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The net acceleration of the point, P, parallel to the instantaneous radius vector, , is identified from Eq. (2.8) to be the following:
The direction of the term is always towards the instantaneous centre of rotation (i.e., along ). The other radial acceleration term, , is caused by the instantaneous change in the radius, , and is positive in the direction of the increasing radius (i.e., away from the instantaneous centre of rotation).
The component of acceleration along the vector in Eq. (2.8) is perpendicular to both and , and is given by
In terms of the polar coordinates, , we have ; hence the motion is resolved in two mutually perpendicular directions, (), where is a unit vector along the direction of increasing (called the circumferential direction), defined by
(2.9)
Thus the rotating frame, , constitutes a right‐handed triad. In this rotating coordinate frame, the motion of the point, P, is represented as follows:
(2.10)
(2.11)
(2.12)
(2.13)
It is clear from Eq. (2.13) that in the rotating coordinate system, , the acceleration along the instantaneous radius vector, , is given by
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, .
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,
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