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We have previously studied
the translational motion

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of rigid bodies by analyzing the
motion of their center of mass.

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We have also studied the
rotation of rigid bodies

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about a fixed axis.

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This week, we will consider
more complicated examples

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of rigid body motion, where
translation and rotation are

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both occurring simultaneously.

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One example is, if I toss a
rigid body through the air

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but give it a spin
at the same time

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so that it tumbles
in space as it falls.

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A second example
is a wheel that is

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rolling along a flat surface.

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Its location is translating,
but at the same time,

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it is also spinning
about its axis.

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A natural way to
analyze such motions

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is to, once again,
take advantage

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of [? Chasles' ?]
theorem, which states

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that one possible
way of describing

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the general motion
of a rigid body

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is by a translation
of its center of mass

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plus a rotation about
its center of mass.

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This separation of
the general case

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into two distinct
type of motions

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will greatly simplify
our analysis.

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One of the most common examples
of simultaneous translational

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and rotational motion is
that of rolling objects,

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and we will concentrate on
this particular application.

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We will see that an important
detail is whether or not

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the object rolls
without slipping,

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and the role that friction
can play in each case.

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The special case of
rolling without slipping

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amounts to a
constraint condition

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that relates the translational
speed of the center of mass

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to the radius and the
rotational angular velocity.