Measuring the Moment of Inertia of a Flywheel


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Measuring the Moment of Inertia of a Flywheel



Objective
=========

Measure the angular velocity of a flywheel and use conservation of
energy to calculate its moment of inertia.



Apparatus
=========

Flywheel

String

Slotted mass on hanger

Stop-watch

Vernier caliper

Metre ruler



Theory
======

The rotational kinetic energy can be defined by the equation K=1/2 I ω2.
Where I is the moment of inertia of the body about the axis of
rotation.

In this experiment, the flywheel rotates freely about a horizontal
axis. The radius of the axle of the flywheel can be measured with a
caliper. As m falls, its gravitational potential energy is transferred
into translational kinetic energy of m, rotational kinetic energy of
the flywheel and work done by friction. As the flywheel completes N
further turns, its original rotational kinetic energy is transferred
into friction loss. Assume the flywheel decelerates uniformly. Thus,
the moment of inertia of the flywheel can be determined.



Procedure
=========

1. The flywheel was set as shown with the axle of the flywheel
horizontal. A polystyrene tile was placed on the floor to avoid the
impact of the mass on the floor.

2. The vernier caliper was used to measure the diameter d of the axle.
The mean of two perpendicular measurements was taken.

3. The hanger with appropriate amount of slotted mass was put on the
tile. Use the balance to measure the total mass m.

4. Sufficient length of string was attached to the hanger so that the
free end wraps once round the axle of the flywheel.

5. The mass was winded up to an appropriate height.

6. Verified that the string fell off the axle when the mass hit the
ground. A label was put on the curved surface of the flywheel. The
mass was winded up again.

7. The height h of the mass was measured. The height h was recorded.
The number of revolutions n1 that the flywheel made was calculated as

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the mass was wound up.

8. The mass was released and at the same time the stop-watch was
started.

9. As soon as the mass hit the ground, timing was stopped and the
number of revolution n2 that the flywheel performed was counted before
it came to rest.

10. The mass was winded up again and steps 7 to 9 were repeated for at
least 3 times. The mean values of the falling time t and n2 were
obtained.



Precautions
===========

1. The mass and the height from which the mass falls should be chosen
so that the falling time is long enough for measurement to be taken
accurately. The mass and the height should not be changed throughout
the experiment once they have been chosen.

2. The first few turns of the string should overlap the others.

3. The mass should be wound up to the same height in all trials.

4. When using the stop-watch, the hand should be held straightly to
minimum the reaction time error.

5. Do not stand too close to the polystyrene tile when releasing the
mass.

6. When choosing the appropriate amount of the slotted mass, a smaller
amount (e.g. 1 slotted mass) should be chose to try first.

7. The later turns of the string should not overlap the others.



Result and calculation
======================

Mass m = 0.2 kg

Height h = 0.80 m

Axle diameter d = 0.0522 m

→ radius r = 0.0261 m

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

mean

Time t/s

1.75

1.85

1.84

1.84

1.81

1.78

1.82

1.78

1.81

No. of revs n1

5

5

5

5

5

5

5

5

5

No. of revs n2

25

27

23

26

26

24

25

23

25

Suppose v is the final velocity of the mass when it reaches the floor
and ω is the angular velocity of the flywheel at this instant. Then

v = 2 × average velocity during fall = 2h / t

= 2 × 0.8 ÷ 1.81

= 0.88 m s-1

ω = v / r

= 0.88 ÷ 0.0261

= 33.9 rad s-1

From the conservation of energy,

Decrease in Increase
in Work done

gravitational potential = kinetic
energy + against

energy of falling mass of mass and
flywheel friction

i.e. mgh = 1/2 mv2 + 1/2 Iω2 + n1W

where W is the work done against friction per revolution.

Since the kinetic energy acquired by the flywheel ( 1/2 Iω2) is
dissipated in n2 revolutions,

n2W = 1/2 Iω2

W = Iω2 / 2n2

Substituting,

mgh = 1/2 mv2 + 1/2 Iω2 + n1W

mgh = 1/2 mv2 + 1/2 Iω2 + n1Iω2 / 2n2

I = mr2 ( n2 / n1 + n2 ) ( gt2 / 2h – 1)

=0.2 × (0.0261)2 ( 25 / (25 + 5) ) ( 10 × (1.81)2 / 2 × 0.8 – 1 )

=2.21 × 10-3 kg m2



Conclusion
==========

The moment of inertia of the flywheel is measured and found to be 2.21
× 10-3 kg m2.



Discussion
==========

Errors and improvement

1. The reaction times error. This can be improved by straighten the
hand when taking the time.

2. The number of revolution n2 that the flywheel performed cannot be
accurately obtained. This can be improved by counting the number of
revolution by two students instant of one and to repeat the experiment
more times.

3. Unsteady hands. When the hand released the mass, force may be push
to the mass. To improve this, student should release the mass slowly
and softly.



Reference
=========

Physic Beyond 2000


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