Page 76 - Dynamics of Mechanical Systems

0593_C03_fm Page 57 Monday, May 6, 2002 2:03 PM

3

Kinematics of a Particle

3.1 Introduction
Kinematics is a study of motion without regard to the cause of the motion. Often this motion
occurs in three dimensions. For such cases, and even when the motion is restricted to two
dimensions, it is convenient to describe the motion using vector quantities. Indeed, the
principal kinematic quantities of position, velocity, and acceleration are vector quantities.
In this chapter, we will study the kinematics of particles. We will think of a “particle”
as an object that is sufficiently small that it can be identified by and represented by a point.
Hence, we can study the kinematics of particles by studying the movement of points. In
the next chapter, we will extend our study to rigid bodies and will think of a rigid body
as being simply a collection of particles. We begin our study with a discussion of vector
differentiation.

3.2 Vector Differentiation

Consider a vector V whose characteristics (magnitude and direction) are dependent upon
a parameter t (time). Let the functional dependence of V on t be expressed as:

V = V() t (3.2.1)

Then, as with scalar functions, the derivative of V with respect to t is defined as:

V( t + ∆ t) − V()
D t
dV dt = Lim (3.2.2)
∆ t→0 t ∆
The manner in which V depends upon t depends in turn upon the reference frame in
ˆ
ˆ
which V is observed. For example, if V is fixed in a reference frame R, then in R, V is
ˆ
independent of t. If, however, R moves relative to a second reference frame, R, then in
R V depends upon t (time). Hence, even though the rate of change of V relative to an
ˆ
observer in R is zero, the rate of change of V relative to an observer in R is not necessarily
zero. Therefore, in general, the derivative of a vector function will depend upon the
reference frame in which that derivative is calculated. Hence, to avoid ambiguity, a super-
script is usually added to the derivative symbol to designate the reference frame in which

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