# Exercise-2

(A)       Theoretical Questions

1.           Define and explain the terms. Principle of equilibrium, force law of equilibrium and moment law of equilibrium.

2.           A number of forces are acting on a body. What are conditions of equilibrium, so that the body is in equilibrium?

3.           Two forces are acting on a body and the body is in equilibrium. What conditions should be fulfilled by these two forces?

4.           How will you prove that a body will not be in equilibrium when the body is subjected to forces which are equal and opposite but are parallel?

5.           Explain the statement “Two equal and opposite parallel forces produces a couple”.

6.           (a) What conditions must be fulfilled by a set of three parallel forces which are acting on a body and body is in equilibrium?

(b) State the graphical conditions that must be satisfied for the equilibrium of a system of coplanar forces.

(c) What are the conditions that must be satisfied in order that a body may have plane motion?

(d) Discuss the various laws governing the equilibrium of coplanar forces.

7.           Three concurrent forces are acting on a body which is in equilibrium, then the resultant of the two forces should be equal and opposite to the third force. Prove this statement.

8.           State and explain the Lami’s theorem.

9.           What do you mean by action and reaction? Give examples.

10.        Explain and define the term : ‘Free-body Diagram’. Draw the free-body diagram of a ball of weight W, placed on a horizontal surface.

11.        State the conditions of equilibrium of a system of forces acting on a body as applicable to

(i)                graphical method, and

(ii)             analytical method.

12.        Explain the term ‘support reactions.’ What are the different types of support?

13.        What is the difference between a roller support and a hinged support?

14.        What are the important types of loading on a beam? Differentiate between uniformly distributed load and uniformly varying load on a beam.

15.        Name the different methods of finding the reactions at the two supports of a beam.

16.        A beam AB of length L is simply supported at the ends A and B. It carries two point loads W1 and W2 at a distance L1 and L2 from the end A respectively. How will you find the reactions RA and RB by analytical method.

17.        Describe in details the different steps involved in finding the reactions of a beam by graphical method.

18.        Define and explain an overhanging beam.

19.        What is the main advantage of roller support in case of the steel trusses of the bridges?

20.        Define and explain the terms: Perfect frame, imperfect frame, deficient frame and a redundant frame.

21.        (a) What is a frame? State the difference between a perfect frame and an imperfect frame.

(b) What are the assumptions made in finding out the forces in a frame?

22.        What are the different methods of analysing (or finding out the forces) a perfect frame? Which one is used where and why?

23.        How will you find the forces in the members of a truss by method of joints when

(i)                the truss is supported on rollers at one end and hinged at other end and carries vertical loads.

(ii)             the truss is acting as a cantilever a.id carries vertical loads.

(iii)           the truss is supported on rollers at one end and hinged at other end and carries horizontal and vertical loads.

(iv)            the truss is supported on rollers at one end and hinged at other end and carries inclined loads.

24.        (a) What is the advantage of method of section over method of joints? How will you use method of section in finding forces in the members of a truss?

(b) Explain with simple sketches the terms (i) method of sections and (ii) method of joints, as applied to trusses.

25.        How will you find the forces in the members of a joint by graphical method? What are the advantages or disadvantages of graphical method over method of joints and method of section?

26.        What is the procedure of drawing a vector diagram for a frame? How will you find out

(i)                magnitude of a force, and

(ii)             nature of a force from the vector diagram?

27.        How will you find the reactions of a cantilever by graphical method?

28.        What are the assumptions made in the analysis of a simple truss.

(B) Numerical Problems

1.           Three forces F1, F2 and F3 are acting on a body as shown in Fig. 2.80 and the body is in equilibrium. If the magnitude of force F3 is 250 N, find the magnitudes of force F1 and F2

[Ans. F1 = 125 N and F2 = 215.6 N]

2.           Three forces of magnitudes P, 100 N and 200 N are acting at a point O as shown in Fig. 2.81. Determine the magnitude and direction of the force P.

[Ans. P = 147 N and θ = 76.8o]

3.           Three parallel forces F1, F2 and F3 are acting on a body as shown in Fig. 2.82 and the body is in equilibrium. If force F1 = 300 N and F3 = 1000 N and the distance between F1 and F2 = 2.0 m, then determine the magnitude of force F2 and distance of F3 from force F2.

[Ans. 1300 N, 0.6 m]

4.           Three forces of magnitude 40 kN, 15 kN and 20 kN are acting at a point O. The angles made by 40 kN, 15 kN and 20 kN forces with x-axis are 60°, 120° and 240° respectively. Determine the magnitude and direction of the resultant force.

[Ans. 30.41 kN and 85.28° with x-axis]

5.           A lamp weighing 10 N is suspended from the ceiling by a chain. It is pulled aside by a horizontal cord until the chain makes an angle of 60° with the ceiling. Find the tensions in the chain and the cord by applying Lami’s theorem and also by graphical method.

[Ans. 11.54 N and 5.77 N]

6.           Draw the free-body diagram of a ball of weight W supported by a string AB and resting on a smooth horizontal surface at C when a horizontal force is applied to the ball as shown in Fig. 2.83.

7.           A circular roller of weight 1000 N and radius 20 cm hangs by a tie rod AB = 40 cm and rests against a smooth vertical wall at C as shown In Fig. 2.84.·Determine the tension in the tie rod and reaction Re at point C.

[Ans. 1154.7 N and 577.3 N]

8.           In problem 6 if radius of ball = 5 cm, length of string AB = 10 cm, weight of ball W = 40 N and the horizontal force F = 30 N, then find the tension the string and vertical reaction RC at point C.

[Ans. 34.64 N and 57.32 N]

9.           A smooth circular cylinder of weight 1000 N and radius 10 cm rests in a right-angled groove whose sides are inclined at an angle of 300 and 600 to the horizontal as shown in Fig. 2.85. Determine the reaction RA and RC at the points of contact.

[Ans. RA = 500 N, RC = 866.6 NJ

10.           If in the above problem, the sides of the groove makes an angle of 450 with the horizontal, then find the reactions RA and RC

[Ans. RA =Rc = 707 N]

11.           Two identical rollers, each of weight 50 N, are supported by an inclined plane and a vertical wall as shown in Fig. 2.86. Find the reactions at the points of supports A, Band C. Assume all the surfaces to be smooth.

[Ans. RA = 43.3 N, RB = 72 N, RC = 57.5 N]

12.           Two spheres, each of weight 50 N and of radius 10 cm rest in a horizontal channel of width 36 cm as shown in Fig. 2.87. Find the reactions on the points of contact A, Band C.

[Ans. RA = RC = 66.67 N, RB = 100 N]

13.           A simply supported beam of length 8 m carries point loads of 4 kN and 6 kN at a distance of 2 m and 4 m from the left end. Find the reactions at both ends analytically and graphically.

[Ans. 6 kN, 4 kN]

14.           A simply supported beam of length 8 m carries a uniformly distributed load of 10 kN/m for a distance of 4 m, starting from a point which is at a distance of 1 m from the left end. Calculate the reactions at both ends analytically and graphically.

[Ans. 25 kN, 15 kN]

15.           A beam 6 m long is simply supported at the ends and carries a uniformly distributed load of 1.5 kN/m and three concentrated loads 1 kN, 2 kN and 3 kN acting respectively at a distance of 1.5 m, 3 m and 4.5 m from the left end. Calculate the reactions at both ends.

[Ans.7 kN, 8 kN]

16.           A simply supported beam of span 10 m carries a uniformly varying load from zero at the left end to 1200 N/m at the right end: Calculate the reactions at both ends of the beam.

[Ans. 2000 N and 4000 N]

17.           A simply supported beam AB is subjected to a distributed load increasing from 1500 N/m to 4500 N/m from end A to end B respectively. The span AB = 6 m. Determine the reactions at the supports.

[Ans. RA = 7500 N, RB = 10500 N]

18.           An overhanging beam carries the loads as shown in Fig. 2.88. Calculate the reactions at both ends

[Ans. RA = 1 kN, RB = 6 kN]

19.           An overhanging beam carries the loads as shown in Fig. 2.89. Calculate the reactions at both ends.

[Ans. RA = 10 kN, RB = 11 kN]

20.           A beam is loaded as shown in Fig. 2.90. Determine the reactions at both ends.

[Ans. RAV =2.875 kN, RAH = 5.196 kN → RB = -7.125 kN]

21.           A beam of span 6 m is hinged at A and supported on rollers at end B and carries load as shown in Fig. 2.91. Determine the reactions at A and B.

[Ans. RAV = 5.87 kN, RAH = 3.222 kN → RB = 7.3 kN]

22.           A beam AB of span 8 m is subjected to the uniformly distributed load of 1 kN/m over the entire length and the moment 32 kN/m at C as shown in Fig. 2.92. Determine the reactions at the both ends.

[Ans. RA = 0, RB = 8 kN]

23.           A simply supported beam AB is subjected to a distributed load increasing from 1500 N/m to 4500 N/m from end A to end B. The span AB = 6 m. Determine the reactions at the supports.

24.  Find the forces in the members AB, AC and BC of the truss shown in Fig. 2.94.

[Ans. AB = 4.33 kN (Comp.)

AC= 2.5 kN (Comp.)

BC= 2.165 kN (Comp.)]

25.    A truss of span 7.5 m carries, a point load of 500 N at joint D as shown in Fig. 2.  Find the reactions and forces in the members of the truss.

[Ans. RA = 106.5 N

RB = 333.5 N

F1 = 333 N (Comp.)

F2 = 288.5N (Tens.)

F3 = 577.5 N (Tens.)

F4 = 667 N (Comp.)

F5 = 577.5 N (Tens.)

26.           A truss of span 7.5 m is loaded as shown in Fig. 2.96. Find the reactions and forces in the members of the truss.

[Ans. AD = 3.464 kN (Comp.)

AC = 1.732 kN (Tens.)

CD = 2.598 kN (Tens.)

CE = 2.598 kN (Comp.)

DE = 3.50 kN (Comp.)

BE = 5 kN (Comp.)

BC = 4.33 kN (Tens.)

27.           Determine the forces in the various members of the truss shown in Fig. 2.97.

[Ans. AB = 1200 N (Comp.)

BC = 800 N (Comp.)

CD = 800 N (Comp.)

DE = 1200 N (Comp.)

EF = 600 N (Tens.)

AF = 600 N (Tens.)

BF = DF = 400 N (Comp.)

FC = 400 N (Tens.)

28.           A plane truss is loaded and Supported as shown in Fig. 2.98. Determine the nature and magnitude of forces in the members 1, 2 and 3.

[Ans. F1 = 833.34 N (Comp.)
F2== 1000 N (Tens.)

F3 = 666.66 (Tens.)]

29.           Determine the forces in all the members of a cantilever truss shown in Fig. 2.99.

[Ans. AC = 1154.7 N (Comp.)

CD = 2309.4 N (Tens.)
BD = 2309.4 N (Tens.)]

30.           A cantilever truss is loaded as shown in Fig. 2.100. Find the force in member AB.

[Ans. AB =15 kN (Tens.)]

31.           Determine the forces in the truss shown in Fig. 2.101 which carries a horizontal load of 16 kN and a vertical load of 24 kN.

[Ans. AC = 24 kN (Tens.)

CD = 24 kN (Tens.)
CB = 24 kN (Tens.)
BD = 30 kN (Comp.)]

32.           Find the forces in the member AB and AC of the truss shown in Fig. 2.94 of question 24, using method of sections.

[Ans. AB = 4.33 kN (Comp.)

AC = 2.5 kN (Comp.)]

33.           Find the forces in the members marked 1, 3, 5 of truss shown in Fig. 2.95 of question 25, using method of sections.

[Ans.F1 = 333 N (Comp.)]

F2 = 577.5 N (Tens.)
F3 = 577.5 N (Tens.)]

34.           Find the forces in the members DE, CE and CB of the truss, shown in Fig. 2.96 of question 26, method of sections.

[Ans. DE = 3.5 kN (Comp.)

CE = 2.598 kN (Comp.)
BC = 4.33 kN (Tens.)]

35.           Using method of section, determine the forces in the members CD, FD and FE of the truss shown in Fig. 2.97 of question 27.

[Ans. CD = 800 N (Comp.)

FD = 400 N (Comp.)
FE = 600 N (Tens.)]

36.           Using method of section, determine the forces in the members CD, ED and EF of the of truss shown in Fig. 2.102

[Ans. CD = 4.216 kN (Comp.)

ED = 3.155 kN (Tens.)

EF = 2.58 (Tens.)]

37.           Find the forces in the members AB, AC and BC of the truss shown in Fig. 2.94 of question 24, using graphical method.

38.           Using graphical method, determine the magnitude and nature of the forces in the members of the truss shown in Fig. 2.95 of question 25.

39.           Determine the forces in all the members of a cantilever truss shown in Fig. 2.99 of question 29, using graphical method. Also determine the sections of the cantilever.