Eleventh Edition Vector Mechanics For Engineers Statics and Dynamics Ferdinand P. Beer Late of Lehigh University E. Russell Johnston, Jr. Late of University. Beer & Johnston Vector Mechanics for Engineers Statics 9th Mazen Nawareg. Page 1 11/27/08 PM user-s As indicated in its preface, Vector Mechanics for Engineers: Statics is designed for the first course in statics offered in the sophomore year of college.

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Vector Mechanics for Engineers: Dynamics. Sev e nth. Editio n. 2 - Introduction. • Rigid Body Dynamics: distance between points is fixed, no size or shape. PDF | On Jul 1, , Arun Kumar Samantaray and others published Lecture Notes on Engineering Mechanics - Vector Mechanics, Equivalent. Vector Mechanics for Engineers: Statics. EighthEdition. 2 Contents. Introduction. Resultant of Two Forces. Vectors. Addition of Vectors. Resultant of Several.

Euler's laws provide extensions to Newton's laws in this area. The concepts of angular momentum rely on the same calculus used to describe one-dimensional motion. The rocket equation extends the notion of rate of change of an object's momentum to include the effects of an object "losing mass". There are two important alternative formulations of classical mechanics: Lagrangian mechanics and Hamiltonian mechanics.

These, and other modern formulations, usually bypass the concept of "force", instead referring to other physical quantities, such as energy, speed and momentum, for describing mechanical systems in generalized coordinates.

The expressions given above for momentum and kinetic energy are only valid when there is no significant electromagnetic contribution. In electromagnetism, Newton's second law for current-carrying wires breaks down unless one includes the electromagnetic field contribution to the momentum of the system as expressed by the Poynting vector divided by c2, where c is the speed of light in free space.

Limits of validity[ edit ] Domain of validity for Classical Mechanics Many branches of classical mechanics are simplifications or approximations of more accurate forms; two of the most accurate being general relativity and relativistic statistical mechanics. Geometric optics is an approximation to the quantum theory of light , and does not have a superior "classical" form.

When both quantum mechanics and classical mechanics cannot apply, such as at the quantum level with many degrees of freedom, quantum field theory QFT is of use. Viscous fluid 3. The mass of a body will not change unless the body is damaged and part of it is physically separated. When a body is taken out in a spacecraft, the mass will not change but its weight may change due to change in gravitational force.

Even the body may become weightless when gravitational force vanishes but the mass remain the same. Time Time is the measure of succession of events.

Vector Mechanics for Engineers ( Dynamics )

The successive event selected is the rota- tion of earth about its own axis and this is called a day. To have convenient units for various activities, a day is divided into 24 hours, an hour into 60 minutes and a minute into 60 seconds. Clocks are the instruments developed to measure time. A point in the space may be referred with respect to a predetermined point by a set of linear and angular measurements.

If coordinates involve only in mutually perpendicular directions they are known as Cartesian coordinates. If the coordinates involve angle and distances, it is termed as polar coordinate system. Length It is a concept to measure linear distances. The diameter of a cylinder may be mm, the height of a building may be 15 m. Actually metre is the unit of length.

However depending upon the sizes involved micro, milli or kilo metre units are used for measurement. A metre is defined as length of the standard bar of platinum-iridium kept at the International Bureau of Weights and Measures.

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To overcome difficulties of accessibility and reproduction, now meter is defined as Referring to Fig. Velocity B The rate of change of displacement with respect to time is defined as velocity.

Acceleration Fig. It is a well known fact that each particle can be subdivided into molecules, atoms and electrons. It is not possible to solve any engineering problem by treating a body as a conglomeration of such discrete particles. The body is assumed to consist of a continuous distribution of matter. In other words, the body is treated as continuum. Rigid Body A body is said to be rigid, if the relative positions of any two particles in it do not change under the action of the forces.

In Fig. Particle A 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 — A bomber aeroplane is a particle for a gunner operating from the ground. 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. Consider the two bodies in contact with each other. Let one body applies a force F on another.

According to this law the second body develops a reactive force R which is equal in magnitude to force F and acts in the line same as F but in the opposite direction.

The force of attraction between any two bodies is directly proportional to their masses and inversely proportional to the square of the distance between them. According to this law the force of attraction between the bodies of mass m1 and mass m2 at a distance d as shown in Fig.

Let F be the force acting on a rigid body at point A as shown in Fig. According to the law of transmissibility of force, this force has the same effect on the state of body as the force F ap- plied at point B.

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In using law of transmissibility of forces it F should be carefully noted that it is applicable only A if the body can be treated as rigid. In this text, the F B engineering mechanics is restricted to study of state of rigid bodies and hence this law is frequently used. 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.

It is subjected to a force F at A. B is another point on the line of action of the force. From the law of superposition it is obvious that if two equal and opposite forces of magnitude F are applied at B along the line of action of given force F, [Ref. Force F at A and opposite force F at B form a system of forces in equilibrium. If these two forces are subtracted from the system, the resulting system is as shown in Fig.

Looking at the system of forces in Figs. This law was formulated based on experimental results. Though Stevinces employed it in , the credit of presenting it as a law goes to Varignon and Newton This law states that if two forces acting simultaneously on a body at a point are presented in magnitude and direction by the two adjacent sides of a parallelogram, their resultant is represented in magni- tude and direction by the diagonal of the parallelogram which passes through the point of intersection of the two sides representing the forces.

Then according to this law, the diagonal AD represents the resultant in the direction and magnitude. Then AD should represent the resultant of F1 and F2. Then we have derived triangle law of forces from fundamental law parallelogram law of forces. The Triangle Law of Forces may be stated as 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.

If more than two concurrent forces are acting on a body, two forces at a time can be combined by triangle law of forces and finally resultant of all the forces acting on the body may be obtained. A system of 4 concurrent forces acting on a body are shown in Fig. Let it be called as R2.

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The units of all other quantities may be expressed in terms of these basic units. The units of length, mass and time used in the system are used to name the systems. Using these basic units, the units for other quantities can be found. From eqn.

Hence the constant of proportionality k becomes units. Unit of force can be derived from eqn. Gravitational acceleration is 9. In all the problems encountered in engineering mechanics the variation in gravitational acceleration is negligible and may be taken as 9.

Hence the constant of proportionality in eqn. Unit of Constant of Gravitation From eqn. Thus if two bodies one of mass 10 kg and the other of 5 kg are at a distance of 1 m, they exert a force 6.

Now let us find the force acting between 1 kg-mass near earth surface and the earth. Hence the force between the two bodies is 6. Thus weight of 1 kg mass on earth surface is 9.

Compared to this force the force exerted by two bodies near earth sur- face is negligible as may be seen from the example of 10 kg and 5 kg mass bodies. Denoting the weight of the body by W, from eqn. Any body falling freely near earth surface experiences this acceleration.

The value of g is 9. The prefixes used in SI system when quantities are too big or too small are shown in Table 1. Table 1. At point C, a person weighing N is standing. The force applied by N B the person on the ladder has the following characters: — magnitude is N C — the point of application is at C which is 2 m from A along the ladder.

Note that the magnitude of the force is written near the arrow. The line of the arrow shows the line of application and the arrow head represents the point of application and the A direction of the force. 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.

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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 1. Coplanar parallel forces All forces are parallel to each other System of forces acting on a beam and lie in a single plane.

Coplanar like parallel All forces are parallel to each other, Weight of a stationary train on a forces lie in a single plane and are acting rail when the track is straight. Coplanar concurrent forces Line of action of all forces pass Forces on a rod resting against a through a single point and forces wall.

Coplanar non-concurrent All forces do not meet at a point, Forces on a ladder resting against forces but lie in a single plane. Non-coplanar parallel All the forces are parallel to each The weight of benches in a class- forces other, but not in same plane.There is certain area of contact, which is, however, small compared to the other dimensions in the problem.

Watch out for typos! Hence there are two school of academicians, one advocating for classical approach and the other advocating for vector approach.

In fluid dynamics, the Navier-Stokes equations are equations, that describe the three-dimensional motion of viscous fluid substances. Referring to Fig. Here is a list of topics in this chapter. At point C, a person weighing N is standing.