Uniform Rotation of Frames 49path r(t) — votr of the point mass relative to the car is a straight line,
since there is no angular displacement of m relative to the car. Thus,
ro(t) = (R + vot)r(t). Note that the polar coordinates of m in frame
O are (r(t),9(t)), where r(t) = R + v 0 t. Hence r(t) = v 0 and f(t) = 0.
Also, Q(t) = OJQ and 6(t) = 0. Formulas (1.3.26) on page 36 provide the
acceleration a = ar + a# of the mass in the frame O, where, with f — Q
and 6 = 0,
ar = -(R + v 0 t)Jl r, ae = 2v 0 wo §• (2.1.14)By (1.3.27) on page 36, r 0 i = -Rw$r. In frame (^0) 1: r^t) = 0. Then
(2.1.14) shows that the acceleration fo(t) is
r 0 (0 = roi(t) + r(t) - v 0 ttj$r + 2v 0 uj 0 d. (2.1.15)
We see that the simple relation (2.1.12) between the accelerations in trans-
lated frames does not correctly describe acceleration of the point mass in
the frame O in terms of the acceleration in the frame 0.
If O is an inertial frame and J^1 is a force acting on the point mass in
O to produce the motion, then, by the Newton's Second Law, m ro = F.
According to (2.1.15),
mf\ = F + (mRul + mvotwl) r — 2mv 0 ujo&- (2.1.16)
Thus, frame 0\ is not inertial. Similar to (2.1.13), inertial forces appear as
correctors to Newton's Second Law: the centrifugal force Fc = {rnRiS^ +
mvoiwg) r and the Coriolis force Fcor = — 2mvou> 0 6. The centrifugal force
prevents the mass from flying off at a tangent because of the rotational
motion of the car and the mass. One component of this force, mRwQ r =
—mroi(t), is related to the motion of the car causing the rotation of the
origin of the frame Oi; the other, mvotuiQ r, takes into account the outward
radial motion of the point. Note that the direction of the centrifugal force
is in the direction of r, and is therefore away from the center and opposite
to the direction of the centripetal acceleration (cf. page 35). Incidentally,
the Latin verb fug ere means "to run away."
The Coriolis force — 2mvocoo Q is somewhat less expected. This force is
perpendicular to the linear path in Oi and ensures that the trajectory of
the point in the rotating frame is a straight radial line despite the rotation
of the frame. This force was first described in 1835 by the French scien-
tist GASPARD-GUSTAVE DE CORIOLIS (1792-1843). His motivation for the
study came from the problems of the early 19th-century industry, such as