**(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 *W _{1} *and

*W*at a distance L

_{2}_{1}and L

_{2}from the end A respectively. How will you find the reactions R

_{A}*and R*by analytical method.

_{B}**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 F_{1}, F_{2} and F_{3} are acting on a body as shown in Fig. 2.80 and the body is in equilibrium. If the magnitude of force *F*_{3} is 250 N, find the magnitudes of force *F _{1}* and

*F*

_{2}[**Ans**. F_{1} = 125 N and *F _{2} *= 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 = 76.8^{o}]

**3. **Three parallel forces F_{1}, F_{2} and F_{3} are acting on a body as shown in Fig. 2.82 and the body is in equilibrium. If force F_{1} = 300 N and *F _{3} *= 1000 N and the distance between

*F*and

_{1}*F*= 2.0 m, then determine the magnitude of force F

_{2}_{2}and distance of F

_{3}from force

*F*

_{2}.[**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 *R _{C} *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 30^{0} and 60^{0} to the horizontal as shown in Fig. 2.85. Determine the reaction *R _{A} *and

*R*at the points of contact.

_{C}[**Ans**. *R _{A} *= 500 N,

*R*= 866.6 NJ

_{C}**10. **If in the above problem, the sides of the groove makes an angle of 45^{0} with the horizontal, then find the reactions *R _{A} *and

*R*

_{C}[**Ans**. R_{A} =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**. *R _{A} *= 43.3 N,

*R*= 72 N,

_{B}*R*= 57.5 N]

_{C}**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**. *R _{A} *=

*R*= 66.67 N,

_{C}*R*= 100 N]

_{B}**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**. R_{A} = 7500 N, R_{B} = 10500 N]

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

[**Ans**. *R _{A} *= 1 kN,

*R*= 6 kN]

_{B}**19. **An overhanging beam carries the loads as shown in Fig. 2.89. Calculate the reactions at both ends.

[**Ans**. R_{A} = 10 kN, R_{B} = 11 kN]

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

[**Ans**. R_{AV} =2.875 kN, R_{AH} = 5.196 kN → R_{B} = -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.** R_{AV} = 5.87 kN, R_{AH} = 3.222 kN → R_{B} = 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**. *R _{A}* = 0,

*R*kN]

_{B}= 8**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**. R_{A} = 106.5 N

R_{B} = 333.5 N

F_{1} = 333 N (Comp.)

F_{2} = 288.5N (Tens.)

F_{3} = 577.5 N (Tens.)

F_{4} = 667 N (Comp.)

F_{5} = 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**. F_{1} = 833.34 N (Comp.)

F_{2}== 1000 N (Tens.)

F_{3} = 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.)

AD = 2309.4 N (Comp.)

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.)

*AD =*10 kN (Comp.)

*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**.F_{1} = 333 N (Comp.)]

F_{2} = 577.5 N (Tens.)

F_{3} = 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.

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