manju P M ,Civil Department SNGCE 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

O

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

O P

OP = 112mm

O

250 KN

Q

OQ = 94mm

Q 19

44. 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 Prepared by Manju P M ,Civil Department SNGCE B A

0.3m 0.5m

C

O

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 A 40º

C O

36KN D B

9.066 KN-m

A 40º

O

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 Prepared by Manju P M ,Civil Department SNGCE

246 KN

62 KN

103.5 KN 1.8m

2.2 30º

2.1m 5.7 m . 46

Draw the FBD

A

Ans : C

B

W (String can pull along its length)

F

Rc

W

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

C

o

Ans : Rc = 57.73 N , F = 115.47 N

OC = 10cm 48 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 Prepared by Manju P M ,Civil Department SNGCE

B

200N

A C 30º

10cm 60º

C 100N 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 C

2m D

4m

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°

B

G - 40kgf C

W1

W2

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

1

Prepared by Manju P M ,Civil Department SNGCE 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 D

1500N

G

E

A F

B D

E

450mm 53 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 N 54 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

Prepared by Manju P M ,Civil Department SNGCE P

B

B

C l

200mm A a Ans 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 Prepared by Manju P M ,Civil Department SNGCE

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 ° A

30°

A 2m

62. 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 C

40cm B

45º A 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 Prepared by Manju P M ,Civil Department SNGCE C 1000 N

2.5m A

B 3m

2m

66. 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

A B

Q

α

W P

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 Prepared by Manju P M ,Civil Department SNGCE

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 A

60º

600 mm 15° 150 mm

C

B

A 60º

45º

B 70 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

100 N

80 N

Ans : 91.9 N , α = 35.86º

G

E

60º A

100 mm

30º C

F 60º

120 N H 72 . 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

2 Ans :Ra = 558.7 N , T = 521 N

Prepared by Manju P M ,Civil Department SNGCE W B C

F

B

D 60º

A

200 KN

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 Prepared by Manju P M ,Civil Department SNGCE