href="#fb3_img_img_fdc18175-78d7-5e6c-8699-cedb2f2b7aa6.png" alt="images"/>. In this special case of the radius vector, , always lying on a fixed plane, its angular velocity vector, , is always perpendicular to the given plane (hence the direction is constant), but can have a time‐varying magnitude, . Hence, Eq. (2.4) yields the following expression for the time derivative of :
(2.7)
When these results are substituted into Eq. (2.3), the following expression for the acceleration of the point, P, is obtained:
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:
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,
href="#fb3_img_img_fdc18175-78d7-5e6c-8699-cedb2f2b7aa6.png" alt="images"/>. In this special case of the radius vector, 