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28. Problem 12.69P(HRW) The particle of mass m as shown in the figure slides down the frictionless
surface and collides with a uniform vertical rod, sticking to it. The rod pivots
about O through the angle
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before
momentarily coming to rest. We have to find
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Solution: Click For PDF Version We will first find the distance,
The speed
The angular momentum
The moment of inertia
Let
The kinetic energy of the combined system of the rod and the particle just after the impact will be
We are given that the rod pivots about O through the angle
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, of the
centre of mass of the combined system of rod of mass
and
the particle of mass
from the
point O, when the mass m has just stuck to the rod. From the definition
of the centre of mass, we get
.
of the
particle just before its impact with the rod can be calculated from the
principle of conservation of energy. We have
of the
particle about O is
of the
combined system of the rod and the particle about the axis passing through O
is
be the
angular speed of the rod and the particle just after the impact. From the
principle of conservation of angular momentum, we have
before momentarily coming to rest. At that position the centre of mass of the
rod and the particle will have risen in height by
.
From the principle of conservation of energy we have