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### Applied mechanics

• 1. CHAPTER 1 RESULTANT OF COPLANAR FORCES CONTENT OF THE TOPIC:  Introduction to Applied Mechanics  Mechanics or Engineering Mechanics  Branches of Mechanics  SI system of Units, Basic units, Derived units  Body, Rigid body, particle  Scalar quantity, vector quantity  Force and Graphical representation of force.  Moment of forces  Couple and moment of couple  Law of Parallelogram of forces  Law of transmissibility of forces  Varignon’s theorem,  Composition of forces  Coplanar force system  Coplanar Non-concurrent force system  Analytical method  Graphical Method: Triangle law of forces, polygon law of forces  Bow’s notation  Problems  Problems on calculation of resultant  Problems on Varignon’s Theorem Applied Mechanics: It is the branch of engineering which studies the effect of external forces applied in any manner on a particle or a body. Engineering Mechanics/ Mechanics: It is the branch of physical science which deals with the behavior of a body when the body is at rest or in motion. Depending upon the body to which the mechanics is applied, the Engineering Mechanics/ Mechanics is classified as a) Mechanics of solids b) Mechanics of fluids Mechanics of solids (rigid bodies) further classified in two groups: CHAPTER NO. 1 Resolution of Coplanar Forces Page 1
• 2. Statics: It is a branch of Mechanics which deals with the studies of the bodies or rigid bodies in equilibrium under the action of external forces. Dynamics: It is a branch of Mechanics which deals with the studies of the bodies or rigid bodies in motion. Dynamics has two parts: a) Kinematics b) Kinetics Kinematics: The study of the body in motion, when the forces which cause the motion are not considered, is called as Kinematics. Kinetics: The study of the body in motion, when the forces which cause the motion are considered, is called as Kinetics. SI system of Units: It is an internal system of units. It is universally approved and accepted. It is adopted by large number of countries. System: Measuring systems are adopted for the measurement of physical quantities. Unit/Quantity: It is standard for the measurement of physical quantities. CHAPTER NO. 1 Resolution of Coplanar Forces Page 2
• 3. Basic Unit/ Fundamental units/ Basic quantities: Basic quantities/ Basic Unit: The quantities which do not depend upon other quantities for their measurement is known as basic quantities and their corresponding units are known as the basic units. Eg. Length, Mass, Time, Temperature, Electric current, plane angle etc. Derived quantities/ Derived Unit: The quantities which depend upon one or more basic quantities for their measurement is known as derived quantities and their corresponding units are known as the derived units. Eg. Velocity, Acceleration, Force, Work & Energy, Power etc. Body: A body is defined as an object, which cannot retain its shape and size under the action of a force system. Rigid body: A rigid body is defined as a body, which can retain its shape and size even if subjected to external forces. In practice, there is small deformation of body under the action of a force system. Such deformation is neglected and the body is treated as rigid body. Particle: A particle is defined as a very small amount of matter, which may be assumed to occupy a single point in space. Practically, any object having very small dimensions as compared to its range of motion can be called as a Particle. Eg. Stars, planets, Rockets, Bullets etc. Scalar quantity: It is the quantity having magnitude only. It has no direction. Eg. Mass, speed etc. CHAPTER NO. 1 Resolution of Coplanar Forces Page 3
• 4. Vector quantity: It is the quantity having magnitude and direction. It is shown by vector. Eg. Force, Velocity, acceleration etc. Force: The external agency, which tends to change the state of a body is known as force. A force is completely defined only when the following four characteristics are specified: - Magnitude - Point of application - Line of action - Direction A force (F) is a vector quantity which is represented graphically by a straight line say ‘ab’ whose length is proportional to the magnitude of force and the arrow shows the direction of force ‘ab’ as shown in Figure above. Unit of force is Newton (N). Force System: When several forces of different magnitude and direction act upon a body, they constitute a system of forces. Main types of force systems are as follows: 1) Coplanar Force System: Lines of action of all the forces lie in the same plane in this system as shown in Fig. (A) below. CHAPTER NO. 1 Resolution of Coplanar Forces Page 4
• 5. 2) Collinear Force System: Lines of action of all the forces lie in the same straight line in this system as shown in Fig. (B) above. 3) Concurrent Force System: Lines of action of all the forces meet at a point in this system. The concurrent forces may not be collinear or coplanar as shown in Fig. (C) above. 4) Parallel Force System: Lines of action of all the forces are in parallel as shown in Fig. (D) above. 5) Non- Coplanar Force System: Lines of action of all the forces does not lie in the same plane as shown in Fig. (E) above. 6) Non- Concurrent Force System: Lines of action of all the forces do not meet at a point in this system as shown in Fig. (E & F) above. 7) Non-Parallel Force System: Lines of action of all the forces are not in parallel as shown in Fig. (H) above. 8) Coplanar Concurrent Force System: Lines of action of all the forces lie in the same plane and meet at a point shown in Fig. (G) above. 9) Coplanar Non-Concurrent Force System: Lines of action of all the forces lie in the same plane, but do not meet at a a point as shown in Fig. (A) above. They may be in parallel. CHAPTER NO. 1 Resolution of Coplanar Forces Page 5
• 6. 10) Coplanar parallel Force System: Lines of action of all the forces are in parallel in the same plane shown in Fig. (D) above. 11) Coplanar, non-concurrent, non-parallel Force System: The lines of action of all the forces are not in parallel, they do not meet at a point but they are in the same plane as shown in Fig. (A) above. 12) Non- Coplanar, non-concurrent Force System: The lines of action of all the forces do not lie in the same plane and do not meet at a point as shown in Fig. (E) above. Fundamental Laws of Mechanics:  Newton’s First Law  Newton’s Second Law  Newton’s Third Law  Newton’s Law of gravitation  Law of transmissibility of Force  Parallelogram law of Forces 1) Newton’s First Law: It states that every body continues in its state of rest or of uniform motion in a straight line unless it is compelled by external agency acting on it.  Newton’s First Law for rotation: Newton’s laws of motion of rotation which state that, “Every body continues in its state of rest or of uniform motion of rotation about an axis unless it is acted upon by some external torque” 2) Newton’s Second Law: It states that the rate of change of momentum of a body is directly proportional to the impressed force and it takes place in the direction of the force acting on it. Force α rate of change of momentum But, Momentum = Mass x velocity As mass do not change, Force α Mass x rate of change velocity Force α Mass x acceleration F α ma F = ma 3) Newton’s Third Law: It states that for every action there is an equal and opposite reaction. CHAPTER NO. 1 Resolution of Coplanar Forces Page 6
• 7. 4) Newton’s Law of gravitation: Everybody attracts the other body. The force of attraction between any two bodies is directly proportional to their masses and inversely proportional to the square of the distance between them. Where, G is the constant of proportionality, it is known as constant of gravitation. Experimentally, it is proved that the value of G = 6.673 x 10-11 Nm2/kg2 F= G 푚1 푚2 푑2 5) Law of transmissibility of Force: Statement: “The point of application of force may be transmitted along its line of action without changing its effect on the rigid body to which the force is applied”. Explanation: A force is acting at point A along line of action AB on rigid body as shown in Fig. (a). Two equal and opposite forces of magnitude ‘P’ are added at point ‘B’ along line of action AB according to the law of superposition as shown in Fig (b). Figure (a) Figure (b) Two equal and opposite forces of the magnitude ‘P’ at point A and B can be subtracted without changing action of original force P according to the law of superposition as shown in Fig (c). Figure (c) Thus the point of application of force P is transmitted along its line of action from A to B. CHAPTER NO. 1 Resolution of Coplanar Forces Page 7
• 8. Varignon’s Theorem of Moments/ Principle of Moments: Statement: “The algebraic sum of the moments of all the forces about any point is equal to the moment of their resultant about the same point”. i.e. ΣM = Σ (Moments of forces) = Moment of R Proof: In above Figure AB and AC represents forces P and Q resp. and ‘O’ is the point about which moment is taken. ABCD represents a parallelogram. A diagonal AD represents resultant of forces P and Q. Now extend CD up to the point ‘O’ which is the line of CD. Join OA and OB. Now, we know that, Moment of force = 2(Area of triangle) Moment of force P = 2 x Area of Triangle AOB And Moment of force Q = 2 x Area of Triangle AOC Algebraic sum of Moments of forces P and Q = Σ M = 2 x (Area of ΔADB + Area of ΔAOC) Now, Area of Δ AOB = Area of ΔADB = Area of ΔACD Since, AB = CD (base is same) and height is same Σ M = 2 x (Area of Δ ACD + Area of Δ AOD) = 2 x (Area of Δ AOD) Σ M = Moment of Resultant Force ‘R’ CHAPTER NO. 1 Resolution of Coplanar Forces Page 8
• 9. Application: 1) It is generally used to locate the point of application of resultant. 2) In case of coplanar non-concurrent system of forces this concept is used to locate the line of action of the resultant. Parallelogram law of Forces Statement: Statement: “If two forces acting simultaneously on a body at a point are presented in magnitude and direction by the two adjacent sides of parallelogram, their resultant is represented in magnitude and direction by the diagonal of the parallelogram which passes through the point of intersection of the two sides representing the forces”. Fig. (a) Fig. (b) The length of diagonal in Fig. (b) will indicate the magnitude of resultant of ‘R’. Derivation: From right angle triangle BCD BD = Q sinθ CD = Q cosθ Using Pythagorus theorem to the ΔOCD OC2 = CD2 + OD2 CHAPTER NO. 1 Resolution of Coplanar Forces Page 9
• 10. OC2 = CD2 + (OB + BD) 2 R2 = (P + Q cosθ)2 + (Q sinθ)2 R2 = P2 + Q2 cos2θ + 2PQ cosθ + Q2 sin2θ R2 = P2 + Q2 + 2PQ cosθ R = √P2 + Q2 + 2PQ cosθ ---------------------------------------(1) Angle α of resultant R with force P is given by, α = tan-1[ Q sinθ 푃+ Q cosθ ] ---------------------------------------(2) Particular cases: 1) When θ = 900 R= √푃2 + 푄2 2) When θ = 00 R= P + Q (acting along Same Direction) 3) When θ = 1800 R= P – Q (acting in Opposite Direction) Moment: The turning effect caused by a force on the body is called as a moment of force. Definition: The moment of a force (M) is equal to the magnitude of the force (F) multiplied by the perpendicular distance (d) between the line of action of the force and the axis of rotation. Moment = Force x Perpendicular Distance M = F x d Sign convention: If the moment of the force producing clockwise rotation is the clockwise moment and it is taken as positive as shown in Fig. (a). If the moment of the force producing anticlockwise rotation is the anticlockwise moment and it is taken as negative as shown in Fig. (b). Figure (a) Figure (b) CHAPTER NO. 1 Resolution of Coplanar Forces Page 10
• 11. Unit: If the force is measured in Newton and the distance in meter, the SI unit of the moment is Newton meter (Nm). Geometrical Representation of Moment: As shown in Fig. below, AB represents force F and O is the point about which the moment of force M is taken. Let OC be the perpendicular distance‘d’. Moment of Force F is given by, M = F x d M = AB x OC M = 2 x (½ AB x OC) M = 2 (Area of triangle OAB) Thus Moment of Force about any point is geometrically equal to twice the area of the triangle having base representing a point about which moment is taken. Couple: Two equal, opposite and parallel (non-collinear) forces are said to form a couple as shown in Fig. below. Figure (a) Arm of couple: The distance ‘a’ between the lines of action of the two forces of a couple is known as ‘arm of couple’. Properties: a) Couple cannot be replaced by a single resultant force. b) Couple cannot produce rotation or moment but it cannot produce straight line motion. CHAPTER NO. 1 Resolution of Coplanar Forces Page 11
• 12. Moment of Couple: From above Fig. moment of couple about any point ‘O’ (moment of centre) is given by F (a +d) – (Fd) = Fa + Fd – Fd = Fa Nm Moment of Couple = Force x Arm Thus the moment of the couple has a constant value irrespective of the point about which moment is taken. Bow’s notation: Any force F divides the space into two parts A and B as shown in Fig. below. This force is named as force AB according to this method. Space Diagram Force Diagram In this method line ‘ab’ is drawn parallel to force F such that the length of line ab represents the magnitude and the direction is from ‘a’ to ‘b’ which indicates the direction of the force ‘F’ as shown in Fig. below. Suitability: This notation is useful for solving the problems in statics by graphical method. Composition of Forces: (Resultant of coplanar concurrent forces) The process of determining the Resultant of number of forces acting simultaneously on a body is known as Composition of Forces. It is the method of reducing the given force system to its equivalent simplest system of single force (or couple). CHAPTER NO. 1 Resolution of Coplanar Forces Page 12
• 13. Combining the forces of any given system is termed as composition of forces. There are two main methods of determining the resultant force: a) Analytical Methods and b) Graphical Methods Analytical Methods: There are two Analytical Methods: 1) Parallelogram law of Forces (as explained above) 2) Component Law or Resolution Methods 1. Component Law of Forces: 1. Forces such as F1, F2, F3 and F4 acting at point ‘O’ as shown in Fig. above. are resolved along x-axis (horizontally). The algebraic sum of horizontal components is ΣH or ΣFX. Figure (a) 2. Similarly, the forces are resolved along y-axis (vertically). The algebraic sum of vertical components is ΣV or ΣFy. CHAPTER NO. 1 Resolution of Coplanar Forces Page 13
• 14. 3. Resultant ‘R’ is given by, R = √ΣF푋2 2 + ΣF푌 4. Angle of inclination θ with x-axis is given by, θ = tan-1[ΣFy/ ΣFX] Particular cases: 1) When θ = 900 R= √푃2 + 푄2 2) When θ = 00 R= P + Q (acting along Same Direction) 3) When θ = 1800 R= P – Q (acting in Opposite Direction) Sign convention: While taking ΣFX, forces acting from left to right are taken as positive and those are acting from right to left are considered negative. While taking ΣFy forces acting upwards are assumed positive and those acting downwards are assumed negative. Resolution of Forces: The process of splitting or subdividing a force into its components without changing its effect on the body is known as Resolution of Forces. It is the replacement of a single force by several components having the same effect as that of single force. Generally, a force ‘F’ is resolved into two components Fx and Fy which are mutually perpendicular to each other as shown in Figure below. Horizontal component Fx = F cos θ Vertical component FY = F sin θ Consider a rigid body as shown in Fig. above. Let F1, F2 and F3 be three forces acting on a rigid body. Let ‘R’ be there resultant. Then we can say that F1, F2 and F3 are resolved parts of R or components of ‘R’ in three different directions. Generally, a force is replaced in to two rectangular components. Graphical Methods: 1) Law of triangle of Forces: CHAPTER NO. 1 Resolution of Coplanar Forces Page 14
• 15. If two forces P and Q (acting at point ‘O’) as shown in Figure below in which they represents the magnitude and direction of the two sides of the triangle taken in order, then the third side taken in opposite sense represents the resultant ‘R’ of the two forces in magnitude and direction. 2) Law of Polygon of Forces: If numbers of forces are acting on a body, are represented in magnitude and direction by the sides of the polygon taken in order, then the closing side taken in opposite sense represents the resultant of all the forces in magnitude and direction. Fig. a Fig. b Fig. c Above Figure (a) shows the system of four forces in magnitude and direction and Figure (b) shows the polygon of same forces. The closing side ‘R’ represents the resultant. We can use the triangle law of forces in this polygon, such that, the resultant of forces F1 and F2 is R as shown in Figure c. Similarly, the resultant of R1 and F3 is R2 and finally, the resultant of R2 and F4 is R by the triangle law of forces. Conclusion: The polygon law of forces is the application of triangle law of forces. Idealizations in mechanics: 1) The body is rigid. 2) The body is treated as continuum. Continuum: when the body is assumed to consist of a continuous distribution of matter is called as continuum. 3) If the size of the body is small as compared to other distances involved in the problem, it may be treated as a particle. 4) If the area over which force is acting on a body is small as compared to the size of the body, it may be treated as a point force. 5) Support conditions are idealized as simple, hinged, fixed etc. CHAPTER NO. 1 Resolution of Coplanar Forces Page 15
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