Читать книгу Foundations of Space Dynamics - Ashish Tewari - Страница 28

As a special case, consider the motion of a point, P, in a fixed plane described by the radius vector, , which is changing in time. The vector is drawn from a fixed point, o, on the plane, to the moving point, P, and hence denotes the instantaneous radius of the moving point from o. The instantaneous rotation of the vector is described by the angular velocity, , which is fixed in the direction given by the unit vector , normal to the plane of motion. Thus we have the following in Eq. (2.4):

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

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, .