7-6: ROTATION ABOUT A FIXED AXIS

It should be stressed at the outset that the equations of rotational
dynamics are generally valid only for a rigid (nondeformable) body rotating
about a fixed axis. For our purposes, a fixed axis may be either an axle or
shaft that is stationary with respect to some inertial reference frame and
about which the body is constrained to rotate or an axis of rotation that
passes through the center of mass of the body and does not change
direction. The significance of this limitation is that a body will not rotate
freely about just any old axis--you cannot, for example, toss a coin in the air
so that it rotates (freely) about an axis tangent to its edge. You could,
however, solder a thin rod to the edge of the coin, mount it between two
supports, and cause it to rotate about the resulting fixed axis. Or, you could
allow the coin to roll down an incline, and as long as it rolled in a straight
line, the motion would again involve rotation about a fixed axis. In either
case, the motion could be analyzed by the techniques presented in this
section.The condition of a rigid body rotating about a fixed axis allows us to
write the equations of rotational dynamics in a simplified scalar form,
corresponding to that of the equations for rectilinear motion. The
combination of angular-motion variables with the moment-of-inertia concept
is what makes this remarkable analogy possible: you can transform a
translational equation into its rotational counterpart by simply replacing
linear variables and mass with angular variables and rotational inertia.There
is a somewhat subtle assumption to be noted in the derivation of the scalar
equations for rotational dynamics, and it concerns the calculation of torque.
The vector definition of torque is actually referred to a point; to apply it to
rotation about a fixed axis, we use only the components of r and F that are
perpendicular to the fixed axis, i.e., that lie in the plane of rotation. The
result for torque is then parallel to the fixed axis and represents the net
external torque on the body. In effect, we assume that the axis exerts a force
equal and opposite to any force that is applied parallel to it; recall that we
made a similar assumption in using only the parallel component of a force
applied to a particle moving along a linear track.Conservation of energy can
be a useful tool for analyzing many instances of rotational motion, particularly
in cases where translational motion is also involved. Systems to which we
may apply conservation of energy are either those in which only conservative
forces act (gravity, a spring, etc.) or those in which "other" forces are not
doing any work on the system. Forces acting at the fixed axis of rotation of a
body, for instance, are not acting through any distance and thus do no
work.

PROBLEM 8: A meter stick of mass M is held vertically with one end on the
floor and is then allowed to fall. Find the angular speed of the stick when it
hits the floor, assuming that the end on the floor does not slip.