Technology & Engineering
Derivative of a Vector
The derivative of a vector is a mathematical operation that calculates the rate of change of a vector with respect to time or another variable. It is commonly used in engineering and physics to describe the motion of objects in space. The derivative of a vector is a vector itself, with each component representing the rate of change of the corresponding component of the original vector.
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eBook - PDFDynamics
Theory and Application of Kane's Method
- Carlos M. Roithmayr, Dewey H. Hodges(Authors)
- 2016(Publication Date)
- Cambridge University Press(Publisher)
Hence, it is meaningless to speak simply of the time derivative of r OP . Clearly, therefore, the calculus used to differentiate vectors must permit one to distinguish between differentiation with respect to a scalar variable in a reference frame A and differentiation with respect to the same variable in a reference frame B . When working with elementary principles of dynamics, such as Newton’s second law or the angular momentum principle, one needs only the ordinary differential calculus of vectors, that is, a theory involving differentiations of vectors with respect to a single scalar variable, generally the time. Consideration of advanced principles of dynamics, such as those presented in later chapters of this book, necessitates, in addition, partial differentiation of vectors with respect to several scalar variables, such as generalized coordinates and motion variables. Accordingly, the present chapter is devoted to the exposition of definitions, and consequences of these definitions, needed in the chapters that follow. 1 2 DIFFERENTIATION OF VECTORS 1.2 1.1 SIMPLE ROTATION Let ˆ a 1 , ˆ a 2 , and ˆ a 3 be a set of right-handed, mutually perpendicular unit vectors fixed in a reference frame A , and let ˆ b 1 , ˆ b 2 , and ˆ b 3 be a similar set of unit vectors fixed in a reference frame B . Suppose that each unit vector ˆ b i initially has the same direction as ˆ a i ( i = 1 , 2 , 3). B is said to undergo a simple rotation relative to A when B is rotated about a line whose orientation relative to A and to B does not change as a result of the rotation. After a particular unit vector parallel to the line of rotation is selected, the angle of rotation q is regarded as positive when a right-handed screw fixed in B , with its axis parallel to the line, advances in the direction of the selected unit vector. Figure 1.1.1 illustrates simple rotation of B in A about ˆ b 1 .
- Kuldip S. Rattan, Nathan W. Klingbeil, Craig M. Baudendistel(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
Derivatives in Engineering CHAPTER 8 8.1 INTRODUCTION This chapter will discuss what a derivative is and why it is important in engineering. The concepts of maxima and minima along with the applications of derivatives to solve engineering problems in dynamics, electric circuits, and mechanics of materials are emphasized. 8.1.1 What Is a Derivative? To explain what a derivative is, an engineering professor asks a student to drop a ball (shown in Fig. 8.1) from a height of y = 1.0 m to find the time when it impacts the ground. Using a high-resolution stopwatch, the student measures the time at impact as t = 0.452 s. The professor then poses the following questions: (a) What is the average velocity of the ball? (b) What is the speed of the ball at impact? (c) How fast is the ball accelerating? 1 m y(t) y = 0 t = 0.452 s t = 0 s Figure 8.1 A ball dropped from a height of 1 m. Using the given information, the student provides the following answers: (a) Average Velocity, v: The average velocity is the total distance traveled per unit time. For example v = Total distance Total time = Δ y Δ t = y 2 − y 1 t 2 − t 1 = − 0 − 1.0 0.452 − 0 223 224 Chapter 8 Derivatives in Engineering = − 1.0 0.452 = −2.21 m/s. Note that the negative sign means the ball is moving in the negative y-direction. (b) Speed at Impact: The student finds that there is not enough information to find the speed of ball when it impacts the ground. Using an ultrasonic motion detector in the laboratory, the student repeats the experiment and collects the data given in Table 8.1. TABLE 8.1 Additional data collected from the dropped ball. t, s 0 0.1 0.2 0.3 0.4 0.452 y(t), m 1.0 0.951 0.804 0.559 0.215 0 The student then calculates the average velocity v = Δy∕Δt in each interval. For example, in the interval t = [0, 0.1], v = 0.951 − 1.0 0.1 − 0 = −0.490 m/s. The average velocity in the remaining intervals is given in Table 8.2. TABLE 8.2 Average velocity of the ball in different intervals.
eBook - PDF- Jerry B. Marion(Author)
- 2013(Publication Date)
- Academic Press(Publisher)
C H A P T E R 2 Vector Calculus 2.1 Introduction The application of vector methods to physical problems most frequently takes the form of differential operations. The rate of change of a vector function with respect to the spatial coordinates or with respect to the time are of particular importance. Such operations allow us, for example, to define the velocity vector of the motion of a particle or to describe the flow properties of a fluid. In this chapter we begin by defining the elemen-tary differential operations which immediately allow us to calculate the velocity and acceleration vectors in the commonly used coordinate systems. Angular velocity is considered next and this leads to a discussion of infinitesimal rotations. Treated next is the important differential operator, the gradient. The fact that the gradient operator may act on vector functions in different ways, leads finally to the divergence and the curl. The chapter concludes with a brief discussion of the simple integral concepts that are necessary in mechanics. 32 2.2 DIFFERENTIATION OF A VECTOR WITH RESPECT TO A SCALAR 33 2.2 Differentiation of a Vector with Respect to a Scalar If a scalar function φ = cp(s) is differentiated with respect to the scalar variable s, then since neither part of the derivative can change under a coordinate transformation, the derivative itself cannot change and must therefore be a scalar. That is, in the x f and x coordinate systems, φ = φ' and s — s', so that άφ = άφ' and ds = ds'. Hence, dcp/ds = d(p'/ds' = (άφ/ds)' (2.1) In a similar manner, we may formally define the differentiation of a vector A with respect to a scalar s.
eBook - PDF- Paul A. Calter, Michael A. Calter(Authors)
- 2011(Publication Date)
- Wiley(Publisher)
No account is taken of any change in direction; hence speed is a scalar quantity. Velocity, on the other hand, is a vector quantity, having both magnitude and direction. The sentence “I drove my car 60 miles per hour on the interstate” is refer- ring to the speed of the car. The statement “I drove my car 60 miles per hour and in the due north direction” is referring to the velocity of the car. For an object moving along a curved path, the magnitude of the velocity along the path is equal to the speed, and the direction of the velocity is the same as that of the tangent to the curve at that point. We will also speak of the components of that velocity in directions other than along the path, usually in the x and y directions. As with average and instantaneous rates of change, we also can have average speed and average velocity, or instantaneous speed or instantaneous velocity. Velocity is the rate of change of displacement and hence is given by the deri- vative of the displacement. If we give displacement the symbol s, then we have the following equation: 1023 The velocity is the rate of change of the displacement. ◆◆◆ Example 9: The displacement of an object is given by where t is the time in seconds. Find the velocity at 1.00 s. Solution: We take the derivative At Graphical Solution: As with other rate of change problems, we can graph the first derivative and determine its value at the required point. Thus the graph for velocity shows a value of at ◆◆◆ Acceleration in Straight-Line Motion Acceleration is defined as the time rate of change of velocity. Like velocity, it is also a vector quantity. Since the velocity is itself the derivative of the displacement, the acceleration is the derivative of the derivative of the displacement, or the second derivative of displacement, with respect to time. We write the second derivative of s with respect to t as follows: 1025 The acceleration is the rate of change of the velocity.
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