Last time we derived 2 vector equations for a multiparticle system's net translational and rotational motion. We now introduce a common multiparticle system whose constraints reduce it to exactly 6 degrees of freedom, which we can hope to solve exactly with our 2 vector equations.
We consider a rigid body: a system of particles constrained so that all interparticle distances remain fixed during all possible motions. That is, the object can move as a unit in space, but cannot deform in shape.
We counted the degrees of freedom, i.e. how many ways we can place the rigid body in space. We pick three reference points (noncollinear) in the body. To locate the body, we 1) specify the location of particle 1 (it could be anywhere, 3 d.f. ); 2) then specify the location of particle 2 (it lies on the sphere at fixed distance from 1, with 2 d.f.); 3) then specify the location of particle 3 (on the circle -- intersection of spheres -- at fixed distance from both 1 and 2, with only 1 d.f.). Any remaining points in the rigid body have locations which are entirely determined by their fixed distances from particles 1, 2, and 3. Thus the rigid body has exactly 6 degrees of freedom.
We then stated Euler's theorem (as generalized by Chasles): that the most general motion possible for a rigid body is a sum of a rotation about some axis fixed in the body, plus a translation of the body as a whole. The proof involves an exploration of orthogonal matrices which we defer to your next mechanics course.
We then specialized to consider motion of a rigid body, when 1 point in the body (which we take to be the origin O) is fixed. Then the most general motion of the rigid body is rotation about some axis through O.
This motion looks different to different observers, particularly to
those who are body-fixed (rotating along with the body) and
those who are space-fixed (stationary in space, and thus in an
inertial frame). We explored how a vector which is stationary relative
to the body looks to an inertial (space-fixed) observer; then how a
more general time-dependent vector, as measured in the body frame,
looks to an inertial observer. In particular, we explored how the
vector's apparent rate of change looks different in the two frames,
since one is rotating with an angular frequency 
with
respect to the other.
We noted that only the inertial observer can expect to see Newton II obeyed -- the rotating (body-fixed) observer uses an accelerated coordinate system. However, such a body-fixed observer (which includes us, since we live on a rotating Earth) may strongly desire to describe the accelerations he measures in terms of an ``effective'' Newton II. To do so, he must account for the rotation of his coordinate frame by introducing ``fictitious'' forces. We derived these -- the Coriolis and centrifugal forces -- from Newton II in the inertial frame, and briefly discussed their physical effects.
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KB