INTRODUCTION TO ENGINEERING MECHANICS

Transcription

1 Chapter-1 INTRODUCTION TO ENGINEERING MECHNICS The state of rest and state of motion of the bodies under the action of different forces has engaged the attention of philosophers, mathematicians and scientists for many years. The branch of physical science that deals with the state of rest or the state of motion is termed as Mechanics. Starting from the analysis of rigid bodies under gravitational force and simple applied forces, the mechanics has grown to the analysis of robotics, aircrafts, spacecrafts. Sir Issac Newton ( ), the principal architect of mechanics consolidated the philosophy and experimental findings developed around the state of rest and state of motion of the bodies and put forth them in the form of three laws of motion and he presented law of gravitation. The mechanics based on these laws is called Classical Mechanics or Newtonian Mechanics. Schrödinger ( ) and roglie ( ) showed that Newtonian mechanics fail to explain the behaviour of particles when atomic distances are concerned. They put forth the theory of Quantum Mechanics. lbert Einstein ( ) proved that Newtonian mechanics fail to explain the behaviour of high speed (speed of light) bodies. He put forth the theory of Relativistic Mechanics. However, engineers are keen to use laws of mechanics to actual field problems. pplication of laws of mechanics to field problems is termed as Engineering Mechanics. or all problems between atomic distances to high speed distances, Newtonian mechanics has stood the test of time for many engineering problems and hence that is the mechanics used by engineers. Newtonian mechanics is commonly known as Classical Mechanics also. In this book classical mechanics is used to solve many field problems INTRODUCTION TO STTICS Depending upon the body to which the mechanics is applied, the engineering mechanics is classified as: (a) Mechanics of Solids and (b) Mechanics of luids. The solid mechanics is further classified as mechanics of rigid bodies and mechanics of deformable bodies. The body which will not deform or the body in which deformation can be neglected in the analysis, is called as Rigid ody. The mechanics dealing with rigid bodies at rest is termed as Statics and that dealing with rigid bodies in motion is termed as Dynamics. If the internal stresses developed in a body are to be studied, the deformation of the body should be considered. This field of mechanics is termed as Mechanics of Deformable odies /Strength of Materials/Solid Mechanics. 1

2 2 Engineering Mechanics 1.2. CONCEPT O PRTICLE ND RIGID ODY particle may be defined as an object which has only mass and no size. Such a body cannot exist theoretically. However in dealing with problems involving distances considerably larger compared to the size of the body, the body may be treated as particle, without sacrificing accuracy. Examples of such situations are bomber aeroplane is a particle for a gunner operating from the ground. ship in mid sea is a particle in the study of its relative motion from a control tower. In the study of movement of the earth in celestial sphere, earth is a particle. body is said to be rigid, if the relative portion of any two particles in it do not change under the action of the forces. igure 1.1 (a) points and are the original position in a body. fter application of a system of forces 1, 2 and 3, the body takes the position as shown in igure 1.1 (b). and are the new positions of and. If the body is to be treated as rigid, the relative positions of should be same as. Thus = Many engineering problems may be solved satisfactorily by assuming bodies rigid ( a ) ( b) ig ORCE Newtons first law is that everybody continues in its state of rest or of uniform motion in a straight line unless it is compelled by an external agency acting on it. This leads to the definition of force as the external agency which changes or tends to change the state of rest or uniform linear motion of the body. Newtons second law is 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. Thus according to this law, orce Rate of change of momentum. ut momentum = mass velocity Since mass do not change, orce mass rate of change of velocity mass acceleration i.e., m a = k m a

3 Introduction to Engineering Mechanics 3 In SI system, unit of force is defined as that force which causes 1 kg mass to move with an acceleration of 1 m/sec 2 and is termed as 1 Newton. Hence the constant of proportionality becomes unity. Thus = m a...(1.1) if is in Newtons m is in kg and a is in m/sec 2. In MKS, the unit of force is defined as that force which makes a mass of 1 kg to move with gravitational acceleration g m/sec 2. This unit of force is called kilogram-weight (kg-wt). Gravitational acceleration g is 9.81 m/sec 2 near the earth surface. In all the problems encountered in engineering mechanics the variation in gravitational acceleration is negligible and hence may be taken as 9.81 m/sec 2. Hence the constant of proportionality in Eqn. 1.1 is 9.81, which means 1 kg-wt = 9.81 Newton...(1.2) It may be noted that in public usage kg-wt force is called as kg only. Characteristics of a orce It may be noted that a force is completely defined only when the following four characteristics are specified: Magnitude Point of application Line of action, and Direction In ig. 1.2, is a ladder kept against a wall. t point C, a person weighing 600 N is standing. The force applied by the person on the ladder has the following characters: 600 N C 2 m ig. 1.2 magnitude = 600 N the point of application is at C which is at a distance 2 m from along the ladder. the line of action is vertical and the direction is downward.

4 4 Engineering Mechanics Note that the magnitude of the force is written near the arrow. The line of the arrow shows the line of action and arrow head represents the point of application and the direction of the force TYPES O ORCES Usual types of forces acting on a body may be classified as concentrated load and distributed load. Concentrated orce: If a force is acting over a small area of the body, it is approximated as a point load/concentrated load. It is represented by an arrow over the centroid of small area over which it acts. In ig. 1.2, weight of the person standing on the ladder is a concentrated load/force. Distributed orce: If the force is acting over a region whose dimension cannot be considered negligible with the dimension of the body, it is a distributed load. To count its effect, we should know its intensity at several points over the region and using mathematical integration its effect is to be accounted. The following are the three types of distributed loads: 1. Line distribution, 2. rea distribution and 3. Volume distribution. 1. Line Distribution: When a force is distributed along a line, in a continuous form, it is called line distribution. In ig. 1.3, the cable of the bridge supports the weight of grinder through a set of closely placed verticle wires. The load on cable is a line distributed force. ig. 1.3 Line distribution of force The distributed line load may be uniform or may uniformly vary or may have general nature. It depends upon the way the load is applied. igure 1.4 (a) shows a uniformly distributed load (udl) on a beam. Similarly ig. 1.4 (b) shows uniformly varying load and ig. 1.4 (c) shows a general loading on beams.

5 Introduction to Engineering Mechanics 5 w kn/m L ( a) Uniformly distributed load (UDL) on a beam w 1 kn/m w 2 kn/m L ( b ) Uniformly varying load (UVL) on a beam ( c) General loading on a beam [ Types of line distribution of force] ig. 1.4 Types of Line Loads on a beam 2. rea Distribution: orce may be distributed over an area. or example, in case of a slab the weight of flooring tiles is a force distributed over an area. This force is uniformly distributed over slab. In case of wall of water tank the water pressure is uniformly varying pressure on wall (ig. 1.5). This force is expressed as kilo-newton per square metre. (kn/m 2 ) or newton per mm 2 (N mm 2 ) etc. H γh kn/m 2 ig. 1.5 Water pressure on a wall of tank [ rea distribution of force] 3. Volume Distribution: force which is distributed over the volume of the body is termed as body force also. Self weight of the body, inertia force of a rotating body are the examples of volume distribution of forces. They may be expressed as kn/m 3 or N/mm 3 etc.

6 6 Engineering Mechanics 1.5. SYSTEM O ORCES When several forces act simultaneously on a body, they constitute a system of forces. If all the forces in a system do not lie in a single plane they constitute the system of forces in space. If all the forces in a system lie in a single plane, it is called a coplanar force system. If the line of action of all the forces in a system pass through a single point, it is called a concurrent force system. In a system of parallel forces all the forces are parallel to each other. If the line of action of all the forces lie along a single line then it is called a collinear force system. Various system of forces, their characteristics and examples are given in Table l.1 and shown in ig Table 1.1: System of orces orce System Characteristics Examples Collinear forces Coplanar parallel forces Coplanar like parallel forces Coplanar concurrent forces Coplanar non-concurrent forces Non-coplanar parallel forces Non-coplanar concurrent forces Non-coplanar non-concurrent forces Line of action of all the forces act along the same line. ll forces are parallel to each other and lie in a single plane. ll forces are parallel to each other, lie in a single plane and are acting in the same direction. Line of action of all forces pass through a single point and forces lie in the same plane. ll forces do not meet at a point, but lie in a single plane. ll the forces are parallel to each other, but not in same plane. ll forces do not lie in the same plane, but their lines of action pass through a single point. ll forces do not lie in the same plane and their lines of action do not pass through a single point. orces on a rope in a tug of war. System of forces acting on a beam subjected to vertical loads (including reactions). Weight of a stationary train on a rail when the track is straight. orces on a rod resting against a wall. orces on a ladder resting against a wall when a person stands on a rung which is not at its centre of gravity. The weight of benches in a class-room. tripod carrying a camera. orces acting on a moving bus.

7 Introduction to Engineering Mechanics 7 Collinear Coplanar parallel Coplanar like parallel Coplanar concurrent Coplanar non-concurrent y y x x z Non-coplanar parallel y z Non-coplanar concurrent x z Non-coplanar non-concurrent ig. 1.6 System of orces 1.6. SCLR ND VECTOR Various quantities used in engineering mechanics may be grouped into scalars and vectors. quantity is said to be scalar if it is completely defined by its magnitude alone. Examples of scalars are length, area, time and mass. quantity is said to be vector if it is completely defined only when its magnitude as well as directions are specified. Hence, force is a vector. The other examples of vectors are velocity, acceleration, momentum etc.

8 8 Engineering Mechanics 1.7. TRNSMISSIILITY O ORCE ccording to law of transmissibility of force the state of rest or motion of the rigid body is unaltered if a force acting on the body is replaced by another force of the same magnitude and direction but acting anywhere on the body along the line of action of the replaced force. Let be the force acting on a rigid body at point as shown in ig ccording to the law of transmissibility of force, this force has the same effect on the state of body as the force applied at point. In using law of transmissibility of forces it should be carefully noted that it is applicable only if the body can be treated as rigid. In this text, the engineering mechanics is restricted to study of state of rigid bodies and hence this law is frequently used. Same thing cannot be done in the subject solid mechanics where the bodies are treated as deformable and internal forces in the body are studied. The law of transmissibility of forces can be proved using the law of superposition, which can be stated as the action of a given system of forces on a rigid body is not changed by adding or subtracting another system of forces in equilibrium. Consider the rigid body shown in ig. 1.8 (a). It is subjected to a force at. is another point on the line of action of the force. rom the law of superposition it is obvious that if two equal and opposite forces of magnitude are applied at along the line of action of given force, [Ref. ig. 1.8 (b)] the effect of given force on the body is not altered. orce at and opposite force at form a system of forces in equilibrium. If these two forces are subtracted from the system, the resulting system is as shown in ig. 1.8 (c). Looking at the system of forces in igs. 1.8 (a) and 1.8 (c), we can conclude the law of transmissibility of forces is proved. ig. 1.7 ( a) ( b) ig. 1.8 ( c) QUESTIONS 1. Explain the terms: (a) Particle (b) Rigid body (c) Concentrated force (d) Distributed force 2. State and explain Law of transmissibility of forces. 3. Explain the term orce and list its characteristics.

9 Introduction to Engineering Mechanics 9 4. Explain the terms: (a) Concurrent and non-concurrent force system. (b) Planar and non-planar system of forces. (c) Collinear force system. 5. Distinguish between a scalar and vector. OJECTIVE TYPE QUESTIONS Select the correct option in the following: 1.1. particle has (a) only mass (b) only size (c) both mass and size (d) neither mass nor size If the distance between two bodies is doubled, the force of attraction between them (a) do not change (b) doubles (c) is halved (d) is 1 th of the original value The unit of mass in SI system is (a) pound (b) gramme (c) kilogramme (d) newton One newton is the force that makes (a) 1 gm mass to move with 1 m/sec 2 acceleration (b) 1 kg mass to move with 1 m/sec 2 acceleration (c) 1 kg mass to move with g m/sec 2 acceleration (d) 1 gm mass to move with g m/sec 2 acceleration Which one of the following is not the essential characteristic of a force? (a) time of application (b) line of action and direction (c) point of application (d) magnitude The weight of benches in a class-room constitutes (a) non-coplanar parallel forces (b) coplanar parallel forces (c) coplanar concurrent forces (d) non-coplanar concurrent forces The force which do not meet at one point and their line of action do not lie on the same plane is known as (a) coplanar non-concurrent (b) coplanar concurrent (c) non-coplanar non-concurrent (d) non-coplanar concurrent The forces which meet at one point and their lines of action also lie on the same plane are known as (a) coplanar concurrent (b) coplanar non-concurrent (c) non-coplanar non-concurrent (d) none of the above.