(^48) Kinematics and Dynamics of a Point Mass
As an EXAMPLE, consider a golf ball in a moving elevator. We fix the
frame O on the ground, and 0\, on the elevator, and select the usual carte-
sian coordinate systems in both frames so that the corresponding coordinate
vectors are parallel. Assume that the elevator is falling down with the grav-
itational acceleration g, so that roi(t) = —gk, and the ball is falling down
inside the elevator, also with the gravitational acceleration g so that ro{t) —
-gk. Then (2.1.12) shows that r(t) = ro(t) — roi(t) = —gk + gk — 0.
Therefore, f*i(t) is constant, and if r(to) = 0, then r[(t) = 0 for all t > to
(or until the elevator hits the ground). An observer in the elevator would
see the ball as fixed in the elevator frame 0\: the translational acceleration
force -mfoi(i) compensates the gravitational force m?o(t), and the ball
behaves as weightless in the elevator frame.
2.1.3 Uniform Rotation of Frames
Note that it is the absence of rotation that allowed us to use the differenti-
ation rule (1.3.3) in the derivation of relation (2.1.12). This and other rules
of differentiation no longer apply if the frames are rotating relative to each
other, and relation (2.1.12) must be modified.
We start with the analysis of uniform rotation, that is, rotation with
constant angular speed around a fixed axis. As a motivational example,
consider a car driving with constant angular speed u>o in a circle with radius
R and center at O. Consider an object (a point mass m) moving inside the
car with constant radial speed VQ relative to O, and rotating together with
the car with constant angular speed UQ around O. Introduce a new (non-
inertial) rotating frame with origin 0\ inside the car and the coordinate
basis vectors i\ = r, j 1 = 0; see Figure 2.1.3. As before, let ro and n be
the position vectors of the point mass in the frames O and Oi, respectively,
and denote 00\ by roi- Assume that T*I(0) = 0.
' * —•-0 0[ r
Fig. 2.1.3 Rotation of FramesBy construction, roi = Rf. For a passenger riding in the car, the