Engineering mechanics_statics of particles, Rigid Body and Friction.pptx

Applied mechanics is the branch of science concerned with the motion of any substance that can be experienced or perceived by humans without the help of instruments.[1] In short, when mechanics concepts surpass being theoretical and are applied and executed, general mechanics becomes applied mechanics. It is this stark difference that makes applied mechanics an essential understanding for practical everyday life.[2] It has numerous applications in a wide variety of fields and disciplines, including but not limited to structural engineering, astronomy, oceanography, meteorology, hydraulics, mechanical engineering, aerospace engineering, nanotechnology, structural design, earthquake engineering, fluid dynamics, planetary sciences, and other life sciences.[3][4] Connecting research between numerous disciplines, applied mechanics plays an important role in both science and engineering.[1]

1. Statics of Particles, Rigid Body, and Friction MODULE-I Presented by: Dr. MAINAKH DAS Assistant Professor, Department of Mechanical Engineering

2. MODULE - I OUTLINE OF PRESENTATION Introduction to mechanics, Basic Terminologies Statics, Principle of Statics and Newton’s laws of mechanics Force, Force system and Resultant Force Parallelogram, Triangle and Polygon Law of Forces Method of Resolution and composition of forces, Equilibrium of Forces and Principle of Transmissibility of forces Law of Superposition, Lami’s Theorem, Law of action and reaction, Tension and compression Free body diagram (FBD) Moment of a force, Varignon’s Theorem and Principle of moment Friction, Limiting friction Laws of static friction, Angle of friction, repose and cone

3. Introduction to Mechanical Engineering (ME131): Text/Reference Books Engineering Mechanics by R. C. Hibbler Engineering Mechanics, 5th EDN by S. Timoshenko, D.H. Young, J.V. Rao, Sukumar Pati . Engineering Mechanics – Statics and Dynamics by A. Nelson

4. Introduction to Mechanics MECHANICS: The branch of state of rest and state of motion of body is called mechanics. OR It is the oldest physical science that deals both static and moving bodies influenced by force. ENGINEERING MECHANICS : Mechanics applied to practical problems in the field is called engineering mechanics.

5. Classification of Engineering Mechanics Without cause of motion With cause of motion

6. Classification of Engineering Mechanics Without cause of motion With cause of motion Resistant Body /Practical Body: A body which shows rigid behavior up to certain amount of load is called resistant body. Rigid Body: A rigid body is a body which does not deform under the influences of forces. Statics : Deals with equilibrium of bodies under action of forces (bodies may be either at rest or move with a constant velocity ). Dynamics : Deals with motion of bodies (accelerated motion ). Kinematic Kinetic

7. Basic Terminologies in Engineering Mechanics The followings are the basic terms which are used in mechanics : Mass: The quantity of matter (solid or fluid) possessed by a body is called mass . Note : The mass of a body remains the same until it is damaged or part of it is physically separated . Time: Time is the measure of succession of events. The successive event selected is the rotation of earth about its own axis and this is called a day. Length: Length is referred to as the distance measured about the largest dimension of an object. Space: It is the geometrical region where the study of a body is conducted, called space . Note: Locate the reference point (origin) and set the coordinate system .

8. Basic Terminologies in Engineering Mechanics Distance : Distance is referred to as a numerical explanation or quantity that identifies how far apart the objects are. Displacement: Displacement is referred to as the shortest distance among the body's initial to the final location. Velocity: It is defined as the rate of change of displacement (i.e., the displacement changes with time). Acceleration : It is defined as the rate of change of velocity (i.e., the velocity changes with time). OR An acceleration is referred to as the second derivative of displacement concerning time. Negative magnitude acceleration is called deceleration.

9. Basic Terminologies in Engineering Mechanics Momentum: The quantity of motion of a moving body is measured as the product of its mass and velocity . Force: Force may be defined as any action that tends to change the state of rest of a body to which it is applied. Force is described as either of push or pull nature. It is a vector quantity The specifications or characteristics of a force are (1) its magnitude , (2) its point of applications , and (3) its direction .

10. Engineering Mechanics: Units

11. Principle of Statics Principles of Statics is a branch of mechanics that studies the effects and distribution of forces on rigid bodies that are at rest. The bodies are considered to be at equilibrium when all the forces sum to zero. To study Statics, we first need to understand the nature of forces, quantify them, and then understand how forces can be applied to bodies . Topics in Statics : Resultant of Force System Equilibrium of Force System Analysis of Trusses Cables Friction Centroids and Centers of Mass Moments of Inertia

12. Scalars and Vectors Scalars: only magnitude is associated. Ex: time, volume, density, speed, energy , mass. Vectors: possess direction as well as magnitude, and must obey the parallelogram law of addition (and the triangle law). Ex: displacement, velocity, acceleration, force , moment, momentum

13. Newton’s laws of mechanics Newton's laws of motion are referred to as fundamental laws of classical mechanics that clarify the association between the motion of bodies and the forces acting upon them. Newton’s First Law of motion: It states that a body at rest will remain at rest, and a body in motion will remain in motion unless an external force acts upon it. In other words, the system's state can only change through the involvement of some external means . First law contains the principle of the equilibrium of forces main topic of concern in Static. Newton’s Second Law of motion: It states that the force acting on an object equals the fractional change of momentum or equal to the product of the system's mass and its gained acceleration. OR A particle of mass “m” acted upon by an unbalanced force “F” experiences an acceleration “a” that has the same direction as the force and a magnitude that is directly proportional to the force. Newton’s Third Law of motion: It states that there is always an equal magnitude and opposite direction reaction for every force effect . Third law is basic to our understanding of Force Forces always occur in pairs of equal and opposite forces.  

14. Force in Engineering Mechanics Force: It may change the motion of a body. i.e. if a body is at rest, the force may set it in motion. And if the body is already in motion, the force may accelerate it . Force is described as either of push or pull nature. It is a vector quantity. Characteristics of a Force: In order to determine the effects of a force, acting on a body, we must know the following Characteristics of a force: Magnitude of the force (e.g., 100 N, 50 N, 20 kN , 5 kN , etc .). The direction of the line along which the force acts. This is also known as the line of action of the force . Nature of the force (i.e., whether the force is a push or a pull). This is denoted by placing an arrowhead on the line of action of the force . The point at which (or through which) the force acts on the body .

15. System of Force When two or more forces act on a body, they are called to form a system of forces. Following systems of forces are important from the subject point of view:

16. System of Force Coplanar forces: The forces, whose lines of action lie on the same plane, are known as coplanar forces. Collinear forces: The forces, whose lines of action lie on the same line, are known as collinear forces. Like collinear forces : Forces acting in same direction, lies on a common line of action and acts in a single plane. Unlike collinear forces : Forces acting in opposite direction, lies on a common line of action and acts in a single plane. Coplanar forces Collinear forces

17. System of Force 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 . Coplanar non-concurrent forces: The forces, which do not meet at one point, but their lines of action lie on the same plane, are known as coplanar non-concurrent forces . Coplanar parallel forces : In this system, the lines of action of all forces lie in the same plane and are parallel to each other. Coplanar non-parallel forces: In this system, the lines of action of all forces lie in the same plane but they are non-parallel to each other .

18. System of Force Non-coplanar forces : Non-coplanar forces are those forces which are not acting from a same plane. Non-coplanar concurrent forces: The forces, which meet at one point, but their lines of action do not lie on the same plane, are known as non-coplanar concurrent forces . Non-coplanar non-concurrent forces: The forces, which do not meet at one point and their lines of action do not lie on the same plane, are called non-coplanar non-concurrent forces.

19. Principle of Physical Independence of Forces and Principle Of Transmissibility of Forces in Engineering Mechanics Principle of Physical Independence of Forces : It states, “If a number of forces are simultaneously acting on a particle, then the resultant of these forces will have the same effect as produced by all the forces .” Principle Of Transmissibility of Forces: The principle of transmissibility states that the point of application of a force can be moved anywhere along its line of action without changing the external reaction forces on a rigid body. Therefore, the points of application of forces may be moved along the line of action to simplify the analysis of rigid bodies.

20. Resultant Force If a number of forces, P, Q, R ... etc. are acting simultaneously on a particle, then 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 called resultant force and the given forces R ... etc. are called component forces. Methods for the Resultant Force: Though there are many methods for finding out the resultant force of a number of given forces, yet the following are important from the subject point of view: Analytical method Graphical method Analytical Methods for the Resultant Force: The resultant force, of a given system of forces, may be found out analytically by the following methods : Parallelogram law of forces Method of resolution

21. Parallelogram Law of Force It states, “If two forces, acting simultaneously on a particle, are 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 that passes through their point of intersection.” Mathematically, resultant force , and where, and = Forces where resultant is required to be found out. = Angle between the forces and . = Angle which the resultant force makes with one of the forces. (i.e., )  

22. Parallelogram Law of Force Consider the parallelogram OACB in which OA and OB represents the force P and Q acting at a point O. The diagonal OC represent the resultant force R. Then consider the , from the Pythagoras theorem, (1) Now consider the , from the Pythagoras theorem , (2) Now, from equation (2) the value of substitute the equation (1),  

23. Parallelogram Law of Force (3) Here, (From figure) Then equation (3), and (here, form )  

24. Parallelogram Law of Force Cases: (1) If i.e., when the forces act along the same line, then (2) If i.e., when the forces act along the same line, then (3) If i.e., when the forces act along the same line, then (4) If , then (since, )  

25. Parallelogram Law of Force Problems: Example 1. Find the magnitude of the two forces, such that if they act at right angles, their resultant is 10 N . But if they Act at 60°, their resultant is 13 N . Example 2 . Two forces act at an angle of 120°. The bigger force is of 40 N and the resultant is perpendicular to the smaller one. Find the smaller force.

26. Triangle Law of Force Triangle Law of Forces is states that “If two forces acting on a body are represented one after another by the sides of a triangle, their resultant is represented by the closing side of the triangle taken from first point to the last point”.

27. Polygon Law of Force It is an extension of Triangle Law of Forces for more than two forces, which states, “If a number of forces acting simultaneously on a particle are 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 the opposite order”.

28. Composition of Forces Composition of Forces: The process of finding out the resultant force, of a number of given forces, is called composition of forces or compounding of forces .

29. Method of Resolution Forces Principle of Resolution: It states, “The algebraic 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 .” Note : The forces are resolved in the vertical and horizontal directions. Method of Resolution for the Resultant Force: Resolve all the forces horizontally and find the algebraic sum of all the horizontal components (i.e ., ). Resolve all the forces vertically and find the algebraic sum of all the vertical components ( i.e., ). The resultant R of the given forces will be given by the equation : The resultant force will be inclined at an angle , with the horizontal, such that Notes : The value of the angle θ will vary depending upon the values of and as discussed below: When is + ve , the resultant makes an angle between 0° and 180°. But when is – ve , the resultant makes an angle between 180° and 360°. 2 . When is + ve , the resultant makes an angle between 0° to 90° or 270° to 360°. But when is – ve , the resultant makes an angle between 90° to 270 °.  

30. Method of Resolution Forces Problems: Example 1. A system of forces are acting at the corners of a rectangular block as shown in Figure below. Determine the magnitude and direction of the resultant force . Example 2 . The following forces act at a point : ( i ) 20 N inclined at 30° towards North of East, (ii) 25 N towards North, (iii) 30 N towards North West, and (iv) 35 N inclined at 40° towards South of West. Find the magnitude and direction of the resultant force .

31. Method of Resolution Forces Example 3. A horizontal line PQRS is 12 m long (as shown in the figure below), where PQ = QR = RS = 4 m. Forces of 1000 N, 1500 N, 1000 N, and 500 N act at P, Q, R, and S respectively, in a downward direction. The lines of action of these forces make angles of 90°, 60°, 45°, and 30° respectively with PS. Find the magnitude, direction, and position of the resultant force. Example 2 . A triangle ABC has its side AB = 40 mm along positive x-axis and side BC = 30 mm along positive y-axis. Three forces of 40 N, 50 N and 30 N act along the sides AB, BC and CA respectively. Determine magnitude of the resultant of such a system of forces .

32. Equilibrium of Forces Equilibrium of Forces : If a body is moving at a constant velocity or the body is at rest then the body is said to be in equilibrium in a state. If a number of forces are acting on the body and its resultant comes out to be zero, then the body is said to be in equilibrium. Such a set of forces, whose resultant is zero, are called equilibrium forces. Principles of Equilibrium: Three important principles of equilibrium are: Two force principle. As per this principle, if a body in equilibrium is acted upon by two forces , then they must be equal, opposite and collinear. Three force principle. As per this principle, if a body in equilibrium is acted upon by three forces , then the resultant of any two forces must be equal, opposite and collinear with the third force. Four force principle. As per this principle, if a body in equilibrium is acted upon by four forces , then the resultant of any two forces must be equal, opposite and collinear with the resultant of the other two forces.

33. Lami’s Theorem It states, “If three coplanar forces acting at a point be in equilibrium , then each force is proportional to the sine of the angle between the other two .” Mathematically, where, P , Q, and R are three forces and α, β, γ are the angles as shown in Figure.  

34. Proof of Lami’s Theorem Consider three coplanar forces P, Q, and R acting at a point O. Let the opposite angles to three forces be α , β and γ as shown in figure. Now complete the parallelogram OACB with OA and OB as adjacent sides as shown in the figure . The resultant of two forces P and Q will be given by the diagonal OC both in magnitude and direction of the parallelogram OACB . Since these forces are in equilibrium, therefore the resultant of the forces P and Q must be in line with OD and equal to R, but in opposite direction . From the geometry of the figure, and  

35. Proof of Lami’s Theorem Then , (1) But Subtracting 180° from both sides of the above equation, (2) From equation (1) and equation (2), we get,  

36. Proof of Lami’s Theorem We know that in , (since Hence Proved   Law of Sines Definition In general, the law of sines is defined as the ratio of side length to the sine of the opposite angle. It holds for all the three sides of a triangle respective of their sides and angles.

37. Lami’s Theorem Problems: Example 1. An electric light fixture weighting 15 N hangs from a point C, by two strings AC and BC. The string AC is inclined at 60° to the horizontal and BC at 45° to the horizontal as shown in Figure. Example 2 . A light string ABCDE whose extremity A is fixed, has weights and attached to it at B and C. It passes round a small smooth peg at D carrying a weight of 300 N at the free end E as shown in Figure. If in the equilibrium position, BC is horizontal and AB and CD make 150° and 120° with BC, determine the, ( i ) Tensions in the portion AB, BC and CD of the string, (ii ) Magnitudes of and ,  

38. Principle of Superposition The principle of superposition of forces states that a single, net, or resultant force has the same effect as the sum of the individual forces acting on an object . The net force, or resultant force, means the vector sum of individual forces. Superposition of Forces Formula: We can express the principle of superposition of forces using the following formula: where is the resultant force or net force equal , to which means we’ve taken the sum of all the forces added together.  

39. Law of Action and Reaction Action means active force. Reaction means reactive force. When a body having a weight W (=mg) is placed on a horizontal plane as shown in Figure, the body exerts a vertically downward force equal to “W” or “mg” on the plane. Then “W” is called action of the body on the plane. According to Newton's 3rd law of motion, every action has an equal and opposite reaction. But action and reaction never act on the same body. So, the horizontal plane will exert equal amount of force “R” on the body in the vertically upward direction. This vertically upward force acting on the body is called reaction force “R” of the plane on the body.

40. Tension and Compression Tension forces pull and stretch material in opposite directions, allowing a rope bridge to support itself and the load it carries. Compression forces squeeze and push material inward, causing the rocks of an arch bridge to press against each other to carry the load .

41. Free body diagram (FBD) Free Body Diagram (FBD ): A Free Body Diagram (FBD) is a graphical representation used in engineering mechanics to visualize the forces and moments acting on a single body or a system of bodies. OR The representation of reaction force on the body by removing all the support or forces act from the body is called free body diagram. Free body diagrams are used to visualize forces and moments applied to a body and to calculate reactions in mechanics problems. These diagrams are frequently used both to determine the loading of individual structural components and to calculate internal forces within a structure. In a free-body diagram, the size of the arrow denotes the magnitude of the force, while the direction of the arrow denotes the direction in which the force acts . Importance: Simplifies Complex Problems: Breaks down complex structures or systems into simpler, manageable parts. Visual Aid: Provides a clear visual representation of all forces and moments, aiding in the understanding of the problem. Foundation for Analysis: Essential for applying fundamental principles of mechanics to solve for unknowns in engineering problems.

42. Free body diagram (FBD) Identify and Represent All Forces : External Forces: Include all forces acting on the body from external sources, such as gravity, applied loads, normal forces, frictional forces, tension in ropes, etc. Reaction Forces: If the body is in contact with other bodies or supports, represent the reaction forces exerted by these bodies or supports. These might include normal forces, frictional forces, or tension/compression in supports. Moments : If there are any moments (torques) acting on the body, represent them with curved arrows . Gravity (Weight): A force acting downward through the center of mass of the block . Normal Force: A perpendicular force exerted by the plane on the block . Frictional F orce: A parallel force exerted by the plane, opposing the motion of the block . Free Body Diagram Problem Diagram

43. Lami’s Theorem Problems: Example 1. A smooth circular cylinder of weight P is lying in a rectangular groove is shown in Figure. Draw the free-body diagram of the cylinder. Example 2 . Two identical rollers each of weight Q= 445N are supported by an inclined plane and a vertical wall as shown in Fig. 2. Assuming smooth surfaces, find the reactions induced at the points of support A , B, and C .

44. Lami’s Theorem Example 3. Two cylinders P and Q rest in a channel as shown in Figure. The cylinder P has diameter of 100 mm and weighs 200 N, whereas the cylinder Q has diameter of 180 mm and weighs 500 N. If the bottom width of the box is 180 mm, with one side vertical and the other inclined at 60 °, determine the Reactions Forces at all the four points of contact. ANS: ; ; ; ;  

45. Moment of a Force Moment of a Force: The moment of a force is equal to the product of the force and the perpendicular distance of the point, about which the moment is required and the line of action of the force . Mathematically, moment , where, P = Force acting on the body, and d = Perpendicular distance between the point, about which the moment is required and the line of action of the force. If the force is in Newton and the distance is in meters, then the units of moment will be Newton-meter (briefly written as N-m). Similarly, the units of moment may be kN -m (i.e. kN × m), N-mm (i.e. N × mm ).  

46. Moment of a Force Types of Moments: The moments are of the following two types: Clockwise moment Anticlockwise moment Clockwise moment: If the force acting on a body rotates the body in the clockwise direction with respect to the axis of rotation, then the moment is called the clockwise moment. Anticlockwise moment: If the force rotates the body in the anti-clockwise direction then the moment is said to be the anti-clockwise moment. Note: The general convention is to take clockwise moment as positive and anticlockwise moment as negative.

47. Varignon’s Theorem Varignon's Theorem : It states, “If a number of coplanar forces are acting simultaneously on a particle, the algebraic sum of the moments of all the forces about any point is equal to the moment of their resultant force about the same point .” Varignon's Theorem, also called the Principle of Moments OR Law of Moments , is a very useful tool in scalar moment calculations. In cases where the perpendicular distance is hard to determine, Varignon's Theorem offers an alternative to finding that distance .

48. Problems Problems: Example 1. A force of 15 N is applied perpendicular to the edge of a door 0.8 m wide as shown in Figure ( a ) Find the moment of the force about the hinge . (b ) If this force is applied at an angle of 60° to the edge of the same door, as shown in Figure, find the moment of this force. Example 2 . A uniform plank ABC of weight 30 N and 2 m long is supported at one end A and at a point B 1.4 m from A as shown in Figure. Find the maximum weight W, that can be placed at C, so that the plank does not topple. ANS:  

49. Friction Friction: Friction is defined as the contact resistance exerted by one body upon a second body when the second body moves or tends to move past the first body . This opposing force, which acts in the opposite direction of the movement of the block, is called force of friction or simply friction . Friction is of the following two types: Static friction: It is the friction experienced by a body when it is at rest. Or in other words, it is the friction when the body tends to move. Dynamic friction: It is the friction experienced by a body when it is in motion. It is also called kinetic friction. The dynamic friction is of the following two types: Sliding friction: It is the friction, experienced by a body when it slides over another body. Rolling friction: It is the friction, experienced by a body when it rolls over another body.

50. Limiting Friction Limiting Friction : This maximum value of frictional force, which comes into play, when a body just begins to slide over the surface of the other body, is known as limiting friction. It may be noted that when the applied force is less than the limiting friction, the body remains at rest, and the friction is called static friction, which may have any value between zero and limiting friction .

51. Normal Reaction It has been experienced that whenever a body, lying on a horizontal or an inclined surface, is in equilibrium , its weight acts vertically downwards through its centre of gravity. The surface, in turn, exerts an upward reaction on the body. This reaction, which is taken to act perpendicular to the plane, is called normal reaction and is, generally, denoted by R or N. It will be interesting to know that the term ‘normal reaction’ is very important in the field of friction, as the force of friction is directly proportional to it .

52. Coefficient of Friction Coefficient of Friction : It is the ratio of limiting friction ( F ) to the normal reaction ( R or N ), between the two bodies, and is generally denoted by μ. Mathematically, coefficient of friction, where, F = Limiting friction, and R or N = Normal reaction between the two bodies  

53. Angle of Friction Angle of Friction: Consider a body of weight W resting on an inclined plane as shown in Figure. We know that the body is in equilibrium under the action of the following forces : Weight (W) of the body, acting vertically downwards, Friction force (F) acting upwards along the plane, and Normal reaction (R) acting at right angles to the plane . The angle of inclination (α) be gradually increased, till the body just starts sliding down the plane. This angle of inclined plane, at which a body just begins to slide down the plane, is called the angle of friction. This is also equal to the angle, which the normal reaction makes with the Vertical.

54. Laws of Static Friction Following are the Laws of Static Friction : The force of friction always acts in a direction, opposite to that in which the body tends to move. The magnitude of the force of friction is exactly equal to the force, which tends to move the body. The ratio of limiting friction (F) and Normal reaction is constant. i.e , The force of friction is independent of the area of contact between the two surfaces. The force of friction depends upon the roughness of the surfaces .  

55. Problems Example 1. A body of weight 300 N is lying on a rough horizontal plane having a coefficient of friction as 0.3. Find the magnitude of the force, which can move the body, while acting at an angle of 25° with the horizontal as shown in the Figure. ANS : Example 2 . A body, resting on a rough horizontal plane, required a pull of 180 N inclined at 30° to the plane just to move it. It was found that a push of 220 N inclined at 30° to the plane just moved the body. Determine the weight of the body and the coefficient of friction . ANS: and   P