alt="images"/>, which is resolved in a fixed coordinate frame, , without resorting to any rotational acceleration terms. Such a coordinate frame whose axes are fixed in space is termed an inertial reference frame, and the acceleration measured by such a frame is termed the inertial (or “true”) acceleration. The inertial acceleration, , can be thought of as being directed towards (or away from) an instantaneous centre of rotation, which itself could be a moving point. For example, a point moving along an arc of a constant radius, , at a constant angular rate, , has its acceleration directed towards the arc's centre, .
2.3 Newton's Laws
In 1687 Newton gave his three famous laws of motion, which are valid for the motion of all objects (unless they are moving at speeds comparable to the speed of light). Stated briefly, they are the following:
1 An object continues to move in a straight line at a constant velocity, unless acted upon by a force applied to it by another object.
2 The time rate of change of the velocity (called the acceleration) of an object is directly proportional to the force applied to the object. The constant of proportionality is a property of the object, called the mass.
3 If an object, A, applies a force on another object, B, then B applies a force on A of the same magnitude, but opposite in direction to that applied by A.
The consequences of these laws are profound, as they govern the motion of all objects (which are moving quite slowly when compared to the speed of light2). The first law implies that there is no absolute position in space, because two observers moving in parallel straight lines at a constant speed, , relative to one another find two events separated in time to take place at different positions. Such observers can be regarded to be located at the origins of two different reference frames, and , such that the relative motion takes place along the parallel axes, , and . The distances travelled by a moving object during the time, , measured in the two frames are different, as given by the following Galilean transformation:
(2.17)
Newton's laws applied to a moving object are equally valid in the two reference frames, and ; hence they are both referred to as inertial reference frames. Another consequence of the first law is that it postulates an absolute quantity called the time, which is the same in all reference frames, and is therefore unaffected by the motion. The second law assigns a property called the mass to all material objects, which can be determined by measuring the force applied to the object and the corresponding acceleration experienced by it, while the third law defines the force applied by two isolated objects upon each other.
2.4 Particle Dynamics
A particle is defined to have a finite mass but infinitesimal dimensions, and is therefore regarded to be a point mass. Since a particle has negligible dimensions, its position in space is completely determined by the radius vector, , measured from a fixed point, o, at any instant of time, . The components of are resolved in a right‐handed reference frame with origin at o, having three mutually perpendicular axes denoted by the unit vectors , and , and are defined to be the Cartesian position coordinates, , of the particle along the respective axes, , as shown in Fig. 2.1. The velocity, , of the particle is defined to be the time derivative of the radius vector, and given by
Figure 2.1 The position vector, , of a particle resolved in an inertial reference frame using Cartesian coordinates, ().
If the reference frame, , used to measure the velocity of the particle is at rest, then the components of the velocity, , resolved along the axes of the frame, , and , are simply the time derivatives of the position coordinates, , and , respectively. However, if the origin, , of the reference frame itself is moving with a velocity, , and the frame, , is rotating with an angular velocity, , with respect to an inertial frame3, then the velocity of the moving reference frame must be vectorially added to that of the particle in order to derive the net velocity of the particle in the stationary frame as follows:
alt="images"/>, which is resolved in a fixed coordinate frame, 