Mathematics of Physics and Engineering

Parallel Translation of Frames 47

starting point is the analysis of the relative motion of frames.
The easiest motion is parallel translation, where the corresponding basis
vectors in the frames stay parallel. Consider two such frames with origins
O and Oi respectively. Denote by roi(t) the position vector of 0\ with
respect to O. Let the position vectors of a point mass in frames O and 0\
be r 0 {t) and ri(i) respectively (Figure 2.1.2).

Fig. 2.1.2 Translation of Frames

Clearly, ro(t) — roi(i) + Vi(t), and the absence of relative rotation
allows us to consider this equality in the frame O for each t We can identify
the parallel vectors that have the same direction and length. Since position
vectors in O and 0\ maintain their relative orientation when there is no
relative rotation of the frames, the coordinate systems in the frames O and
0\ are the same. Then we can apply the rule for differentiating a sum
(1.3.3) to obtain simple relations between the velocities and accelerations
in the frames O and 0\

ro(*)=r 0 i(t) + fi(t), r(t)=roi(*) + ri(t). (2.1.12)

If the frame O is inertial, then, by Newton's Second Law, mr(t) = F(t),
where m is the mass of the point, and F is the sum of all forces acting on
the point. The second equality in (2.1.12) then implies

mri(t) = F(t) - mroi(t) (2.1.13)

In effect, there are two forces acting on m in the frame 0. One is the force
F. The other, — mroi, is called a translational acceleration force.
It is an example of an apparent, or inertial, force, that is, a force that
appears because of the relative motion of frames and is not of any of the
four types described on page 43. If roi(t) is constant, then roi(t) = 0 and
the Second Law of Newton holds in 0\, that is, Oi is also an inertial frame.
Thus, all frames moving with constant velocity relative to an inertial frame
are also inertial frames.