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13. Problem 11.87P (HRW) A tall, cylinder-shaped chimney falls over when its
base is ruptured. Treating the chimney as a thin rod with height h, express the
(a) radial (b) tangential components of the linear acceleration of
the top of the chimney as a function of the angle
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made by the chimney with the vertical. (c) At what angle
does
the linear acceleration equal g?|
Solution: Click For PDF Version
Let the mass of the chimney be M. When the chimney is
standing vertically its weight Mg will act at its centre of mass, which
is at a height
Substituting for I we find
The radial acceleration is the centripetal acceleration and
is
For finding the linear acceleration of the top end of the
chimney we first calculate its angular acceleration
we find
The linear acceleration of the top end
The linear acceleration will be equal to g, at an
angle
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from the base. At the instant when the chimney is at angle
with the vertical, the drop in the height of the centre of gravity will be
. The
change in the potential energy of the chimney will be
.
The moment of inertia
of the
chimney about the end at the ground is
.
Therefore, if we denote the angular speed of the chimney treated as a rigid body
when it is inclined at an angle
with the
vertical as
, the
rotational energy of the chimney will be
. Equating
the change in potential energy to the change in kinetic energy, we get
.
.
.
The torque,
, on the
chimney about its rotational axis, when it is at an incline
, will be
. Using
the relation
,
.
.
given by
the equation