# Troublshooting your pc for dummies - Bourg D.M.

ISBN 0-596-00006-5

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[(M)/(L3)][(L)/(T)]2[L2]

and collecting the dimensions in the numerator and denominator, we get the following form:

(ML2L2)/(L3X2) .

Canceling dimensions that appear in both the numerator and denominator yields

M(L/T2)

which is consistent with the form shown earlier for resistance, Rf. This exercise also reveals that the empirical term, Cf, for the coefficient of friction must be nondimensional, that is, it is a constant number with no units.

With that, let’s take a look at some more common physical quantities that you will be using, along with their corresponding symbols, component dimensions, and units in both the English and SI systems. This information is summarized in Table 1-L

Units and Measures 3

Table 1-1. Common Physical Quantities and Units

BH&i ‘^Symbol ' ' ? Dimensions ' ''tUn'rts,Englisib " ‘ h ' -j" Units, SI

Acceleration, linear a ’ lit1 ft/s1 '! m/s2

Acceleration, angular a radian/7"1 radian/s2 radian/s2

Density P MlI3 slug/ft3 kg/m3

Force F M(UT2) pound, lb newton, N

Kinematic viscosity u L2/T tf/s m2/s

Length 2{orjf,y,z) L feet,ft meters, m

Mass m M Slug kilogram, kg

Moment (torque) M (see footnote3) M(L2/T2) ft-lb N-m

Mass Moment of Inertia I ML1 1 b-ft-s2 kg-m2

Pressure P MULT1) lb/ft2 N/m2

Time T T seconds, s seconds, s

Velocity, linear V LIT ft/s m/s

Velocity, angular w radian/7" radian/s radian/s

Viscosity M MI(LT) lbs/ft2 Ns/m2

a In general, I will use a capital M to represent a moment (torque) acting on a body and a lowercase m to represent the mass of a body. If I'm referring to the basic dimension of mass ina general sense, that is, referring to the dimensional components of derived units of measure, I'll use a capita! Usually, the meanings of these symbols will be obvious based on the context in which they are used; however, I will specify their mea nings in cases in which ambiguity may exist.

Coordinate System

Throughout this book I will refer to a standard right-handed Cartesian coordinate system when specifying position in 2D or 3D space. In two dimensions I will use the coordinate system shown in Figure 1-la, in which rotations are measured positive counterclockwise.

(0,0) I

? x

a. Two dimensions

Figure 1-2. Right-Handed Coordinate System

4 j Chapter 1: Basic Concepts

In three dimensions 1 will use the coordinate system shown in Figure 1-lb, in which rotations about the x-axis are positive from positive y to positive z, rotations about the y-axis are positive from positive z to positive x, and rotations about the z-axis are positive from positive x to positive y.

Vectors

Let me take you back for a moment to your high school math class and review the concept of vectors. Essentially, a vector is a quantity that has both magnitude and direction. Recall that a scalar, unlike a vector, has only magnitude and no direction. In mechanics, quantities such as force, velocity acceleration, and momentum are vectors, and you must consider both their magnitude and direction. Quantities such as distance, density and viscosity ate scalars.

With regard to notation, I’ll use boldface type to indicate a vector quantity such as force, F. When referring to the magnitude only of a vector quantity I’ll use lightface type. For example, the magnitude of the vector force, F, is F with components along the coordinate axes, Fx, Fy, and Fz. In the code samples throughout the book, I’ll use the * symbol to indicate vector dot product or scalar product operations, depending on the context, and I’ll use the " symbol to indicate vector cross product operations. .

Because we will be using vectors throughout this book, it is important that you refresh your memory on the basic vector operations, such as vector addition, dot product, and cross product. For your convenience, so that you don’t have to drag out that old math book, I’ve included a summary of the basic vector operations in Appendix A. This appendix provides code for a Vector class that contains all the important vector math functionality Further, I explain how to use specific vector operations, such as the dot product and cross product operations, to perform some common and useful calculations. For example, in dynamics you’ll often have to find a vector perpendicular, or normal, to a plane or contacting surface; you use the cross product operation for this task. Another common calculation involves finding the shortest distance from a point to a plane in space; here you use the dot product operation. Both of these tasks are described in Appendix A, and I encourage you to review it before delving too deeply in the example code presented throughout the remainder of this book.

Mass, Center of Mass, and Moment of Inertia

The properties of a body, mass, center of mass, and moment of intertia, collectively called mass properties, are absolutely crucial to the study of mechanics, as the linear and angular* motion of a body and a body’s response to a given force are functions of these mass properties. Thus, to accurately model a body in motion, you need to

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