4 NEWTON’S LAWS OF MOTION 4.9 Frames of reference
of our coordinate system, or rotate the coordinate axes. Clearly, in general, the
components of vectors r, r 1 , and r 2 are going to be modified by this change in
our coordinate scheme. However, Fig. 12 still remains valid. Hence, we conclude
that the vector equation (4.26) also remains valid. In other words, although
the individual components of vectors r, r 1 , and r 2 are modified by the change
in coordinate scheme, the interrelation between these components expressed in
Eq. (4.26) remains invariant. This observation suggests that the independence
of the laws of physics from the arbitrary choice of the location of the underlying
coordinate system’s origin, or the equally arbitrary choice of the orientation of
the various coordinate axes, can be made manifest by simply writing these laws
as interrelations between vectors. In particular, Newton’s second law of motion,
f = m a, (4.27)is clearly invariant under shifts in the origin of our coordinate system, or changes
in the orientation of the various coordinate axes. Note that the quantity m (i.e.,
the mass of the body whose motion is under investigation), appearing in the
above equation, is invariant under any changes in the coordinate system, since
measurements of mass are completely independent of measurements of distance.
We refer to such a quantity as a scalar (this is an improved definition). We con-
clude that valid laws of physics must consist of combinations of scalars and vec-
tors, otherwise they would retain an unphysical dependence on the details of the
chosen coordinate system.
Up to now, we have implicitly assumed that all of our observers are stationary(i.e., they are all standing still on the surface of the Earth). Let us, now, relax
this assumption. Consider two observers, O and OJ, whose coordinate systems
coincide momentarily at t = 0. Suppose that observer O is stationary (on the
surface of the Earth), whereas observer OJ moves (with respect to observer O)
with uniform velocity v 0. As illustrated in Fig. 34 , if r represents the displacement
of some body P in the stationary observer’s frame of reference, at time t, then the
corresponding displacement in the moving observer’s frame of reference is simply
rJ = r − v 0 t. (4.28)The velocity of body P in the stationary observer’s frame of reference is defined