Department of

PHYSICS
Experiment 103
Moment of Inertia
Name:
Course/Year: ME/1
Subject/Section: PHYS101L/A23
Date of Performance:
Date of Submission:
Criteria Score
Data Sheet with sample computation (40 points)
Guide Questions (GQ) (10 points)
Interpretation of Results (Analysis) (15 points)
Error Analysis (5 points)
Conclusion (15 points)
Application (10 points)
Graph/figure (5 points)

Instructor
Guide Questions

1. If a disk and a ring both have the same mass and radius, which one has the greater
moment of inertia? Why is this so?

Moment of inertia is the tendency of an object to withstand rotational motion or

remain rotating unless an external torque or force is applied. Moment of inertia depends
on the mass of the object and relative distribution of that mass to the axis of rotation. The
disk has a uniform distribution of mass all throughout thus it has mass near its axis of
rotation. Mass near its axis of rotation decreases the moment of inertia while a ring has
mass only on its outmost part therefore increases rotational inertia. Mathematically, the
moment of inertia of a disk has coefficient of ½ which halves the rotational inertia.

2. A figure skater doing pirouettes spins faster when she tucks her arms close to her chest
and spins slower when she spreads out her arms. Explain this in terms of moment of
inertia and newton’s second law for rotational motion.

Rotational inertia depends on mass and distribution of that mass along the axis of
rotation. Tucking the arms of the skater to her chest lets the mass being concentrated at the
axis of rotation. Angular acceleration has an inverse relationship with moment of inertia.
As the moment of inertia increases wherein the concentration of mass goes farther from
the rotational axis with a constant mass the angular acceleration decreases, however an
opposite effect occurs when the skater tucks her arms decreases the rotational inertia thus
increasing angular acceleration.

Analysis

The experiment illustrated Newton’s law of acceleration for rotational motion. Moment of
inertia is the counterpart of the linear inertia, torque is for the force, and angular acceleration for
the linear acceleration. The moment of inertia is the tendency of an object to remain in motion
unless an external force or toque is applied or remain at rest. The moment of inertia of an object
rotating at an axis was mathematically expressed in equation 1 where I is the rotation inertia, m is
mass of the object, r is the perpendicular distance relative to the axis of rotation.

𝐼 = 𝑚𝑟 2 (eqn. 1)

The experiment determined the moment of inertia of a disk with a rotational axis at its
radius and diameter. The experiment used rigid disk with continuous distribution of mass. Its
moment of inertia computed by integrating the perpendicular distance to the axis of rotation and
its differential mass expressed in elemental volume (dV) and density (r) shown in equation 2. The
derived equation for the disk rotating at its radius and diameter shown in equation 3 and 4,
respectively. Thus acquiring the accepted value of the rotational inertia of a disk rotating at its
radius and diameter.

𝐼 = ∫ 𝑟 2 r 𝑑𝑉 (eqn. 2)
1
𝐼 = 2 𝑚𝑟 2 (eqn. 3)
1
𝐼 = 4 𝑚𝑟 2 (eqn. 4)

Newton’s law of acceleration for rotational motion determined the experimental value to
in acquiring moment of inertia of the disk rotating along its radius and diameter. Mathematically
expressed in equation 5. Objects in motion tend to have kinetic friction. A friction mass was used
to overcome the small amount of kinetic friction.

𝑚(𝑔−𝑎)𝑟 2
𝐼= (eqn. 5)
𝑎

where I is the rotational inertia, m is the sum of the mass of pan and mass added , g is the
acceleration due to gravity (9.81 m/s2), a is the acceleration of the disk read from the smart timer,
and r is the radius of the shaft where thread is wound.
The computed accepted value of the rotational inertia was compared in the acquired
experimental value using percent difference shown in equation 6.

|𝐴𝑐𝑐𝑒𝑝𝑡𝑒𝑑 𝑉𝑎𝑙𝑢𝑒−𝐸𝑥𝑒𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 𝑉𝑎𝑙𝑢𝑒|

𝐴𝑐𝑐𝑒𝑝𝑡𝑒𝑑 𝑉𝑎𝑙𝑢𝑒+𝐸𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡𝑎𝑙 𝑉𝑎𝑙𝑢𝑒 𝑥 100 (eqn. 6)
2

The mass and radius of the disk were 1470.6 g and 11.4 cm, respectively. The computed
accepted value of the rotational inertia of the disk rotating along its radius and diameter were
95,559.59 g cm2 and 47, 779. 794 g cm2, respectively. The thread was wound in a shaft with radius
0.8375 cm. The recorded acceleration of the disk rotating along its radius and diameter were 0.53
cm/s2 and 0.75 cm/s2. By this, the experimental rotational inertia of the disk rotating along its
radius and diameter were 97, 317.14 g cm2 and 45,836.898 g cm2, respectively. The deviation of
the experimental value on the accepted value can be concluded in the percent difference of 1.82 %
for the axis along radius and 4.15 % along diameter. The both cases validated the effect of the axis
of rotation to the rotational inertia of a rigid object. The difference in the read acceleration of the
disk gave an inverse relationship of the disk’s acceleration with its moment of inertia.

Error Analysis

The assumption for the accepted value of the rotational inertia of the disk were the disk
rotates freely negligible of any kind of friction. Every object that moves either rotating or linear
kinetic friction will be generated. In this experiment the small amount of kinetic friction was
overcame using a 25g used as friction mass. However, friction in a form of air resistance still
existed. It contributed in the percent difference. The friction in the bearings of the apparatus will
affect the data. The setup, suspension of the thread, and its alignment will affect the error in the
experiment. If the thread is not perpendicular to the rotating axis a x component of the force may
exist. The availability of the masses added may provide a closer approximation for the
experimental value since the rotational motion of the disk is caused by the force due to gravity.

Conclusion

The experiment’s object which were determining the moment of inertia of a disk about an
axis through its center perpendicular to its plane; and determining the moment of inertia of a disk
about an axis through tis diameter. All objectives were met. The findings in the experiment also
concluded the inverse relationship of disk’s acceleration to its rotational inertia. The experiment
held a constant mass of the disk and radius, and constant radius of the shaft where thread was
wound. Thus, the axis of rotation has a significant effect in the moment of inertia. A higher moment
of inertia, a lower acceleration and vice-versa.

Application

Moment of inertia is the capability of an object to resist rotational motion or remain

rotating. It is dependent on the mass of the object, the relative distance of the distribution of mass
of that object to its rotational axis, and orientation of the its rotational axis. Moment of rotation
can be seen in every object who experienced torque or rotating. As a mechanical engineering
student, we deal a lot of rotating gears. The size of the gear which can affect the angular velocity
that cause a linear motion or work. A notable application of moment of inertia is the fly wheel. A
fly wheel is attached to a rotating shaft to flow power from a motor to a machine efficiently. The
high dense mass of a fly wheel gives a higher inertia to oppose and moderate fluctuations in the
engine speed also stores excess energy for intermittent use. The high rotational inertia of the fly
wheel comes from its weight from out of the axis. It is presented in car engines to smooth out
fluctuations provided by the internal combustion of the cylinders.