Ashish Tewari - Foundations of Space Dynamics

and consists of the acceleration towards the instantaneous centre of rotation, , as well as that away from the instantaneous centre, . Of the acceleration normal to the instantaneous radius vector , the term is caused by a change of the radius in the rotating coordinate frame, , whereas the other term, , is due to the variation of the angular velocity of rotation, , in the same rotating frame.

An alternative representation of the motion of the point P is via Cartesian coordinates , , measured in a reference frame whose axes are fixed in space. Let us consider as such a fixed, right‐handed coordinate system with , and being the constant plane of rotation. The radius vector and its time derivatives in the fixed frame are then given by

(2.14)

(2.15)

(2.16)

In general, a time variation of the radius vector, , gives rise to a radial acceleration, , 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, .

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 light 2). 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.