(^64) Kinematics and Dynamics of a Point Mass
Hint: direct computation, (h) Consider the vector OP = ro(i) = x(t) i +
y(t)j + z(t) k rotating in the frame O. Verify that, with the above definition
of u, we have
r 0 (t) = w(t) x r 0 (i). (2.1.41)
Hint: write relation (2.1.37) as ro(t) = U{t)f\, where f\ = x\ i + j/i j + z\ k.
Then rx = UT{t)r 0 (t) and r 0 {t) = U{t)ri.
Note that (2.1.41) agrees with (2.1.18) on page 51 when u(t) is constant
and justifies the above definition of u>(t) as a rotation vector.
Denote by ro(t) the position vector of a point P in the frame O, and
by TI(£), the position of the same point in the frame 0. Since the frames
have the same origin, we have ro(t) = ri(t) for all t. On the other hand,
because of the relative rotation of the frames, the values of ro(t) and ri(i)
are different. As we did earlier on page 52, denote by Do and D\ the
derivatives with respect to time in the frames O and 0\, respectively. If the
point P is fixed in the rotating frame 0\ and 0\P = r\ = x\ £1+2/1 Ji+zi ki
is the position vector of P in 0\, then Dir 0 (t) = r(t) = 0, and (2.1.41)
implies Doro(t) = ro(t) — UJ x ro(t).
Similar to the derivation of (2.1.21), we can show that if the point P
moves relative to the frame 0\, then
r 0 (t) = ri(t)+u>xr 0 (t). (2.1.42)
Therefore, for every vector R — R(t), expressed as functions in frames O
and Oi, both denoted by R
D 0 R(t) = DiR(t) + u(t) x R(t). (2.1.43)
EXERCISE 2.1.18? Verify (2.1.43). Hint: write R(t) = x(t)i 1 (t)+y(t)j 1 (t) +
z(t) ki(t) and differentiate this equality using the product rule. Since the vectors
i> Ji! ki are fixed in the frame 0\, you can use equality (2.1.41) to compute
the time derivatives of these vectors. Also, by definition, D\R(t) = x(t)i(t) +
»()5i(t)+ ()«i ()•
Remark 2.1 Let us stress that the vector ro(t) = x(t)i+y(t)j+z(t) k is
the same as the vector ri(t) = zi(£) ?i(£) +2/1 (£)&() +21 (£) ki(t): both are
equal to OP even if the point P moves relative to the frame 0. As a result,
ro(i) =/= U(t)ri(t). There is no contradiction with (2.1.37), because the
components of the vectors TQ, r\ are defined in different frames and cannot
be related by a matrix-vector product. What does follow from (2.1.37) is
coco
(coco)
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