Mechanics of Solids Chap-1 Introduction PDF

# Mechanics of Solids ## 1. Introduction ### 1.1 Engineering Mechanics - Mechanics is the branch of science which deals with the state of rest or motion of particles and bodies under the action of forces. - In applied mechanics, the approach is to study systematically the laws of mechanics and t...

# Mechanics of Solids ## 1. Introduction ### 1.1 Engineering Mechanics - Mechanics is the branch of science which deals with the state of rest or motion of particles and bodies under the action of forces. - In applied mechanics, the approach is to study systematically the laws of mechanics and their applications to engineering problems. **Mechanics** *(Branch of Science)* ↓ **Applied Mechanics** *(Application of Mechanics to Engineering Problems)* - **Mechanics of rigid bodies** *(Engineering Mechanics)* - **Mechanics of deformable bodies** *(Strength of materials)* - **Mechanics of fluids** - **Statics** *(body at rest)* - **Dynamics** *(bodies in motion)* - **Kinetics** *(Study of motion of a body and forces responsible for the motion)* - **Kinematics** *(Study of motion of a body, causes of motion are not studied)* - **Statics: **It is that branch of Engineering Mechanics, which deals with the forces and their effects, while acting upon the bodies at rest. - **Dynamics: **It is that branch of Engineering Mechanics, which deals with the forces and their effects, while acting upon the bodies in motion.. - **Kinetics: **It is that branch of dynamics, which deals with the bodies in motion due to the application of forces. - i.e. We study motion of a body and forces responsible for the motion. - **Kinematics: **It is that branch of dynamics, which deals with the bodies in motion without any reference to the forces which are responsible for the motion. - i.e. We study only motion of a body, causes of motion (forces) are not studied. ### 1.2 Fundamental Concepts 1. **Space:** Space is the region in all directions and contains everything within. e.g. sun and stars, planets etc. - OR It is the unlimited expanse of physical dimensions in which all material objects are located. 2. **Time:** It is a measure of duration between successive events. - The S.I. unit of time is Second. - One second is a fraction of $1/86,400$ of an average solar day. 3. **Matter:** The substance that possesses weight, occupies space and that can be apprehended by sense is called matter. 4. **Body:** the matter bounded by a closed surface is called body. - It is an accumulation of large number of articles. - The body is formed by surfaces - e.g. a bus, earth etc. 5. **Inertia:** It is the Property by Virtue of which a body offers resistance to any change of its state of rest or motion. 6. **Mass:** The quantity of matter contained by a body is called mass of the body. It is a quantitative measure of inertia. Consider two identical blocks of same size and shape, one made of wood and the other of steel. Keep these blocks on identical surfaces. Push them by applying same force (i.e. equal in magnitude and direction) It will be observed that the wooden block will move much faster than the steel block. It shows that steel block possesses greater inertia and thus greater mass than the wooden block. 7. **Rest:** A body is said to be at rest if it does not change its position with respect to a reference point at different intervals of time. - e.g. a car parked in garage. 8. **Motion:** A body is said to be in motion if it changes its position with respect to a reference point at different intervals of time. e.g. a car moving on road. ### 1.3 Idealizations in Mechanics 1. **Particle:** It is an idealized (assumed) body which may have negligible mass and whose size and shape can be neglected. It is viewed as mass point. - It is an idealized body whose size does not affect its response to the forces acting on it. e.g. a car moving on a road. - Here, car is considered as a particle. If we are interested in finding its velocity and acceleration then size of car has no effect on the calculations. 2. **Rigid body:** A body is said to be rigid if the relative distance between the particles of the body remain same before and after application of force. 3. **Deformable body:** A body is said to be deformable if its size and shape change under application of external forces. - Generally all bodies are deformable to smaller or greater extent depending upon its rigidity and the magnitude of applied force. 4. **Continuum:** It is defined as an idealized body whose matter is assumed to be totally continuous, homogeneous and non-porous. e.g. surface of a wing of an aircraft. - The solid body consists of particles arranged in particular sequence and consists of voids in between. The idealization of a body as continuum ignores the presence of Voids and assumes the solid part to be present everywhere in the body. ### 1.4 Fundamental Principles of Mechanics The elementary mechanics rests on a few fundamental principles based on experimental observations. These are : 1. Parallelogram law of forces. 2. Principle of Transmissibility. 3. Principle of super position of forces. 4. Newton's First Law of motion. 5. Newton's second Law of motion. 6. Newton's Third Law of motion. 7. Newton's Law of Gravitation. **1. Parallelogram Law of forces:** "If two forces, acting simultaneously on a particle be represented in magnitude and direction by the two adjacent sides of a parallelogram; their resultant may be represented in magnitude and direction by the diagonal of the parallelogram which passes through their point of intersection." **2. Principle of Transmissibility:** "If a force acts at any point on a rigid body, it may also be considered to act at another point on its line of action, provided the point is rigidly connected with the body." **3. Principle of Superposition of forces:** If two equal, opposite and collinear forces are added to or removed from the system of forces, there will be no change in the position of the body. This is known as principle of superposition of forces. **4. Newton's First law of motion:** "Every body continues in a state of rest or uniform motion unless it is compelled to change that state by some external force." **5. Newton's Second law of motion:** "The acceleration of body is proportional to the impressed force and takes place in the direction in which the force acts." $Force \propto Acceleration$ $F \propto a$ $F = ma$ Where m is the constant representing mass of the body. This law helps us to measure a force quantitatively. **6. Newton's Third Law of motion:** "For every action there is always an equal and opposite reaction." Which means that the forces of action and reaction between two bodies are equal in magnitude but opposite in direction. **7. Newton's Law of Gravitation:** "Two bodies are attracted towards each other along the line connecting them with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them." $F \propto m_1m_2$ $F \propto 1/r^2$ $F = G. m_1m_2 / r^2$ Where, - $m_1$ = mass of body 1 - $m_2$ = mass of body 2 - $r$ = distance between two bodies - $F$ = Force of attraction between two bodies. - $G$ = Universal Gravitational constant = 6.67 x 10-11 N.m²/kg2 Weight of the body can be defined with the help of Law of Gravitation. Weight of a body on earth is defined as the force exerted by earth on a body. For example, Let, - $M$ = mass of earth - $m$ = mass of body lying on earth. - $R$ = radius of the earth = 6.374 x 106 m - $F$ = Weight of the body (W) $F = G. M. m/ R^2$ Where, $F = m.g$ $g = G. M/ R^2 = 9.81 m / s^2$ OR $W = m.g$ ### 1.5 Fundamental Units All the physical quantities used in Engineering Mechanics are expressed in terms of three fundamental quantities, i.e. 1. length 2. mass 3. time The units of these fundamental quantities are called fundamental units or base units. **Derived units:** The units of some physical quantities are derived from fundamental units. Such units are called derived units. e.g. Units of area, velocity, acceleration etc. ### 1.6 System of Units There are four systems of units, which are commonly used and universally recognised. 1. F.P.S. System 3. M.K.S. System 2. C.G.S. System 4. S. I. System ### 1.7 S.I. System (International System of Units) S.I. System of units was recommended by the Eleventh General Conference of Weights and Measures in 1960. In India, the Standards of Weights and Measures Act of 1956, has been revised to recognise all the S.I.units in industry and commerce. As shown in table 1.1, the fundamental units in S.I. System are metre (m), kilogram (kg) and second (s). In S.I. Units, there are 7 fundamental units, 2 supplementary units and a number of derived units. **Table 1.1 System of units** System of Units | unit of fundamental quantities | |---|---| | Length | mass | time | | F.P.S. System | Foot | Pound | Second | | C.G.S. System | Centimetre | gram | Second | | M.K.S. System | metre | kilogram | Second | | S.I. System | metre | kilogram | Second | **Table 1.2 Fundamental units of S.I. System** Sr. No. | Physical quantities | unit | symbol | |---|---|---|---| | 1. | Length | metre | m | | 2. | Mass | kilogram | kg | | 3. | Time | second | S | | 4. | Temperature | kelvin | K | | 5. | Electric current | Ampere | A | | 6. | Luminous Intensity | candela | cd | | 7. | Amount of Substance | mole | mol | **Table 1.3 Supplementary units of S. I. System** Sr.. | Physical quantities | unit | symbol | |---|---|---|---| | 1. | Plane angle | Radian | rad | | 2. | Solid angle | Steradian | sr | **Table 1.4 Principal S.I.units** Sr.No. | Physical quantity | Unit | Symbol | |---|---|---|---| | 1. | Force | Newton | N | | 2. | Work | Joule | J. N.m | | 3. | Power | Watt | W | | 4. | Energy | Joule | J.N.m | | 5. | Area | Square metre | m2 | | 6. | Volume | Cubic metre | m³ | | 7. | Pressure | Pascal | Pa | | 8. | Velocity/speed | metre per second | m/s | | 9. | Acceleration | metre/second2 | m/s2 | | 10. | Angular Velocity | radian/second | rad/s | | 11. | Angular acceleration | radian/second2 | rad/s2 | | 12. | Momentum | kilogram metre/second | kg.m/s | | 13. | Torque | Newton metre | N.m | | 14. | Density | kilogram/metre³ | kg/m³ | | 15. | Couple | Newton.metre | N.m | | 16. | Moment | Newton.metre | N.m | **Table 1.5 S. I. Prefixes** Multiplication factor | Prefix | Symbol | |---|---|---| | 10^12 | Tera | T | | 10^9 | Giga | G | | 10^6 | Mega | M | | 10^3 | kilo | k | | 10^2 | hecto | h | | 10^1 | deca | da | | 10^-1 | deci | d | | 10^-2 | centi | c | | 10^-3 | milli | m | | 10^-6 | micro | μ | | 10^-9 | nano | n | | 10^-12 | pico | p | ### 1.8 Conversion of Units - 1 m = 100 cm - 1 m = 1000 mm = 10^3 mm - 1 cm = 10 mm - 1 km = 1000 m = 10^3 m - 1 cm² = 100 mm² - 1 m² = 10^6 mm² - 1 kgf = 9.81 N = 10 N - 1 kN = 10^3 N - 1 MN = 10^6 N = 10^3 kN - 1 GN = 10^9 N = 10^6 kN = 10^3 MN - 1 MPa = 1 N/mm² - 1 GPa = 10^3 N/mm² - 1 Pascal = 1 N/m² - 1 N.m = 1 Joule - 1 Kilo Joule = 10^3 Joule. - 1 Watt = 1 Joule/Second = 1 N.m/Second - 1 kW = 10^3 Watt - 1 hp = 736 Watt - 1 degree = $π/180$ radian ### 1.9 Scalar Quantities A scalar quantity is one which can be completely specified by its magnitude only. For example, - Length, - distance, - area, - Volume, - time, - work, - mass, - density, - temperature, - speed, - Energy. - mass moment of Inertia. ### 1.10 Vector Quantities A vector quantity is one which requires magnitude and direction both to completely specify it. For example, - displacement, - Velocity, - acceleration, - Momentum, - Moment, - Force, weight - Angular displacement - Angular Velocity - Angular acceleration, - Impulse ## 2. Coplanar Concurrent Forces ### 2.1 Introduction This chapter deals with different types of forces, its characteristics, Force systems. Different methods of calculating resultant of a force system are discussed in detail. This chapter particularly deals with coplanar concurrent forces 1.e. forces meeting at one point and acting along a plane. These forces are assumed to act at the geometric centre of the body and therefore, they do not produce any rotational effect on the body. Analytical methods as well as Graphical methods for calculating resultant are described. Equilibrium of body, free body diagrams and Lami's theorem has been covered with a lot of numericals. ### 2.2 Force "An agent which produces or tends to produce, destroys or tends to destroy motion of a body is called force." Unit of force is Newton (N). Force is a vector quantity. **1 N Force:** A force which can produce an acceleration of 1 m/sec² in a mass of 1 kg is called 1 N force. **1 kg (f) force:** A force which can produce an acceleration of g m/sec2 (gravitational acceleration) in a mass of 1 kg is called 1 kg(f) force. 1 kg(f) = 9.81 N **Characteristics of a force:** Followings are the characteristics of a force. 1. **Magnitude:** Magnitude of a force may be 10 kN, 50 kN, 200 N etc. 2. **Direction:** - i.e. along OX. - towards North. - at 30° West of North. 3. **Nature:** The nature of force may be tensile or compressive. (Push or Pull) 4. **Point of application:** The point at which the force acts on the body is called point of application. ### 2.3 Types of Forces 1. **Contact force (surface force):** It is defined as the force produced by direct contact of bodies. It may be pull type or push type. - For example, - The weight force exerted by one body lying over the another body. - Force exerted by air on an aeroplane. 2. **Body force:** The force produced without contact of the bodies is called body force. It is the force which holds together the particles forming the rigid body. - For example, - If a force P is applied on a body, the body will develop internal resistance (R) - This R is a body force. - Other examples of body force are, - (i) **Gravitational force:** It is the force exerted by the earth on a body. It does not require any medium. It is always directed vertically downwards. - (ii) **Magnetic force and electric force:** Forces developed between two magnetically charged bodies are called magnetic forces. Forces developed between two electrically charged bodies are called electric forces. 3. **Point force and distributed force:** - **Point force:** The force acting at a point is called point force. When the force acts on very small area compared to total surface area, it can be considered as point force. - For example, - Force exerted by a man standing on a platform. - Force exerted by tyres of a car on the road. - **Distributed force:** When force is distributed over large area, it is called distributed force. - For example, - Force exerted by water tank resting on ground. 4. **External force and internal force:** - The force exerted by weight of block A (W) on block B is external force on B. While the resistance offered by block B to retain its shape is called internal force. - A force is considered internal or external depending upon the boundary of the body. 5. **Action and Reaction:** As per Newton's third law, - whenever a body exerts force (action) on other body, the other body exerts similar force on the former body, known as reaction. - In Fig. 2.3, W = Action, R = Reaction. 6. **Friction or friction force:** When a body slide or tends to slide on a surface, a resisting force opposing the motion is produced at the contact surface. This resisting force is called friction or friction force. - Where, - P = external force - F = Friction force. 7. **Wind force:** The force exerted by wind when it is obstructed by a body is called wind force. 8. **Hydrostatic force:** It is the force exerted by water on a body when it is obstructed by a body. 9. **Cohesion and Adhesion:** - **Cohesion:** It is the force (attraction) developed between molecules of same material e.g. attraction between molecules of mercury. - **Adhesion:** It is the force (attraction) developed between molecules of different materials. - e.g. attraction between molecules of glass and water. 10. **Thermal forces:** These are the forces developed due to heating or cooling of bodies. ### 2.4. Mass and Weight **Mass (m)** - It is the matter contained in a body. - Mass is scalar quantity. - S.I. unit is kg. **Weight (w)** - It is the gravitational force acting at the centre of gravity of a body. - Weight is a vector quantity. - S.I. unit is N or kN. - $W = m.g$ - where, $W$ = weight (N) - $m$ = mass (kg) - $g$ = 9.81 m/s² ### 2.5 System of Forces When two or more forces act on a body, they are called to form a System of forces. Following system of forces are important. 1. **Coplanar forces:** The forces whose line of action lie on the same plane, are known as Coplanar forces. - Here, forces P1, P2, P3 are coplanar forces. - co-means same. - planar means along a plane. 2. **Concurrent forces:** The forces which meet at one point, are known as concurrent forces. - Forces P1, P2, P3 are concurent forces. 3. **Collinear forces:** The forces whose lines of action lie on the same line, are known as Collinear forces. - Forces P1, P2, P3 are collinear forces. 4. **Coplanar concurrent forces:** The forces which meet at one point and their lines of action also lie on the same plane are known as coplanar concurrent forces. - Forces P1, P2, P3 are coplanar concurrent forces. 5. **Coplanar Non-concurrent forces:** The forces whose lines of action lie on the same plane but they do not meet at one point are known as coplanar nonconcurrent forces. 6. **Non-coplanar concurrnet forces:** The forces whose lines of action do not lie on the same plane, but they meet at one point are called non coplanar concurrent forces. 7. **Non-coplanar non-concurrent forces:** The forces whose lines of action do not lie on the same plane and they do not meet at one point are known as non-coplanar non-concurrent forces. 8. **Like parallel forces:** The forces whose lines of action are parallel to eachother and all of them act in the same direction are known as like parallel forces. 9. **Unlike parallel forces:** The forces whose lines of action are parallel to eachother but all of them do not act in the same direction are known as unlike parallel forces. 10. **Spatial forces:** Forces acting in the space are known as Spatial forces. 11. **Spatial concurrent forces:** The forces acting in space but meeting at one point are known as Spatial concurrent forces. - e.g. Forces acting in the strings of a parachute. 12. **Spatial non-concurrent forces - like force system:** This type of force system consists of forces acting in space, but are parallel and act in same direction system. - e.g. forces in columns of a building. 13. **Spatial non-concurrent forces-unlike force system:** This type of force system consists of forces acting in space, parallel but act in opposite direction. ### 2.6 Principle of Physical Independence of Forces It states, "If a number of forces acting simultaneously on a particle, then each one of them will produce the same effect which it would have done while acting alone". ### 2.7 Principle of Superposition of Forces It, states, "If two equal, opposite and collinear forces are added to or removed from the system of forces, there will be no change in the system and position of the body." P = 100 N acting at A. If two equal, opposite and collinear forces ### 2.8 Principle of Transmissibility of Forces It states, " If a force act at any point on a rigid body it may also be considered to act at any other point on its line of action, provided the point is rigidly connected with the body." According to this law, a force acting on a rigid body can be shifted along its line of action. A push force can be converted in to a pull force. 1.e. nature of force can be changed. This principle is valid for rigid body only. By shifting the force, only the static state of the body remains same, but the internal stresses are definitely changed. ### 2.9 Resultant Force If a number of forces P1, P2, P3. P4. ... etc. are acting simultaneously on a particle, it is possible to find out a single force which could replace them i.e. which would produce the same effect as produced by all the given forces. This single force is capable of producing same effect on a body is known as resultant force and the given forces P1, P2, P3 P4.... etc. are called component forces. R is the resultant of three concurrent forces P1. P2 and P3. 0 is the angle of resultant with P₁. ### 2.10 Composition of Forces The Process of finding out resultant force of a number of given forces is called composition of forces. **Methods of resultant force** **Analytical Methods** - (i) Parallelogram law of forces - (ii) Resolution of forces - (iii) Triangle law of forces **Graphical Methods** - (1) Triangle law of forces - (ii) Polygon law of forces ### 2.11 Parallelogram Law of Forces It states, "If two forces, acting simultaneously on a particle, be represented in magnitude and direction by the two adjacent sides of a parallelogram; their resultant may be represented in magnitude and direction by the diagonal of the parallelogram which passes through their point of intersection" $R = √P^2 + Q^2 + 2PQ cos θ$ $tan α = (Q sin θ) / (P + Q cos θ)$ Where, - $R$ = Resultant force - $θ$ = angle between P and Q - $α$ = angle between P and R ### 2.12 Triangle Law of Forces It states, "If two forces acting at a point be represented in magnitude and direction by two sides of a triangle taken in order, their resultant may be represented in magnitude and direction by the third side of triangle, taken in opposite order." ### 2.13 Resolution of a Force The process of splitting up the given force in to two or more components. in particular direction, without changing effect on the body is called resolution of a force. There are two types of components. - (a) Orthogonal Components - (b) Non-Orthogonal Components. **(a) Orthogonal Components:** (Rectangular Components) The components which are perpendicular to each other or the components which make 90° with each other are known as orthogonal components. Generally a given force is split up in to two mutually perpendicular components - (i) Horizontal component (x-component) - (ii) Vertical component (y-component) **(b) Non-orthogonal components:** The components those are not perpendicular to eachother are know as non orthogonal components. The components make either obtuse 0 > 90° or make acute angle 0 < 90°, The Corthogonal components can be calculated law of parallelogram of forces or triangle law of forces. Using parallelogram law of forces : $R = √P^2 + Q^2 + 2PQ cos θ$ $tan α = (Q sin θ) / (P + Q cos θ)$ Using Triangle law of forces : $R = √P^2 + Q^2 - 2PQ cos β$ $α = sin^-1 (Q sin β / R)$ Where, - $θ$ = angle between P and Q - $β$ = 180° - θ - $α$ = angle between P and R ### 2.14 Principle of Resolution "The algebric sum of the resolved parts of a number of forces in a given direction is equal to the resolved part of their resultant in the same direction." Consider two forces P and Q, which are represented in magnitude and direction by the two adjacent sides OA and OB of a parallelogram OACB as shown in figure-2.43. R is the resultant of P & Q. will be represented in magnitude and direction by diagonal OC of the parallelogram. Let OX be the direction in which the forces are to be resolved. ### 2.15 Resolution of More Than Two Concurrent Forces Consider P1, P2, P3, P4 ... etc. are acting at a point as shown in figure 2.61. ΣΗ = Algebraic sum of horizontal forces V = Algebraic sum of vertical forces. R = Resultant of all forces. $R = √(ΣH)^2 + (ΣV)^2$ $tan θ = (ΣV) / (ΣH)$ θ = angle of resutant with horizontal. (i) If, ΣΗ = +Ve, ΣV = + Ve - R will be in the first quadrant. (ii) If ΣΗ = -Ve, ΣV = +Ve - R will be in the second quadrant. (iii) If ΣΗ = -Ve, V = -Ve - R will be in the third quadrant. (iv) If ΣΗ = +Ve, ΣV = -Ve - R will be in the fourth quadrant. ### 2.16 Conditions of Equilibrium of Coplanar Concurrent Forces: A system of coplanar concurrent forces is said to be in equilibrium if the following conditions are satisfied. - (1) ΣΗ = 0, i.e. Algebric sum of all the horizontal forces must be zero. - (ii) ΣV = 0, i.e. Algebric sum of all the vertical forces must be zero. - (iii) Since, ΣΗ = 0, ΣV = 0 ### 2.17 Polygon Law of Forces "If a number of forces acting at a point be represented in magnitude and direction by the sides of a polygon taken in order, then the resultant of all these forces may be represented in magnitude and direction by the closing side of the polygon taken in opposite order." ### 2.18 Graphical Method **Space diagram:** If forces acting in a system are represented by direction in the form of diagram using Bow's notation the diagram is known as space diagram. **Vector diagram:** To find resultant force of a coplaner concurrent system, each force of space diagram is represented by magnitude and direction. The diagram so obtained is called vector diagram. **Bow's Notation:** While drawing space diagram each force is denoted by an english character on either side of it. This system is known as Bow's notation. For example, 60 kN force is denoted by two characters A on one side and B on other side. Hence, force AB = 60 kN. ### 2.20 Lami's Theorem "If three coplanar forces acting at a point be in equilibrium, then each force is proportional to the sine of angle between the other two." $P/sin α = Q/ sin β = R/ sin γ$ Consider three forces P, Q and R acting at a point O, let opposite angles to the forces be a, ẞ and γ. Now draw two lines OA equal to force P and OB equal to force to some scale meeting at O. Now complete the parallelogram OACB with OA and OB as two adjacent sides and OC as a diagonal. In A OAC, according to sine Rule, $OA/sin∠ACO = CA/sin∠COA = OC/sin∠OAC$ … (1) ∠ACO = 180° – α ∠COA = 180° – β ∠OAC = 180° - ∠ACO - ∠COA = 180° - (180° - α) – (180° – β) = 180° - 180° + α - 180° + β = α + β - 180° = (360° – γ) – 180° ∠OAC = 180° - γ Substitute values in eq. (1) $OA/sin (180° -a) = CA/sin (180° - β) = OC/sin (180° – γ)$ $OA/sin α = CA/sin β = OC/sin γ$ $P/sin α = Q/ sin β = R/ sin γ$ ## 3. Moment of a Force ### 3.1 Moment of a Force: The moment of a force is equal to the product of the force and the perpendicular distance of line of action of force from the point about which the moment is required. $M = P x x$ Where, - $M$ = moment - $P$ = Force - $x$ = Perpendicular distance between the line of action of force and the point about which moment is required. Moment produce the turning effect of the body. ### 3.2 Couple Two equal and opposite forces whose lines of action are different form a couple. Resultant or net force of couple is zero. Hence, couple acting on a body do not create any translatory motion of the body. Couple produces only rotational motion of the body. **Types of couple:** There are two types of couple. - (a) Clockwise couple - (b) Anticlockwise couple. **Arm of a couple:** The perpendicular distance between the lines of action of two forces forming couple is known as the arm of couple. In figure 3.2, a = arm of couple. ### 3.3 Characteristics of a Couple A couple has the following characteristics. - (1) The algebric sum of the forces, forming the couple is zero. - (2) The algebric sum of the moment of the forces, forming the couple, about any point is the same and equal to the moment of the couple itself. Moment of couple M = Pa Now, taking moment @ O, ### 3.4 Differentiate: **Moment** - (i) Moment = Force x Perpendicular distance - $M = P.x$ - (ii) It is produced by a single force not passing through c.g. of the body. - (iii) The force move the body in the direction of force and rotate the body. - (iv) To balance the force causing moment, equal and opposite force is required. - (v) For example, - To tight the nut by spanner - To open or close the door. **Couple** - (i) Two equal and opposite forces whose lines of action are different form a couple. - (ii) It is produced by two equal and opposite parallel, non-collinear forces. - (iii) Resultant force of couple is zero. Hence, body does not move. but rotate only. - (iv) Couple can not be balanced by a single force. It can be balanced by a couple only. - (v) For example, - To rotate key in the lock. - To open or close the wheel valve of water line. - To rotate the steering wheel of car. ### 3.5 Equivalent Couples Couples are said to be equivalent couples, if they produce similar effects on a body. ### 3.8 Conditions of Equilibrium for Coplanar Non-Concurrent Forces If a body is acted upon by a number of co-planar non-concurrent forces, it may have one of the following states: 1. The body may move in any one direction. 2. The body may rotate about itself without moving. 3. The body may move in any one direction, and at the same time it may also rotate about itself. 4. The body may be completely at rest. Now we will discuss the above four states. (1) If the body moves in any direction, it means that there is a resultant force acting on it. A little consideration will show, that if the body is to be at rest or in equilibrium, the resultant force causing movement must be zero i.e. ΣΗ = 0 and EV = 0 (2) If the body rotates about itself without moving it means that there is a single resultant couple acting on it. A little consideration will show that if the body is to be at rest or in equilibrium, the moment of the couple causing rotation must be zero. i.e. ΣΜ = 0 (3) If the body moves in any direction and at the same time it rotates about itself, it means that there is a resultant force and also a resutant couple acting on it. A little consideration will show that if the body is to be at rest or in equilibrium the resultant force causing movement and the resultant moment of the couple causing rotation must be zero. i.e. ΣΗ = Ο ΣV = 0 ΣΜ =