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26. Problem 12.60P (HRW) Two children, each with mass, sit on opposite ends of a narrow board with length L and mass M (the same as the mass of each child). The board is pivoted at its centre and is free to rotate in a horizontal circle without friction. (Treat is as a thin rod). What is the rotational inertia of the board plus the children about a vertical axis through the centre of the board? What is the angular momentum of the system if it is
rotating with angular speed
While the system is rotating, the children pull
themselves toward the centre of the board until they are half as far
from the centre as before. What is the resulting angular speed in terms
of What is the change in the kinetic energy of the system as a result of the children changing their positions? (What is the source of the added kinetic energy?)
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in a clockwise direction as seen from above? What is the direction of
the angular momentum?
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Solution: Click For PDF Version (a) As the moment of inertia of a thin rod of length L and mass M
about a axis perpendicular to it and passing through its centre is
(b) The angular momentum of the system as the children and the board are rotating
with angular speed
(c)
The rotational inertia,
Let
The kinetic energy of a system rotating with rotational inertia I and
angular momentum L is
The kinetic energy is added to the system as children move inward and do work against Coriolis force acting on them.
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,
the total rotational inertia, I ,of the board and the children sitting as
shown in the figure will be
.
in a
clockwise direction as seen from above will be
, and
it points downward.
,
of the system when the children have changed positions as shown in the figure
will be
be the
changed angular speed of the system. As the angular momentum of the system
remains unchanged when there is no external torque acting on the system, we have
. This gives
.
. Change
in kinetic energy of the system will be
.