Ashish Tewari - Foundations of Space Dynamics

The net velocity of the point, P , is defined to be the time derivative of the radius vector, , which is expressed as follows according to the chain rule of vector differentiation:

(2.6)

and consists of the radial velocity component, , and the circumferential velocity component, . Similarly, when the chain rule is applied to the velocity, , the result is the net acceleration of the moving point, P , which is defined to be the time derivative of , or the second time derivative of . 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:

(2.8)

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