Determination of moment of inertia of a
flywheel about its axis of rotation
Theory
The flywheel
is a big sized wheel. Most of its mass is distributed over the peripheral
region. A thick cylindrical rod, called the axle, passes through the centre of
mass of the wheel. The axis of the axle is perpendicular to the circular
surface of the flywheel. The axle is kept horizontally by means of a holder
hung on the wall. The wheel with the axle can rotate about the axis of the
axle. There is a peg joined with axle. Objective of this experiment is to
deturmine the moment of inertia of the flywheel about the axis of rotation,
i.e., the axis of the axle.
(a) Flywheel
when the rope and load is about to be detached from the axle. (b)
The same flywheel after rotating it for nl number of times (here n1
is 6) There is a small peg on the axle, as shown in figure 1. We make a
loop on one end of a rope round this peg. A load of mass, M, is connected to
the other end of the rope. We hold the flywheel in such a way that the.
load is
about to be detached from the axle (figure 1 b). Then we keep a straight meter
scale at The bottom surface of the load, see where on the wall the end of the
meter scale touches and put a mark over there. Next, we rotate the flywheel for
n1 times. Consequently the load moves upward. Again, we keep the straight meter
scale at the bottom surface of the load; we see where on the wall the
end of the meter scale touches and put a mark over there. The separation between
the two marks is h.
Now, if the
flywheel is made free to rotate, then its angular velocity increases uniformly
and the linear velocity of the load also increases uniformly. The flywheel
completes n1 revolutions after the release of the load and
the load traverses a distance h vertically.
The work
done by gravity on the load = M g h A part of this work is used to
increase rotational kinetic energy of the flywheel, and part of it supplies the
linear kinetic energy to the load and the rest is used to work against the
friction between the flywheel and the holder.
Let, I is
the moment of inertia of the flywheel about its axis of rotation..
When the load is just detached from the axle, the angular velocity
of the flywheel be ω and the linear velocity of the load is v. So the
rotational kinetic energy of the flywheel
and the
kinetic energy of the load
Let, the
work done against friction to complete a single revolution = Wf ,
therefore, the work done against friction to complete n1
revolutions = n1Wf So,
we can write,
Let, t is
the time between the moment of detachment of the load from the axle and the
moment when the flywheel comes into rest. In time t the flywheel completes n2
revolutions.
During this
second part of the motion of the flywheel, all of the rotational kinetic energy
will be used to work against the friction which is n2wf, and hence
Since
in one complete revolution the angular displacement is π2radian, in n2 revolutions, total angular displacement is 2n2π radian. The average angular velocity,
Now,
during
this second part of the motion, the angular velocity decreases uniformly form
By using
equations (4) and (5) we can determine the moment of inertia of the flywheel
about its axis of rotation by measuring M, h, r, n1 and n2
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