2.4 Components of Vectors

In Section 2.3 we added vectors by using a scale diagram and by using properties of right triangles. Measuring a diagram offers only very limited accuracy, and calculations with right triangles work only when the two vectors are perpendicular. So we need a simple but general method for adding vectors. This is called the method of components. To define what we mean by the components of a vector A, we begin with a rectangular (Cartesian) coordinate system of axes (Figure 1). We then draw the vector with its tail at O, the origin of the coordinate system. We can represent any vector lying in the xy-plane as the sum of a vector parallel to the x-axis and a vector parallel to the y-axis. These two vectors are labeled A_x and A_y in Fig. 1.17a; they are called the component vectors of vector of A, and their vector sum is equal to A. In symbols,

Figure 1

A = A_x + A_y

Since each component vector lies along a coordinate-axis direction, we need only a single number to describe each one. When A_x points in the positive x-direction, we define the number A_x to be equal to the magnitude of A_x. When A_x points in the negative x-direction, we define the number A_x to be equal to the negative of that magnitude (the magnitude of a vector quantity is itself never negative). We define A_y in the same way.

Figure 2

We can calculate the components of the vector A if we know its magnitude A and its direction. We’ll describe the direction of a vector by its angle relative to some reference direction. In Fig. 1.17b this reference direction is the positive x-axis, and the angle between vector A and the positive x-axis is (theta). Imagine that the vector A originally lies along the postive x-axis and that you then rotate it to its correct direction, as indicated by the arrow in Fig. 2 on the angle theta. If this rotation is from the postive x-axis toward the postive y-axis, then the angle is postive, otherwise negative.

2.3 Vector Addition And Subtraction

Suppose a particle undergoes a displacement A followed by a second displacement B. The final result is the same as if the particle had started at the same initial point and undergone a single displacement C. We call displacement C the vector sum, or resultant, of displacements A and B. We express this relationship symbolically as

C = A + B

The boldface plus sign emphasize that adding two vector quantities require a geometrical process and is not same process as adding two scalar quantities as 2 +3 =5. In vector addition we usually place the tail of second vector at the head, or tip, of first vector. If we make the displacement A and B in reverse order, with B first and A second, the result is same. Thus,

C = B + A and A + B = B + A

This shows that the order of terms in a vector sum doesn’t matter. In other words, vector addition obeys the commutative law.

If A and B are both drawn with their tails at the same point, vector C is the diagonal of a parallelogram constructed with A and B as two adjacent sides.

It’s a common error to conclude that if C = A + B, then the magnitude C should equal the magnitude A plus the magnitude B. In general, this conclusion is wrong; for the vectors shown in figure above, you can see that C < A +B. The magnitude of A + B depends on the magnitude of A and B and on the angle between A and B. Only in the special case in which A and B are parallel is the magnitude of A + B equal to the sum of the magnitudes of A and B. When the vectors are anti-parallel, the magnitude of C equals the difference of the magnitudes of A and B. Be careful about distinguishing between scalar and vector quantities, and you’ll avoid making errors about the magnitude of a vector sum. When we need to add more than two vectors, we may first find the vector sum of any two, add this vectorially to the third, and so on.

We can subtract vectors as well as add them. To see how, recall that vector -A has has the same magnitude as A but the opposite direction. We define the difference A - B of two vectors A and B to be the vector sum of A and -B:

A - B = A + (-B)

A vector quantity such as a displacement can be multiplied by a scalar quantity (an ordinary number). The displacement 2A is a displacement (vector quantity) in the same direction as the vector A but twice as long; this is the same as adding A to itself. In general, when a vector A is multiplied by a scalar c, the result cA has magnitude |c|A (the absolute value of c multiplied by the magnitude of the vector A). If c is positive, cA is in the same direction as A; if c is negative, cA is in the direction opposite to A.

2.2 Conventions for Equations And Diagrams

A vector quantity V is represented by a line segment, having direction of the vector and having an arrowhead to indicate the sense. The length of the directed line segment represent to some convenient scale the magnitude of vector |V|and is printed with light-face italic type V. For example, we may choose a scale such that an arrow one inch long represents a force of twenty pounds.

In scalar equations, and frequently on diagrams where only the magnitude of a vector is labeled , the symbol will appear light-face italic type. Boldface type is used for vector quantities whenever the directional aspect of is  a part of its mathematical representation. When writing vector equations, always be certain to preserve the mathematical distinction between vectors and scalars. In handwritten work, use a distinguishing mark for each vector quantity such as an underline, V, or an arrow over the symbol to take the place of boldface type in print.

2.1 Scalars And Vectors

We use two kinds of quantities in mechanics - scalars and vectors. Scalar quantities are those with which only a magnitude is associated. Examples of scalar quantities are time, volume, density, energy, and mass. Vector, quantities on the other hand, possess direction as well as magnitude and must obey the parallelogram law of addition (to be discussed later). Examples of vector quantities are displacement, velocity, acceleration, force, moment, and momentum. Speed is a scalar. It is the magnitude of velocity, which is vector. Thus, velocity is defined by a direction as well as speed. 

Vectors representing physical quantities can be classified as free, sliding or fixed.

A free vector is one whose action is not confined to or associated with a unique line in space. For example, if a body moves without rotation, then the movement or displacement of any point in the body may be taken as a vector. This vector describes equally well the direction and magnitude of displacement of every point in the body. Thus, we may represent the displacement of such a body by a free vector.

A sliding vector has a unique line of action in space but not a unique point of application. For example, when an external force acts on a rigid body, the force can be applied at any point along its line of action without changing its effect on the body as whole, and thus it is a sliding vector.

A fixed vector is one for which a unique point of application is specified. The action of force on a deformable or non-rigid body must be specified by a fixed vector at the point of application of the force. In this instance of the forces and deformations within the body depend on the point of application of the force, as well as on its magnitude and line of action.

1.10 Rounding Off Number

For numerical calculations, the accuracy obtained from solution of problem generally can never be better than the accuracy of the problem data. This is what is to be expected, but often handheld calculators or computer involve more figures in the answer than the number of significant figures used for the data. For this reason, a calculated result should always be "rounded off" to an appropriate number of significant figures.

To convey appropriate accuracy, the following rules for rounding off a number to n significant figures apply:

• If the n + 4 digit is less than 5, n + 1 digit and other following it are dropped. For example, 2.326 and 0.451 rounded to n = 2, significant figures would be 2.3 and 4.5.
• If n + 1 digit is equal to 5 with zero following it, then round off the nth digit to an even number. For example, 1.245 and 0.8655 rounded to n = 3 significant figures become 1.24 and 0.866.
• If the n + 1 digit is greater than 5 or equal to 5 with any nonzero digits following it, increase the nth digit by 1 and drop n + 1 digits and other following it. For example, 0.723 87 and 565.500 3 rounded off to n = 3 significant figures become 0.724 and 566. 

1.9 A Remark on Significant Digits

In engineering calculations, the information given is not known to more than a certain number of significant digits, usually three digits. Consequently, the results obtained cannot possibly be accurate to more significant digits. Reporting results in more significant digits implies greater accuracy than exists, and it should be avoided. For example, consider a 3.75-L container filled with gasoline whose density is 0.845 kg/L, and try to determine its mass. Probably the first thought that comes to your mind is to multiply the volume and density to obtain 3.16875 kg for the mass, which falsely implies that the mass determined is accurate to six significant digits. In reality, however, the mass cannot be more accurate than three significant digits since both the volume and the density are accurate to three significant digits only. Therefore, the result should be rounded to three significant digits, and the mass should be reported to be 3.17 kg instead of what appears in the screen of the calculator. The result 3.16875 kg would be correct only if the volume and density were given to be 3.75000 L and 0.845000 kg/L, respectively. The value 3.75 L implies that we are fairly confident that the volume is accurate within ±0.01 L, and it cannot be 3.74 or 3.76 L. However, the volume can be 3.746, 3.750, 3.753, etc., since they all round to 3.75 L (Fig. 1–62). It is more appropriate to retain all the digits during intermediate calculations, and to do the rounding in the final step since this is what a computer will normally do. When solving problems, we will assume the given information to be accurate to at least three significant digits. Therefore, if the length of a pipe is given to be 40 m, we will assume it to be 40.0 m in order to justify using three significant digits in the final results. You should also keep in mind that all experimentally determined values are subject to measurement errors, and such errors will reflect in the results obtained. For example, if the density of a substance has an uncertainty of 2 percent, then the mass determined using this density value will also have an uncertainty of 2 percent. You should also be aware that we sometimes knowingly introduce small errors in order to avoid the trouble of searching for more accurate data. For example, when dealing with liquid water, we just use the value of 1000 kg/m3 for density, which is the density value of pure water at 0°C. Using this value at 75°C will result in an error of 2.5 percent since the density at this temperature is 975 kg/m3. The minerals and impurities in the water will introduce additional error. This being the case, you should have no reservation in rounding the final results to a reasonable number of significant digits. Besides, having a few percent uncertainty in the results of engineering analysis is usually the norm, not the exception.

1.8 Prefixes And Rules For Their Use

When a numerical quantity is either very large or very small, the units used to define its size may be modified using a prefix. Some of the prefixes used in the SI system are shown in table.  Each represent a multiple or submultiples which, if applied successively, moves the decimal point to every third place. Except for some volume and area measurement, the use of these prefixes is to be avoided in sciences and engineering.

Multiple	Exponential Form	Prefix	SI Symbol
1 000 000 000	109	giga	G
1 000 000	106	mega	M
1 000	103	kilo	K

Submultiple
0.001	10-3	milli	M
0.000 001	10-6	micro	µ
0.000 000 001	10-9	nano	N

Rules for Use

The following rules are given for the proper use of various SI symbols:

• A symbol is never written with a plural “s”, since it may be confused with the unit for second (s).
• Symbols are always written in lowercase letters, with the following exceptions: symbols for the two largest prefixes, giga and mega, are capitalized as G and M, respectively, and symbols named after an individual are also capitalized.
• Quantities defined by several units which are multiple of one another are separated by a dot to avoid confusion with prefix notation, as indicated by N = 1 kg.m.s^-2. Also, m.s (meter-second), whereas ms (milli-second).
• The exponential power represented for a unit having a prefix refers to both the unit and its prefix.
• With the exception of the base unit the kilogram, in general avoid the use of a prefix in the denominator of composite function.

1.7 The International System of Units

The International System of Units, abbreviated SI, is accepted in the United States and throughout the world, and is modern version of metric system. By international agreement, SI units will in time replace other systems. In SI, the kilogram (kg) for mass, meter (m) for length, and second (s) for time are selected as the base units, and the unit Newton (N) for force is derived from the preceding three by F = ma. Thus force (N) = mass (Kg) × acceleration (ms^-2) or

N = kg. ms^-2

Thus, 1 newton is the force reqiured to give a mass of 1 kilogram an acceleration of 1 meter per second per second.

1.6 Units of Measurement

In mechanics we use four fundamental quantities called dimensions. These are length, mass, time and force. The units used to measure these quantities cannot all be chosen independently because they must be consistent with Newton' second law. Because of this, the units used to measure these quantities cannot be all selected arbitrarily. The equality F = ma is maintained only if three of the four basic units, called base units, are arbitrarily defined and the fourth is then derived from the equation. Although there are many different systems of units, only SI system of units will be used in this blog.

1.5 Weight And Mass

According to Newton's law of gravitational attraction, any two particles or bodies have a mutual attractive or gravitational force acting between them. In the case of a particle located at or near the surface of the earth, however, the only gravitational force having sizable magnitude is that between the earth and the particle. Consequently, this force, termed the weight, will be the only gravitational force considered in our study of mechanics. 

From Law of gravitation, we can develop an approximate expression for finding the weight of a particle having a mass m₁ = m. If we assume the earth to be a nonrotating sphere of constant density and having a mass of m₂ = M_e, then if r is the distance between the earth’s centre and the poarticle we have

Letting g = GM_e/r², yields

W = mg

By comparison with F = ma, we term g the acceleration due to gravity. Since it depnds on r, it can be seen that the weight of a body is not an absolute quantity.  However, for most engineering calculations, g is determined at sealevel amd at a latitude of 45^o, which is considered as “standard location.”

1.4 Newton's Law of Motion and Law of Gravitational Attraction

The entire subject of rigid-body mechanics is formulated on the basis of Newton's three laws of motion, the validity of which is based on experimental observation. Sir Issac Newton was the first to state correctly the basic laws governing the motion of a particle and to demonstrate their validity. They apply to the motion of particle as measured from non-accelerating reference frame.

First Law. A particle originally at rest, or moving in a straight line with constant velocity, will remain in this state provided the particle is not subjected to an unbalanced force.

Second Law. A particle 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 F. If F is applied to particle of mass m, this law may be expressed mathematically as                                                                                                                                F = ma
Third Law. The mutual forces of action and reaction between two particles are equal, opposite and collinear.

Law of Gravitational Attraction. Shortly after formulating his three laws of motion, Newton postulated a law governing the mutual attraction between any two particles. Stated mathematically,

where

F = force of gravitation between the two particles

G = universal constant of gravitation; according to experimental evidence, G = 66.73×10^-12 N.m².kg^-2

m₁, m₂ = mass of each of the two particles.
r = distance between the two particles.

1.3 Basic Concepts

The following concepts and definitions are basic to the study of mechanics, and they should be understood at the outset.

·        Space is the geometric region occupied by the bodies whose positions are described by linear and angular measurements relative to a coordinate system. For three-dimensional problems, three independent coordinates are required. For two dimensional problems, only two coordinates are required.

·        Time is the measure of succession of events and is a basic quantity in dynamics. Time is not directly involved in the analysis of statics problem.

·        Mass is a measure of the inertia of a body, which is resistance to a change of 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. This force appears in many applications of statics.

·        Force is the action of body on another.A force tends to move a body in the direction of its action. The action of a force is characterized by it magnitude, by the direction of its action, and by the point of its application.

·        Idealizations or Models are used in mechanics in order to simplify application of theory. A few of the more important idealizations will now be defined. Others that are noteworthy will be discussed at points where they are needed. 

·        A particle has a mass, but size that can be neglected. For example, the size of mars is insignifcant compared to the size of its orbit, and therefore the mars can be modeled as a particle when studying its orbital motion. When a body is idealized as a particle, the principles of mechanics reduce to a rather simplified form since the geometry of the body will not be involved in the analysis of the problem.

·        A rigid-body can be considered as a combination of a large number of particles in which all particles remain at a fixed distance from one another both before and after applying a load. As a result, the material properties of any body that is assumed o be rigid will not have to be considered when analyzing the forces acting on the body. In most cases the actual deformations occuring in structures, machines, mechanisms, and the like are relatively small, and the rigid-body assumption is suitable for analysis.

·        A concentrated force represents the effect of a loading which is assumed to act at a point on a body. We can represent a load by a concentrated force, provided the area over which the load is applied very small compared to the overall size of the body. An example would be the contact force between a wheel and the ground.

1.2 Historical Development of Mechanics

Mechanics is the oldest of physical sciences. The earliest recorded writing on the subject of mechanics are those of Aristotle and Archimedes some two thousand years ago on the principle of lever and buoyancy. Substantial progress come later with the formulation of the laws of vector combination of forces by Stevinus who also formulated most of the principles of statics. The first investigation of a dynamics problem is credited to Galileo (1564-1620) for his experiments of falling stones.  The accurate formulation of laws of motion, as well as law of gravitation, was made by Newton (1642-1727), who also conceived the idea of infinitesimal in mathematical analysis. Substantial contributions to the development of mechanics were also made by da Vinci, Varignon, Euler, D'Alembert, Langrage, Laplace and others.

You may visit http://www.sciencebits.com/MR_Short_History for details.

1.1 Mechanics

Mechanics can be defined as that branch of physical sciences concerned with the state of rest or motion of bodies that are subjected to the action of forces. In general, this subject is subdivided into three branches.

• Rigid Body Mechanics
• Deformable Body Mechanics
• Fluid Mechanics

In this blog, we start from rigid-body mechanics since it forms a suitable basis for the design and analysis of many structural, mechanical and electrical devices encountered in engineering. Also, rigid-body mechanics provides part of the necessary background for the study of mechanics of deformable bodies and the mechanics of fluid.

Rigid-body mechanics itself is divided into two areas, statics and dynamics. Statics deals with the equilibrium of bodies, that is, those that are either at rest or move with constant velocity; where dynamics is concerned with the accelerated motion of bodies. Although statics can be considered as a special case of dynamics in which acceleration is zero, statics deserves a separate treatment since many objects are designed with the intention that they remain in equilibrium.

In writing of this post, we took definitions from "Engineering Mechanics 11th Edition by R.C Hibbeler".