(i) Introduction

- Engineering mechanics is divided into statics and dynamics. The study of a body at rest is known as statics whereas the study of a body in motion is known as dynamics.
- A quantity which is completely specified by magnitude and direction is known as vector quantity.
- A particle is a body of infinitely small volume and is considered to be concentrated at a point.
- Law of parallelogram of forces states that “If two forces, acting at a point be represented in magnitude and direction by the two adjacent sides of a parallelogram, then their resultant is represented in magnitude and direction by the diagonal of the parallelogram passing through that point.

5. If two forces *P *and *Q *act at a point and the angle between the two forces be ex, then the resultant is given by

R=√(P^2+Q^2+2PQ cosα )

and the angle made by the resultant with the direction of force *P *is expressed as

tanθ=(Q sinα)/(P+Q cosα )

6. If the two *forces P *and Q are equal and are acting at an angle a between them, then the resultant is given by

R=2P cos〖 α/2〗

and angle made by the resultant is expressed as θ=α/2

7. According to Lame’s theorem, “If three forces acting at a point are equilibrium, each force will be proportional to the sine of the angle between the other two forces.”

8. The relation between newton and dyne is given by One newton = 10^{5} dyne.

9. Moment of a force about a point = Force × perpendicular distance of the line of action of the force from that point.

10. The force causes linear displacement while moment causes angular displacement. A body will be in equilibrium if *(i) *resultant force in any direction is zero and *(ii) *the net moment of the forces about any point is zero.

11. Gravitational law of attraction is given by,

F=G (m_1×m_2)/r^2

where G = Universal gravitational constant

m1, m2 = mass of bodies

r = Distance between the bodies

F = Force of attraction between the bodies.

**(ii) ****Coplanar Concurrent Forces**

12. Coplanar forces means the forces are acting in one plane.

13. Concurrent forces means the forces are intersecting at a common point.

14. Collinear forces means the forces are having same line of action.

15. The resultant of coplanar forces are determined by analytical and graphical methods.

16. The resultant *(R) *of three collinear forces F_{1}, F_{2} and F_{3} acting in the same direction, is given by R = F_{1} + F_{2} + F_{3} If the force F_{2} is acting in opposite direction then their resultant will be, R = F_{1} + F_{2} + F_{3}

17. The resultant of three or more forces acting at a point is given by R =√((ΣH)^2+(ΣV)^2 ) , where ƩH = Algebraic sum of horizontal components of all forces, ƩV = Algebraic sum of vertical components of all forces, The angle made by the resultant with horizontal is given by, tan θ = ((ΣV))/((ΣH) )

18. The resultant of several forces acting at a point is found graphically by using polygon law of forces.

19. Polygon law of forces states that if a number of coplanar forces are acting at a point such that they can be represented in magnitude and direction by the sides of a polygon taken in the same order, then their resultant is represented in magnitude and direction by the closing side of the polygon taken in the opposite order.

**(iii) Coplanar Parallel Forces**

20. Parallel forces are having their lin.es of action parallel to each other.

21. The moment of a force about any point is the product of force and perpendicular distance between the point and line of action of force.

22. Anti-clockwise moment is taken +ve whereas clockwise moment is taken – ve.

23. Varignon’s principle states that the moment of a force about any point is equal to the algebraic sum of moments of its components about that point.

24. Like parallel forces are parallel to each other and are acting in the same direction, whereas the unlike parallel forces are acting in opposite direction.

25. The resultant of two like parallel forces is the sum of the two forces and acts at a point between the line in such a way that the resultant divides the distance in the ratio inversely proportional to the magnitudes of the forces.

26. When two equal and opposite parallel forces act on a body at some distance apart, the two forces form a couple which has a tendency to rotate the body. The moment of this couple is the product of either one of the forces and perpendicular distance between the forces.

27. A given force F applied to a body at any point *A *can always be replaced by an equal force applied at another point *B *in the same direction together with a couple.

28. If the resultant of a number of parallel forces is not zero, the system can be reduced to a single force, whose magnitude is equal to the algebraic sum of all forces. The point of application of this single force is obtained by equating the moment of this single force about any point to the algebraic sum of moments of all forces acting on the system about the same point.

29. If the resultant of a number of parallel forces is zero, then the system may have a resultant couple or may be in equilibrium. If the algebraic sum of moments of all forces about any point is not zero, then system will have a resultant couple. But if the algebraic sum of moments of all forces about any point is zero, the system will be in equilibrium.

**(A) Theoretical Questions**

**(i) ****Introduction **

- What do you mean by scalar and vector quantities?
- Define the law of parallelogram of forces. What is the use of this law?
- State triangle law of forces and Lame’s theorem.
- Two forces P and Q are acting at a point in a plane. The angle between the forces is ‘α’. Prove that the resultant
*(R)*of the two forces is given by - Define the following terms: dyne, newton, meganewton and moment of a force.
- Prove that one newton is equal to 10
^{5}dyne. - Explain the terms: clockwise moments and anti-clockwise moments,
- What is the effect of force and moment on a body?
- Indicate whether the following statement is true or false

The resultant components of the forces acting on a body along any direction is .zero but the net moment of the forces about any point is not zero, the body will be in equilibrium’.

[Ans. False]

10. Write the S.I units of: Force, moment and velocity.

11. What do you mean by resolution of a force? /

12. A number of coplanar forces are acting at a point making different angles with x-axis. Find an expression for the resultant force. Find also the angle made by the resultant force with z-axis.

13. State and explain the pinciple of transmissibility of forces.

14. State and explain the following laws:

(i) Newton’s laws of motion.

(ii) The gravitational law of attraction.

15. Using gravitational law of attraction, prove that W = m × g.

16. Explain fully the following terms:

(i) Resolved part of a given force in a given direction.

(ii) Lame’s theorem.

**(ii) Coplanar Concurrent Forces **

17.Define and explain the following terms;

*(i) *Coplanar and non-coplanar forces

*(ii) * Collinear and concurrent forces

*(iii) * Parallel and non-parallel forces.

18. What is the difference between collinear and concurrent forces?

19. State and explain the following laws of forces;

I. Law of parallelogram of forces

II. Law of triangle of forces

III. Law of polygon of forces.

20. Derive an expression for the resultant in magnitude and direction of two coplanar concurrent forces using cosine law method.

21. Explain in detail the method of finding resultant in magnitude and direction of three or more

forces acting at a point by analytical and graphical method.

22. Explain the procedure of resolving a given force into two components at right angles to each other.

23. Three collinear forces F_{1}, F_{2} and F_{3}* *are acting on a body. What will be the resultant of these forces, if

* (a) *all are acting in the same direction

* (b) *force F_{3}* *is acting in opposite direction.

24. State the law of parallelogram of forces and show that the resultant R = √(P^2+Q^2 ) when the two forces P and Q are acting at right angles to each other. Find the value of R if the angle between the forces is zero.

**(iii) Coplanar Parallel Forces **

25. Define the terms ; Coplanar parallel forces, like parallel forces and unlike parallel forces.

26. Define and explain the moment of a force. Differentiate between clockwise moment and anti- clockwise moment

27. (a) State the Varignon’s principle. Also give the proof of Varignon’s principle.

(b) Differentiate between;

(i) Concurrent and non-concurrent forces,

(ii) Coplanar and non-coplanar forces,

(iii) Moment of a force and couple.

28. Define moment of a force about a point and show that the algebraic sum of the moments of two coplanar forces about a point is equal to the moment of their resultant about that point.

29. What are the different types of parallel forces? Distinguish between like and unlike parallel forces.

30. Prove that the resultant of two like parallel forces F1 and F2 is F1 + F2. Also prove that the resultant divides the line of joining the points of action of Fl and F2 internally in the inverse ratio of the forces.

31. Prove that in case of two unlike parallel forces the resultant lies outside the line joining the points of action of the two forces and on the same side as the larger force.

32. Describe the method of finding the line of action of the resultant of a system of parallel forces.

33. The resultant of a system of parallel forces is zero, what does it signify?

34. Describe the method of finding the resultant of two unlike parallel forces which are equal in magnitude.

35. Indicate whether the following statements are True or False;

i. Force is an agency which tends to cause motion.

ii. The value of g reduces slightly as we move from poles towards the equator.

iii. Coplanar forces are those which have the same magnitude and direction.

iv. A couple consists of two unequal and parallel forces acting on a body, having the same line of action.

v. A vector diagram of a force represents its magnitude, direction, sense and point of application.

vi. The force of gravitation on a body is called its weight.

vii. The centre of gravity of a body is the point, through which the resultant of parallel forces passes in whatever position may the body be placed.

**[Ans.** (i) True (ii) True (iii) False (iv) False (v) False (vi) True (vii) True]

**(B) ****Numerical Problems**

i) **Introduction **

- Determine the magnitude of the resultant of the two forces of magnitude 12 N and 9 N acting at a point when the angle between the two forces is 30°.

[**Ans**. 20.3 N]

2. Find the magnitude of two equal forces acting at a point with an angle of 60° between them, if the resultant is equal to 30 × N.

[**Ans**. 30 N]

3. The resultant of two forces when they act at right angles is 10 N, whereas when they act at an angle of 60° the resultants is √148. Determine the magnitude of the two forces.

**[Ans.** 8 Nand 6 N]

4. Three forces of magnitude 30 kN, 10 kN and 15 kN are acting at a point O. The angles made by 30 kN force, 10 kN force and 15 kN force with x-axis are 60°, 120° and 240° respectively. Determine the magnitude and direction of the resultant force.

**[Ans.** 21.79 kN, 83° 24]

5. A weight of 800 N is supported by two chains as shown in Fig. 1.89. Determine the tension in each chain.

[**Ans**. 273.5 N, 751.7 N]

Fig 1.89

Fig. 1.90

6. An electric light fixture weighing 20 N hangs from a point C, by two strings AC and BC.AC is inclined at 60° to the horizontal and BC at 30° to the vertical as shown in Fig. 1.90. Using Lame’s theorem or otherwise determine the forces in the strings AC and BC. .

[Ans. 8.929 N, 13.05 N]

7. A beam AB of span 6 m carries a point load of 100 N at a distance 2 m from A. Determine the

beam reaction.

[Ans. RA = 66.67 N; RB = 33.33 N]

8. Four forces of magnitudes 20 N, 30 N, 40 N and 50 N are acting respectively along the four sides of a square taken in order. Determine the magnitude, direction and position of the resultant

Ans.20×2,〖225〗^o,7a/(2×√2)

9. Two forces magnitude 15 N and 12 N are acting at a point. If the angle between the two forces is 60°, determine the resultant of the forces in magnitude and direction.

[Ans. 23.43 N, 26.3°]

10. Four forces of magnitude P, 2P, 3 × √3 P and 4P are acting at a point O. The angles made by these forces with x-axis are 0o, 60o, 150° and 300o respectively. Find the magnitude and direction of the resultant force.

[Ans. P, 1200]

*(ii) ***Coplanar Concurrent Forces **

11. Three collinear horizontal forces of magnitude 300 N, 100 N and 250 N are acting on rigid body. Determine the resultant of the forces analytically and graphically when: (i) all the forces are acting in the same direction; (ii) the force 100 N acts in the opposite direction.

[Ans. (i) 650 N, (ii) 450 N]

12. Two forces of magnitude 15 N and 12 N are acting at a point. The angle between the forces is 60o Find the resultant in magnitude.

[Ans. 20.43 N]

13. A force of 1000 N is acting at a point, making an angle of 60o with the horizontal. Determine the components of this force along horizontal and vertical directions.

[Ans. 500 N, 866 N]

14. A small block of weight 100 N is placed on an inclined plane which makes an angle of 600 with the horizontal. Find the components of this weight (i) perpendicular to the inclined plane and (ii) parallel to the inclined plane.

[Ans. 50 N, 86.6 N]

15. Two forces *P *and *Q *are acting at a point O as shown in Fig. 1.91. The force *P *= 264.9 N and force *Q *= 195.2 N. If the resultant of the forces is equal to 400 N then find the values of angles β,γ,α

[**Ans**. P = 35°, Y = 25^{o}, a = 60°]

16. A small block of unknown weight is placed on an inclined plane which makes an angle of 30° with horizontal plane. The component of this weight parallel to the inclined plane

is 100 N. Find the weight of the block.

[Ans. 200 N]

17. In question 16, find the component of the weight perpendicular to the inclined plane.

[Ans. 173.2 N]

18. The four coplanar forces are acting at a point as shown in Fig. 1.92. Determine the resultant in magnitude and direction analytically and graphically.

[**Ans**. 1000 N, e == 60° with OX]

19. The four coplanar forces are acting at a point as shown in Fig. 1.93. One of the forces is unknown and its magnitude is shown by *P. *The resultant is having a magnitude 500 N and is acting along x-axis. Determine the unknown force *P *and its inclination with x-axis.

[**Ans**. *P *= 286.5 Nand 9 = 53° 15']

**(iii) Coplanar Parallel Forces **

20. Four forces of magnitudes 20 N, 40 N, 60 N and 80 N are acting respectively along the four sides of a square *ABCD *as shown in Fig. 1.94. Determine the resultant moment about point *A. *Each side of square is 2 m.

[Ans. 200 Nm anti-clockwise]

21. A force of 50 N is acting at a point *A *as shown in Fig. 1.95. Determine the moment of this force about O.

[Ans. 100 Nm clockwise]

22. Three like parallel forces 20 N, 40 N and 60 N are acting at points, *A, Band *C respectively on a straight line *ABC. *The distances are *AB *= 3 m and *BC *= 4 m. Find the resultant and also the distance of the resultant from point A on line *ABC.*

[Ans. 120 N, 4.5 m]

23. The three like parallel forces 101 N, *F *and 300 N are acting as shown in Fig. 1.96. If the resultant *R *= 600 N and is acting at a distance of 45 cm from *A, *then find the magnitude of force *F *and distance of *F *and *A.*

[Ans. 200 N, 30 cm]

24. Four parallel forces of magnitudes 100 N, 200 N, 50 N and 400 N are shown in Fig 1.97. Determine the magnitude of the resultant and also the distance of the resultant from the *A. *

[Ans. *R *= 350 N, 3.07 m]

25. A system of parallel forces are acting on a rigid bar as shown in Fig. 1.98. Reduce this system to:

(i) a single force,

(ii) a single force and a couple at A

(iii) a single force and ‘a couple at B.

[Ans. (i) R = 120 N at 2.83 m from A

(ii) R = 120 N and MA = - 340 Nm

(iv) R = 120 N and MB = 120 Nm]

26. Five forces are acting on a body as shown in Fig. 1.99. Determine the resultant.

[Ans. R = 0, Resultant couple = 10 Nm]

27. Determine the resultant of the parallel forces shown in Fig. 1.100.

[Ans. Body is in equilibrium]

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