Experiment 203

Moment of Inertia

Flores, Arvin Christian S.

PHY11L/A7

November 6, 2015

Introduction

It is accepted that Inertia by definition is a resistance of an object in change of motion. It is

conceptualized by Galileo and Isaac Newton. For example when you slide a book, it will not stop because of the

lack of force but because of the presence of the force which is friction, without it the book may go on forever to

slide. Newton's first law, which describes the inertia of a body in linear motion, can be extended to the inertia of a

body rotating about an axis using the moment of inertia. The relationship between the net external torque and the

angular acceleration is of the same form as Newton's second law and is sometimes called Newton's second law for

rotation. Here in the experiment there will be a discussion with the determination of moment of inertia of disk and

a ring including the factors affecting it to vary. The setup of the apparatus was first done followed by the

experiment proper. The ability of a certain body to rotate or the moment of inertia of the two objects in

combination and the disk alone, it will be done in the same time. The object is allowed to rotate about its axis with

the aid of mass loads which serves as the tension causing it to rotate. The I of a ring is attained from the difference

of the total I and the Idisk. Afterwards, the I of a disk is computed again but this time, it is rotated about its

diameter.

State Problem/Purpose

The purpose of this experiment is the determination of moment of inertia of disk and a ring including the variables

influencing it to rotate. The capacity of a certain body to turn or the moment of inertia of the two things combined

is done at the same time. The object is permitted to turn in its axis with the help of loads which serves as the strain

making it turn. The Inertia of a circle is to apply Newton's second law regarding the change in volume. This

clarifies the dissimilarity of the volume and density of an object can be the reason why inertia decreases or

increases. The equation below is used:

In order to get the actual value of moment of inertia of disk and ring,

I TOTAL=12MDISK R

2+ 12MRING (R1

2+R22 )

To get the experimental value of moment of inertia in table 2

I=m( g−a ) r2

a

To get the Actual value of moment of inertia of disk in table 2

IDISK=12M DISK R

2

For the actual value of moment of inertia of disk

IDISK=14M DISK R

2

Hypothesis

After analyzing the experiment to be done. If we determine the mass moment of inertia of the disk and the

ring then which has the greater moment of Inertia. Trial and error will play a big role here because in real life

standpoint air resistance and other external forces are included. In the latter part of the experiment the hypothesis

in this area if we compare the moment of inertia of solid disk rotated at two different axes: at the center and along

its diameter then maybe again we can identify which has the greater moment of Inertia.

Materials

1 set rotating platform 1 pc disk 1 pc ring 1 pc photogate 1 pc smart timer 1 pc smart pulley 1 set weights 1 pc mass hanger 1 pc vernier caliper

Caption: This the materials need for the experiment. Source: Group mate’s cellphone

Procedure The experiment is separated into three parts. The first part is the setting up the apparatus. The

second part is the determination of Moment of Inertia of Disk and Ring (rotated about the center). The last part is

the determination of Moment of Inertia of Disk (rotated about the diameter). The gathered data is correlated with

other experiments. Setting-up the apparatus: The apparatus were making sure it is complete. Smart timer is

connected to the head photogate and a 220V outlet. The rod was attached to the cylinder along the shaft and the

other end is to the mass hanger. The rod must be placed tangent to the smart pulley. The disk was placed

horizontally leveled in the shaft. Other gadget was used in order to make the disk leveled. Determination of

moment of inertia of disk and ring: First, the diameter of the shaft was measured by a vernier calliper where the

rod is attached and diameter of the disk and the inner and outer ring was also measured. The mass of the ring and

the disk was already given. Double checking is advised especially to the position or level of the disk. The ring

was placed to the disk. Friction mass was determined by adding load into the mass hanger. Friction mass would

be constant for all the parts of the experiment. Then, the experiment first trial was begun. The acceleration was

measured by a smart timer. Smart timer must be set to acceleration, linear pulley. Constant increase of mass must

be load for every trial must be done to obtain uniformity of the acceleration. Solve for the experimental moment

of inertia of the disk and ring. The friction mass was not included in the computation. From then, the ring was

removed and the procedure was applied again until the data will be complete. The Determination of moment of

inertia of ring will be based on calculation since IRing is the difference of the ITotal to IDisk. Determination of Moment

of Inertia of Disk rotated about the diameter: To begin with, relocate the disk from the vertical shaft. Rotate the

disk and embed the D-shaped opening on the hole to the shaft. Repeat the procedures done in the second part of

the experiment. Add a load to the mass hanger, set the smart timer to acceleration, linear pulley, drop the mass

hanger and the smart timer will do its place. Added mass must be put in every trial. Again, constant increase of

mass must be load for every trial to obtain uniformity increase of the acceleration.

Caption: Setup for Part 4, Source: Groupmate's cellphone

Results/Observations

Higher moments of inertia demonstrate that more force must be connected or put on in order to

cause a turn whereas lower moments of inertia means that only small forces are required. Masses that are further

far from the axis of rotation have the greatest moment of inertia. The moment of inertia can be calculated by its

volume and mass. By the application of Newton's Second Law of Motion, It determine the numerical estimation

of Inertia (I). The force present to the hanging mass is its own particular weight and tension plus the gravitational

force while the force that makes the disk rotating is the tension due to the mass hangs on the hanger. The heavier

the load in the mass hanger, the faster it rotates. On the data observed, adding constant amount of mass in every

trial will result to a consistent increase of acceleration. (Using the data sheet, mass of pan + mass added equates to

20 g, the acceleration equates to .9 cm/s2 and when the mass of pan + mas added increase to 30 g, the acceleration

increase 1.3 cm/s2) As the hanging mass increases, the acceleration increases.

Analysis/Calculations

Table 1 & Table 2:

I = 5.31

0 = 11.76 + .939 = 6.3495

I=m(g−a ) r2

a

Trial 1: I=55(980−1.2 ) (1.4575 )2

1 .2=95299 .91889

gcm2

Trial 2: I=105 (980−2. 2 ) (1 . 4575)2

2 . 2=99136 . 54472

gcm2

Trial 3: I=155 (980−3.2 ) (1 .4575)2

3 .2=100508 .8948

gcm2

Table 3:

1 ring =

12MRING (R1

2+R22 )

= 149304.84

1 ring = 1total – 1disk

= 144952.3580 – 98313.11947

= 96637.23853

Table 4:

I=m(g−a ) r2

a

Trial 1: I=20 (980−.09 ) (1. 4575 )2

. 9=46220 . 18332

gcm2

Trial 2: I=30 (980−1.3 ) (1 . 4575)2

1. 3=47978. 2737

gcm2

Trial 3: I=40 ( 980−1 .7 ) (1.4575 )2

1 .7=48899 . 03069

gcm2

After doing the experiment, it proves that the moment of inertia of the disk (rotated about the center) is greater

than the moment of inertia of the ring (rotated about the center). Even though the ring has a greater mass than the

disk, but the disk have greater radius than of the ring. In computing the values using the formula, the disk has a

greater value of moment of inertia. The moment of inertia on the disk is greater than that of the ring because the

mass of the disk is evenly distributed. Also the moment of inertia of the disk is greater when it is rotated about the

center is because the mass distributed is far from the center mass of the disk and also the axis of rotation.

Conclusion

Even though the mass of a particular rigid body constant the moment of inertia is not constant because of the

external forces which act upon the system and also the change in the acceleration of the rotation for every trial.

The factors that affect the moment of inertia can be the change in acceleration of a certain body is distance from

the axis, density of the rigid body, the position of the object and the axis of rotation of the system and etc.

The cause of the changes on the rotational motion of a rigid body is because of the net torque.

Literature Cited Fundamentals of Physics Extended, 10th Edition

http://www.engineeringtoolbox.com/moment-inertia-torque-d_913.html

http://hyperphysics.phy-astr.gsu.edu/hbase/mi2.html

http://emweb.unl.edu/NEGAHBAN/EM223/note19/note19.htm

http://www.physicsclassroom.com/class/newtlaws/Lesson-1/Inertia-and-Mass

TABLE 2. Determination of Moment of Inertia of Disk (rotated about the center)

mass of disk, MDISK = 1394.40 grams

radius of disk, RDISK = 11.761 cm

Actual value of moment of inertia of disk,

IDISK=12M DISK R

2

= 96437.48556 g-cm2

friction mass = 30 grams radius, r = 1.4575 cm

TRIAL(mass of pan + mass

added), m acceleration,aexperimental value of moment of inertia,

1 55 grams 1.2 cm/s2 95299.91889 gcm2

2 105 grams 2.2 cm/s2 99136.54472 gcm2

3 155 grams 3.2 cm/s2 100508.8948 gcm2

average 98315.11947 gcm2

% difference 0.48 %

TABLE 3. Determination of Moment of Inertia of Ring (rotated about the center)

Mass of ring, Mring = 1439.3 grams

Inner radius of ring, R1 = 5.31 cm

Outer radius of ring, R2 = 6.3495 cm

Actual value of moment of inertia of ring

Iring = 1/2Mdisk(R2+R2)

Iring = 49304.84 gcm2

Experiment value of moment of inertia (by difference),

Iring = 46637.23853 gcm2

% difference 0.8268 %

TABLE 1. Determination of Moment of Inertia of Disk and Ring (rotated about the center)

mass of disk, MDISK = 1394.40 grams

mass of ring, MRING= 1439.30 grams

radius of disk, RDISK = 11.761 cm

inner radius of ring, R1 = 5.31 cm

outer radius of ring, R2 = 6.3495 cm

Actual value of moment of inertia of disk and ring

ITOTAL=IDISK+ IRINGITOTAL=

12MDISK R

2+ 12MRING (R1

2+R22 )

I TOTAL = 145,742.3265 g-cm2

friction mass = 30 grams radius, r = 1.4575 cm

TRIAL(mass of pan + mass

added), m acceleration,aexperimental value of moment of inertia,

I=m(g−a ) r2

a1 55 grams 0.8 cm/s2 143008.2968 gcm2

2 105 grams 1.5 cm/s2 145504.3564 gcm2

3 155 grams 2.2 cm/s2 146349.4232 gcm2

Average 144952.3588 gcm2

% difference 0.30 %

TABLE 4. Determination of Moment of Inertia of Disk (rotated about the diameter)

mass of disk, MDISK = 1394.40 grams

radius of disk, RDISK = 11.761 cmActual value of moment of inertia of disk,

Idisk= 48218.74278 g-cm2

friction mass = 30 grams radius, r = 1.4575 cm

TRIAL(mass of pan + mass

added), m acceleration,aexperimental value of moment of inertia,

1 20 grams .9 cm/s2 46220.18332 gcm2

2 30 grams 1.3 cm/s2 47978.2737 gcm2

3 40 grams 1.7 cm/s2 48899.03069 gcm2

average 47699.16257 gcm2

% difference 1.08 %