We have previously studied the translational motion
of rigid bodies by analyzing the motion of their center of mass.
We have also studied the rotation of rigid bodies
about a fixed axis.
This week, we will consider more complicated examples
of rigid body motion, where translation and rotation are
both occurring simultaneously.
One example is, if I toss a rigid body through the air
but give it a spin at the same time
so that it tumbles in space as it falls.
A second example is a wheel that is
rolling along a flat surface.
Its location is translating, but at the same time,
it is also spinning about its axis.
A natural way to analyze such motions
is to, once again, take advantage
of [? Chasles' ?] theorem, which states
that one possible way of describing
the general motion of a rigid body
is by a translation of its center of mass
plus a rotation about its center of mass.
This separation of the general case
into two distinct type of motions
will greatly simplify our analysis.
One of the most common examples of simultaneous translational
and rotational motion is that of rolling objects,
and we will concentrate on this particular application.
We will see that an important detail is whether or not
the object rolls without slipping,
and the role that friction can play in each case.
The special case of rolling without slipping
amounts to a constraint condition
that relates the translational speed of the center of mass
to the radius and the rotational angular velocity.