MODULE 1

MODULE 1: FUNDAMENTAL CONCEPTS IN MECHANICS SCIENCE OF MECHANICS same manner; they also offer the same resistance to a change in translational • Mechanics – is a branch of the physical motion. sciences that is concerned with the state of rest or motion of bodies that are subjected to the Force action of forces. • Force – represents the action of one body to • Three (3) Branches: another. A force can be exerted by: o Mechanics of Rigid Bodies o Actual contact – push or pull o Mechanics of Deformable Bodies o At a distance – gravitational and o Fluid Mechanics magnetic forces. • Divided into Two (2) Areas: • Characterized by its point of application, its o Statics – deals with bodies that are magnitude and its direction. either at rest or move with a constant velocity. • Represented by a vector. o Dynamics – is concerned with the accelerated motion of bodies. Note: The concept of force is not independent of the other three. The resultant force acting on a body is • Additional Information about Mechanics: related to the mass of the body and to the manner in o Mechanics – is defined as the science which its velocity varies with time. that describes and predicts the conditions of rest or motion of bodies 3 IMPORTANT IDEALIZATIONS IN MECHANICS: under the action of forces. o Mechanics – is an applied physical Particle science since it aims to explain and predict physical phenomena and thus • A particle has a mass, but a size that can be to lay the foundations for engineering neglected. applications. o Example: The size of the earth is insignificant compared to the size of its 4 FUNDAMENTAL CONCEPTS AND PRINCIPLES orbit, and therefore the earth can be modeled as a particle when studying its • The basic concepts used in mechanics are orbital motion. space, time, mass, and force. • When a body is idealized as a particle, the Space principles of mechanics reduce to a rather simplified form since geometry of the body will • Space – The concept of space is associated not be involved in the analysis of the problem; with the position of a point P. We can assume that it occupies a single point o How can we determine point P? in space. ▪ We can define the position P by providing three lengths Rigid Body measured from a certain reference point, or origin, in the • A rigid body can be considered as a three given directions. combination of a large number of particles in ▪ These lengths are known as which all the particles remain at a fixed distance the Coordinates of P. from one another, both before and after applying a load. Time • This model is important because the body’s • Time – To define an event, it is not sufficient to shape does not change when a load is applied, indicate its position in space. We also need to so we do not have to consider the type of specify the time of the event. material from which the body is made. Mass • In most cases the actual deformation occurring in structures, machines, • Mass – the concept of mass is used to mechanisms, and the like are relatively small, characterize and compare bodies on the basis and the rigid-body assumption is suitable for of certain fundamental mechanical analysis. experiments. o Example: Two bodies of the same Concentrated Force mass are attracted by the earth in the

• A concentrated force represents the effect of 4) Newton’s Law of Gravitation a loading which is assumed to act at a point on a body. – two particles of mass M and m are mutually attracted with equal and opposite forces F and -F of magnitude F, • We can represent a load by a concentrated given by the formula: force, provided the area over which the load is applied is very small compared to the overall F = GMm/r2 size of the body. o Example: A contact force between a o Where: wheel and the ground. ▪ r – distance between the two particles 6 FUNDAMENTAL PRINCIPLES ▪ G – a universal constant called the Constant of Gravitation 1) Parallelogram Law for the Addition of Forces – this introduces the idea of an action exerted at a – two forces acting on a particle may be replaced by a distance and extends the range of application of single force, called their resultant, obtained by drawing Newton’s third law: the diagonal of the parallelogram with sides equal to the given forces. o The action F and the reaction -F in the figure are equal and opposite, they 2) Principle of Transmissibility have the same line of action. – the conditions of equilibrium or of motion of a rigid SYSTEM OF UNITS body remain unchanged if a force acting at a given point of the rigid body is replaced by a force of the same • Associated with the four fundamental concepts magnitude and same direction, but acting on a different discussed above are the so-called kinetic point, provided that the two forces have the same line units (i.e., the units of lenth, time, mass, and of action. force). 3) Newton’s Three Laws of Motion • Basic Units: units of length, time, and mass. • Derived Units: unit of force. – formulated by Sir Isaac Newton. International System of Units (SI Units) First Law – Law of inertia • The base units are: • If the resultant force acting on a particle is zero, o Length – meter (m) the particle remains at rest (if originally at rest) ▪ It was originally defined as one or moves with a constant speed in a straight line ten-millionth of the distance (if originally in motion). from the equator to either pole, ▪ It is now defined as 1, 650, Second Law – Acceleration 763.73 wavelengths of the orange -red light • If the resultant force acting on a particle is not corresponding to a certain zero, the particle has acceleration proportional transition in an atom of to the magnitude of the resultant and in the krypton-86. direction of this resultant force. This law can be o Mass – kilogram (kg) stated as: ▪ The kilogram which is approximately equal to the F = ma mass of 0.001 m3 of water, is o Where: defined as the mass of a platinum-iridium standard kept ▪ F – resultant force acting on at the International Bureau of the particle Weights and Measures at Sevres, near Paris, France. ▪ m – mass of the particle o Time – second (s) ▪ a – acceleration of the particle ▪ It was originally chosen to represent 1/86,400 of the Third Law – Action-Reaction mean solar day. ▪ It is now defined as the • The forces of action and reaction between the duration of 9, 192, 631, 770 bodies on contact have the same magnitude, cycles of the radiation same line of action, and opposite sense. corresponding to the transition between two levels of the

fundamental state of the cesium-133 atom. o Force – Newton (N) ▪ The unit of force is a derived unit. It is called the newton (N) and is defined as the force that gives acceleration of 1 m/s² to a body of mass 1 kg. • SI units – are said to form an absolute system of units. o This means that the three base units chosen are independent of the location where measurements are made. The meter, the kilogram, and the second may be used anywhere on the earth; they may even be used on another planet and still have the same significance. VECTOR QUANTITIES • Mathematical expressions possessing magnitude and direction, which add according to the parallelogram law. • Represented by arrows in diagram • Its magnitude defines the length of the arrow used to represent it. • Used to represent force acting on a given particle has well-defined point of application— namely, the particle itself Fixed or Bound Vector • Cannot be moved without modifying the conditions of the problem Free Vector • May be freely moved in space (e.g. couples)

Sliding Vector • The sum of the two vectors can be found by arranging P and Q in tip-to-tail fashion and then • Can be moved along their lines of action. connecting the tail of P with the tip of Q. Equal Vector Coplanar Vectors • Two vectors that have the same magnitude and • Vectors that are contained in the same plane. the same direction are said to be equal, whether • Their sum can be obtained graphically. or not they also have the same point of • It is simpler to use the triangle rule than application. applying parallelogram method. • May be denoted by the same letter. Note: The repeated applications of triangle rule could be Negative Vector eliminated and obtain the sum of the three vectors directly. The result would be unchanged if the vectors Q • The negative vector of a given vector P is and S is replaced by their sum Q + S which expresses defined as a vector having the same magnitude the fact that vector addition is associative. In the case of as P and a direction opposite to that of P. two vectors, we can write P + Q + S = S + (P + Q) = S + (Q + P) = S + Q +P. This expression shows that the order • Denoted by -P. in which several vectors are added together is ▪ P and -P are commonly reffered to as immaterial. equal and opposite vectors. ▪ P+(-P)=0 SUBTRACTION OF VECTORS SCALAR QUANTITIES • Defined as the addition of the corresponding negative vector. • Physical quantities that have magnitude but not direction such as volume, mass, or energy • represents the action of one body on another • characterized by its point of application, its • Represented by plain numbers to distinguish from vectors magnitude, and its direction ADDITION OF VECTORS FORCE • The sum of the two vectors P and Q is - represents the action of one body on another obtained by attaching the two vectors to the same point A and constructing a parallelogram, - characterized by its point of application, its using P and Q as two adjacent sides. magnitude, and its direction o The diagonal that passes through A represents the sum of the vectors P FORCE ON A PARTICLE and Q. o Denoted by P + Q. • Have the same point of application • The sum of three vectors (or more vectors) MAGNITUDE OF A FORCE P, Q, and S is obtained by first adding the vectors P and Q and then adding the vectors S • characterized by a certain number of units to the vector P+Q o Segment of Line – represents the force; may be chosen to represent the Note: The magnitude of the vector P+Q is NOT, in magnitude of force general, equal to the sum P+Q of the magnitudes of the vectors P and Q. Since the parallelogram constructed on DIRECTION OF A FORCE the vectors P and Q does not depend upon the order in which P and Q are selected, we conclude that the • defined by its line of action and the sense of addition of two vectors is commutative, and we write the force P+Q=Q+P. o Line of Action - the infinite straight line along which the force acts; Triangle Rule characterized by the angle it forms with some fixed axis • An alternative method for determining the sum o Sense of the Force – should be of two vectors. indicated by an arrowhead • Derived from the parallelogram law. Note: It is important in defining a force to indicate its sense. Two forces having the same magnitude and the same line of action but different sense, such as the forces shown above, will have directly opposite effects on a particle.

Space Diagram • sketch showing the physical conditions of the problem Free-Body Diagram • method to choose a significant particle and draw a separate diagram showing this particle and all forces acting on it • name derives from the fact that when drawing the chosen body, or particle, it is “free” from all other bodies in the actual situation.