alt="images"/>, and lies in the instantaneous plane of rotation normal to . The rotation of is indicated by the right‐hand rule, where the thumb points along , and the curled fingers show the instantaneous direction of rotation,1. The time derivative of is therefore expressed as follows:
where the term represents a unit vector in the original direction of , and is the change normal to caused by its rotation. Equation (2.2) will be referred to as the chain rule of vector differentiation in this book.
Similarly, the second time derivative of is given by the application of the chain rule to differentiate as follows:
where is the instantaneous angular velocity at which the vector is changing its direction. Hence, the second time derivative of is expressed as follows:
The bracketed term on the right‐hand side of Eq. (2.5) is parallel to , while the second term on the right‐hand side is perpendicular to both and . The last term on the right‐hand side of Eq. (2.5) denotes the effect of a time‐varying axis of rotation of .
2.2 Plane Kinematics
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:
