CHAPTER 1 INTRODUCTION DEFINITION OF MECHANICS Mechanics may be defined as the physical science which describes and predicts the conditions of rest or motion of bodies under the action of force systems. In other words, where there is motion or force, there is mechanics. In engineering, mechanics is generally based on Newton s Laws and is often called Newtonian (or Classical) Mechanics after the English scientist Sir Isaac Newton (1642-1727). Conditions involving speed of bodies close to the speed of light (about 300 x 10 6 m/s) and conditions requiring consideration of bodies with extremely small mass and size (such as subatomic particles with sizes in the order of 10-12 m and smaller) cannot be adequately described by Newton s Laws. These extreme conditions are treated in Relativistic Mechanics and Quantum Mechanics. However, for a vast range of problems between these extremes, Newton s Laws give accurate results and are far simpler to apply. Although the fundamental principles of Newtonian Mechanics are surprisingly few in numbers, they have exceedingly wide range of applications. Modern research and development in the fields of vibrations, stability and strength of machines and structures, rocket and spacecraft design, robots, automatic control, engine performance, fluid flow, electrical machines and apparatus are highly dependent upon the basic principles of mechanics. Mechanics is divided into three parts as shown below: Mechs. of Rigid Bodies Mechs. of Deformable Bodies Mechs. of Fluids Statics Theory of Elasticity Compressible Fluids Dynamics Strength of Materials Incompressible Fluids Theory of Plasticity Theory of Failure As seen, mechanics of rigid bodies is divided into two parts as Statics and Dynamics. Statics, is the branch of mechanics that deals with the bodies that are acted on by balanced forces. A force system acting on a body is said to be balanced if it has no tendency to change the state of rest or motion of the body in any way. If a body is in equilibrium, the force system acting on it must be balanced.
Furthermore, a body in a state of equilibrium must be either at rest or moving along a straight path with a constant velocity. Most problems in Statics concern bodies at rest. Statics constitutes a very important part of mechanics since it presents solution methods for the determination of support forces at bodies in equilibrium and establishment of relationships between external loads and internal force distribution. Numerous practical engineering problems involving load carrying members can be solved by using the Principles of Statics. Dynamics is concerned with moving bodies. It is subdivided into two parts as Kinematics and Kinetics. Kinematics deals with the geometry of motion without taking into consideration the loading that causes this motion. Kinetics considers the loads that cause the motion. In rigid body mechanics, it is assumed that bodies are completely rigid nondeformable. In reality though, no structure or machine is completely rigid and it will somewhat deform under the effect of forces it is subjected to. However these deformations are generally extremely small and they neither affect the body in concern nor the equilibrium conditions appreciably. On the other hand, if the fact considered in the analysis is the amount of deformation in the member or its resistance against failure, these deformations gain importance and in this case, such bodies become the subject of mechanics of deformable bodies. The third part of mechanics, mechanics of fluids is concerned with liquids and gases at rest or in motion. A comprehensive knowledge of mathematics is essential for the solution of almost all problems in mechanics. FUNDAMENTAL CONCEPTS Certain concepts and definitions are basic to the study of mechanics and they should be understood at the outset. The basic concepts in mechanics are space, time, mass and force. These concepts cannot be truly defined. They should be accepted on the basis of our intuition and experience, and used as a mental frame of reference for our study of mechanics. In Newtonian mechanics, space, time and mass are absolute quantities, which mean that they are independent of each other (this is not true in Relativistic Mechanics, where the time of an event depends upon its position and the mass of a body varies with its velocity) and cannot be defined in terms of other quantities or in simpler terms. Force is a derived quantity.
Space: is the geometric region occupied by bodies whose positions are described by linear or angular measurements relative to a specific coordinate system. For three dimensional problems, three independent coordinates are needed. For two dimensional problems only two coordinates will be required. Time: is a concept for measuring the succession and the duration of events. Time is not directly involved in the analysis of problems in Statics. Mass: is a measure of the translational inertia of the body, which is its resistance to a change in velocity. Mass can also be thought of as the quantity of matter in a body. The mass of a body affects the gravitational attraction force between it and other bodies. The concept of mass is used to characterize and compare two bodies on the basis of certain fundamental mechanical experiments depending on the definitions given above. For example: 1) Two bodies of the same mass will be attracted by the Earth in the same manner. 2) They will also offer the same resistance to a change in translational motion. Force: A force represents the action of one body on another. Force can be generated either by the direct contact of bodies or by their effect at a distance. Forces always occur in pairs. Forces of a pair are always equal in magnitude and opposite in direction. Force is a vector quantity. The action of a force is characterized by its magnitude, by the direction of its action and by its point of application. A force tends to move a body in the direction of its action (the push pull effect of the force). In addition, a force tends to rotate the body about any axis which does not intersect the line of action of the force and which is not parallel to it (the moment effect of the force). Idealization in Mechanics: The mathematical description of a real engineering problem can become very complex. Thus, idealization (or models) and assumptions are used in mathematics in order to simplify the application of the theory. 1 Particle: is a body of negligible dimensions. In the mathematical sense, a particle is a body whose dimensions are considered to be near zero so that we may analyze it as a mass concentrated at a point. We often choose a particle as a differential element of the body. We may treat a body as a particle when its dimensions are irrelevant to the description of its position or the action of forces applied to it. A body considered as a particle is taken as a unique point, which is generally the mass center of the body. A particle has mass but no shape and dimensions. In so doing, the principles of mechanics are reduced to a rather simplified form, since the geometry of the body will not be involved in the analysis of the problem. The
line of actions of all the forces applied to the body must pass through this point. Forces can only exert push pull effects on a particle. Rigid Body: is an idealized body composed of a large number of particles all of which always remain at fixed distances from each other. In addition to the tendency to move a body in the direction of its application, a force may also tend to rotate a body about an axis. A rigid body is assumed to undergo no deformation under the action of applied forces. Its shape and dimensions remain fixed under all loading conditions and at all times. Point Force: is an idealized force assumed to act at a point on a body. A constant force exerted on a body by another is actually distributed over the area of contact between two bodies. If the area of contact is relatively small, the contact force between the two bodies may be considered as a point (concentrated) force. LAWS OF MECHANICS 1) The Parallelogram Law: Two vectors A v and B v, treated as free vectors, can be replaced by their equivalent R v, which is the diagonal of the parallelogram formed by A v and B v as its two sides, as shown. R v is called the resultant of A v and B v. Hence, the combined effect of two forces A v and B v (for example acting on a particle) is equivalent to the effect of their resultant. O B v A v B v = R v = A v + B v A v R v 2) The Principle of Transmissibility: The effect of a force on a rigid body will remain unchanged if the forced is moved to act on its line of action. In other words, a force may be applied at any point on its given line of action without altering the resultant effects on the rigid body on which it acts. C x x B F v rigid body 3) Newton s First Law: If the resultant force acting on a particle is zero, then the particle is in equilibrium. Stated mathematically as ( F v = 0 ), where F v is the vector sum (the resultant) of all the forces acting on the particle.
4) Newton s Second Law: The acceleration of a particle is proportional to the resultant force acting on it and is in the direction of this force. v v F = ma 5) Newton s Third Law: The forces of action and reaction between interacting bodies are equal in magnitude, opposite in direction and collinear. 6) Newton s Law of Gravitation: This law states that two particles of mass m 1 and m 2 are mutually attracted with equal and opposite forces F v and F v of magnitude F, given by the formula. v m 1 F v F m 2 r m1m2 F = G 2 r where, r is the centroidal distance between the two particles G is the universal constant of gravitation equal to 11 6.673 10 3 m 2 kg s A particular case of great importance is that of the attraction of the Earth on a particle located at its surface. The force F exerted by the Earth on the particle is then defined as the weight W of the particle. W = mg Where, m s the mass of the particle g is the gravitational acceleration of the Earth INERTIAL REFERENCE FRAME To describe accurately the state of rest or motion of particles or rigid bodies, a proper reference frame must be used. An inertial reference frame is one in which Newton s Laws of Motion are valid or highly accurate. Experiments indicate that a nonaccelerating reference frame, which is ether stationary or translating uniformly relative to the Sun is an approximate inertial reference frame in most situations. DIMENSIONS, UNITS AND DIMENSIONAL HOMOGENEITY A dimension is a qualitative description of a physical quantity, which can be quantified by a certain standard of measure called unit. The seven base
dimensions are length, time, mass, electric current, thermodynamic temperature, amount of matter and luminous intensity. We use only length, time and mass in Statics. A unit of a dimension is a standard of measure for the quantitative description of dimension. The primary unit of a base dimension is called a basic unit. The SI uses meter (m), second (s), kilogram (kg), ampere (A), Kelvin (K), mole (mol) and candela (cd), respectively for the above seven base dimensions. A useful check of our computations may be obtained if we carry out the computations with the units as well as the numerical values. The Law of Dimensional Homogeneity states that because natural phenomena proceed with no regard for man-made units, basic equations representing physical phenomena must be valid for all systems of units. And all equations derived analytically from these fundamental laws must also be dimensionally homogeneous. According to this Law: Every grouping in a physical equation must have the same dimensional representation. In other words, Apples cannot be equal to oranges. Numerical Accuracy: The accuracy of the solution of a problem depends upon two items: 1) The accuracy of the given data 2) The accuracy the computations performed The solution cannot be more accurate than the less accurate of these two items. For example: If the loading of a bridge is known to be 375000 N with a possible error of 500 N either way, the relative error which measures the degree of accuracy of the data is 500 = 0.13% 375000 It would then, be meaningless in computing the reaction at one of the bridge supports, to record it as 71610 N. The accuracy of the solution cannot be greater than 0.13%, no matter how accurate the computations are, and the possible error in the answer may be as large as 0.13 71600 = 93 N 100 The solution should be properly recorded as 71600 ± 93 N 71600 ± 100 N