Mathematics of Physics and Engineering

General Accelerating Frames 65

the equality ro(t) = U{t)r{t), where f{t) = x(t)i + y{t) j + z{t) k.
Also keep in mind that, despite the equality of the vectors ro(t) = ri(t), the
curve C defined by ro in the frame O is different from the curve C\ defined
by 7*1 in the frame 0. For example, if P is fixed in 0\, then C\ is jus a
single point.

EXERCISE 2.1.19r Consider the special case of a uniform rotation of frame
0\ relative to frame O so that the origins of the frames coincide, k = ki,
and the rotation vector is u> = W3K. Calculate U(t) and U{t). Show that
the matrix fl = U(t)UT(t) has the form

/O -w 3 0\

n = w 3 0 0.

\o 0 0/

For the point P fixed in the rotating frame and having the position vector
in the fixed frame ro(t) = x(t) i + y(t) j + z(t) k\ show that

r 0 {t) = Cl,ro{t) = -u 3 y{t)i + u> 3 x(t)3 = w x r 0 (t).
As a result, you recover relation (2.1.18) we derived geometrically on page
51.
To continue our analysis of rotation, assume that the vector function
w = u){t) is differentiable in t. Then we can set R = ro = A)f*o in (2.1.43)
and use (2.1.42) to derive the relation between the accelerations of the point
in the two frames:

ro(*) = ri(t) + 2w(t)xfi(t) + w(t)x (u(t)xr 0 (t))+w(t)xr 0 (t); (2.1.44)

as before, ro and ri are the position vectors of the point in the frames O
and 0\, respectively.

EXERCISE 2.1.20? (a) Verify (2.1.44). Hint: r 0 (t) = D 0 r 0 (t) = D 0 (fi +u> x
r 0 ) = D\r\ +uxri+uxro+ux(fi+ux ro). Note that both u) and ro are
defined in the same frame O, so the product rule (1.3.6), page 26, applies, (b)
Verify that (2.1.44) can &e written as
Mt)=ri(t) + 2u(t)xri(t)+w(t)x(u(t)xri(t))+w(t)xr 1 (t). (2.1.45)

Finally, assume that the point 0\ is moving relative to O so that the
function roi(t) = 00\ is twice continuously differentiable. Then we have
r(t) = roi(t) + ri(t). Consider the parallel translation of the frame O
with the origin O' at 0\, and define the rotation vector u to describe the