52 Kinematics and Dynamics of a Point Mass
on page 62. Let the frame 0\ rotate relative to the frame O so that the
corresponding rotation vector u> is fixed in the frame O. Because of this
rotation, the basis vectors in Oi depend on time when considered in the
frame O: ti = »i(t), Jx =J 1 (), i = M*)- % (2-1.18),
dii/dt = u> x ii, djx/dt = u> x j 1 , dki/dt = u> x k\. (2.1.19)Denote by ro(t) and ri(t) the position vectors of the point mass in O and
Oi, respectively. If P is the position of the point mass, then ro(t) = OP,
r(t) = 0\P, and, with O = 0\, we have ro(t) — r(t) for all t. Still, the
time derivatives of the vectors are different: ro(t) ^ ri(t) because of the
rotation of the frames. Indeed,
ri {t) = xx (t) §i + yi (t) jx + zx (t) ku
r 0 (t) = x 1 (t)i 1 {t)+y 1 (t)j 1 {t)+zl(t)K 1 {t);recall that the vectors i%, jx, k\ are fixed relative to Oi, but are moving
relative to O. Let us differentiate both equalities in (2.1.20) with respect to
time t. For the computations of ri(t), the basis vectors are constants. For
the computations of ro(t), we use the product rule (1.3.4) and the relations
(2.1.19). The result is
r 0 (t) = ri(t)+u;xr 1 (t). (2.1.21)
EXERCISE 2.1.1? Verify (2.1.21).
There is nothing in the derivation of (2.1.21) that requires us to treat ro
as a position vector of a point. Accordingly, an alternative form of (2.1.21)
can be stated as follows. Introduce the notations Do and D\ for the time
derivatives in the frames O and 0\, respectively. Then, for every vector
function R = R(t), the derivation of (2.1.21) yields
D 0 R{t) = DiR(t) +UJX R(t). (2.1.22)Relation (2.1.21) is a particular case of (2.1.22), when R is the position
vector of the point. We now use (2.1.22) with R = r 0 , the velocity of the
point in the fixed frame, to get the relation between the accelerations. Then
(i) ZVo = r 0 ; (ii) by (2.1.21), Dxr 0 = f 1 +wxt,i; (hi) also by (2.1.21)
w x r 0 = w x (ri + w x n). Collecting the terms in (2.1.22),
ro = ri + 2uxri+wx(wxj'i). (2.1.23)Therefore, the acceleration in the fixed frame has three components: the
acceleration fi in the moving frame, the Coriolis acceleration acor =