8 ROTATIONAL MOTION 8.11 Combined translational and rotational motion
object can roll over such a surface with hardly any dissipation. For instance, it
is far easier to drag a heavy suitcase across the concourse of an airport if the
suitcase has wheels on the bottom. Let us investigate the physics of round objects
rolling over rough surfaces, and, in particular, rolling down rough inclines.
Consider a uniform cylinder of radius b rolling over a horizontal, frictionalsurface. See Fig. 83. Let v be the translational velocity of the cylinder’s centre
of mass, and let ω be the angular velocity of the cylinder about an axis running
along its length, and passing through its centre of mass. Consider the point of
contact between the cylinder and the surface. The velocity vJ of this point is
made up of two components: the translational velocity v, which is common to
all elements of the cylinder, and the tangential velocity vt = −b ω, due to the
cylinder’s rotational motion. Thus,
vJ = v − vt = v − b ω. (8.88)Suppose that the cylinder rolls without slipping. In other words, suppose that
there is no frictional energy dissipation as the cylinder moves over the surface.
This is only possible if there is zero net motion between the surface and the
bottom of the cylinder, which implies vJ = 0 , or
v = b ω. (8.89)It follows that when a cylinder, or any other round object, rolls across a rough sur-
face without slipping—i.e., without dissipating energy—then the cylinder’s trans-
lational and rotational velocities are not independent, but satisfy a particular
relationship (see the above equation). Of course, if the cylinder slips as it rolls
across the surface then this relationship no longer holds.
Consider, now, what happens when the cylinder shown in Fig. 83 rolls, with-out slipping, down a rough slope whose angle of inclination, with respect to the
horizontal, is θ. If the cylinder starts from rest, and rolls down the slope a verti-
cal distance h, then its gravitational potential energy decreases by −∆P = M g h,
where M is the mass of the cylinder. This decrease in potential energy must be
offset by a corresponding increase in kinetic energy. (Recall that when a cylin-
der rolls without slipping there is no frictional energy loss.) However, a rolling
cylinder can possesses two different types of kinetic energy. Firstly, translational