Please copy and paste this embed script to where you want to embed

E203: MOMENT OF INERTIA

DE VEGA, Kim Lawrence M.

OBJECTIVE The purpose of this experiment is to measure the object’s resistance to change in a rotation direction. For the first part, the solid disk was rotated at the center. The moment of the disk and ring was determined as well as the moment of the disk alone. From the data gathered, the moment of the disk alone is smaller compared to the moment of the disk and ring. For the latter part of the experiment, the solid disk is rotated along its diameter. The computed moment was reduced in half due to the distribution of weight was less compared when the disk was positioned horizontally. MATERIALS AND METHODS The materials used for this experiment were a set of rotating platform, disk, ring, a photo gate, a smart timer, a smart pulley, set of weights, a mass hanger and a vernier caliper. The experiment was divided into four parts. Before the experiment started, the mounting rod was attached to the smart pulley and photo gate head. The mass hanger was connected to a thread that passed over the smart pulley and was looped around the cylinder along the vertical shaft. The smart timer was connected to the photo gate head and was set to accel, linear

pulley. The first part was the determination of moment of inertia of disk and ring. the ring was placed on the disk and a vernier caliper was used to measure the diameter of the shaft, from the diameter measured the radius was computed. A small amount of mass was added to the hanger to overcome kinetic friction which was the friction mass. Different masses were added to the hanger and the reading from the smart timer was considered its acceleration. Three trials were made. The actual value of moment of inertia of the disk and ring was computed using the formula:

R

2 1

2 1 2 1 I TOTAL = M DISK R + M RING ¿ +R2 ) 2 2

(1)

The experimental value of moment of inertia was computed using the formula:

I=

m ( g−a ) r a

2

(2)

For the second part, the moment of inertia of disk rotated about the center was determined. For this part the ting was removed from the disk and same procedure was done. The experimental value was calculated using equation (2) while the actual value was computed using the formula: 1|Page

1 I DISK = M DISK R 2 2

(3)

For the determination of the actual moment of inertia ring, it was calculated using the formula: 2

R1 2 1 I RING = M RING ¿ +R2 ) 2

(4)

The experimental value for the moment of inertia ring was calculated using the formula:

I RING(EXPTL)=I TOTAL(EXPTL)−I DISK( EXPTL)

(5)

For the last part which was the determination of moment of inertia disk rotated about its diameter, the disk was positioned vertically and same procedure as the first part was done. The experimental value was computed using equation (2).The actual value was computed using the formula:

1 I DISK = M DISK R 2 4

Fig. 2 Moment of Inertia of the Disk rotated about the center set-up

(6)

The mean experimental inertia was compared with the actual value using the percentage difference formula:

¿ actual+ experimental ( ) 2 %diff =¿

¿ actual−experimental∨

x100%

Fig. 3 Moment of Inertia of the Disk & Ring Setup

(7)

Fig. 1 Materials used

Fig. 4 Moment of Inertia of the Disk rotated about the diameter set-up

2|Page

OBSERVATIONS AND RESULTS Table 1: Weights and Lengths Measured Constant Values MDISK 1480 g MRING 1428.2 g RDISK 11.35 in R1 5.37 in R2 6.375 in Radius, r Table 2: Moment of Inertia of Disk and Ring Trial Mass,m Acceleration Inertia 1 00 g 00 cm/s2 gcm2 2 00 g 00 cm/s2 gcm2 2 3 00 g 00 cm/s gcm2 average gcm2 % difference % The actual value of inertia of the disk and ring was computed using the values on table 1 and using equation (1):

5.37 ¿

1 1 I TOTAL = (1480) ( 11.35 )2+ (1428.2) ¿ +6.3752) 2 2 ¿ =141721.9881 The mass needed to overcome the friction was 20 grams. The reading from the smart timer was recorded and the experimental value of inertia was computed using equation (2):

I=

35 ( 980−0.2 ) 0.2252 0.2

=116703.3656

It can be observed from the table that as the mass added increases, the acceleration increases. But for the computed inertia, as the mass and acceleration increases, it decreases. From equation (2), inertia is directly proportional to the product of the mass, the difference between the gravitational acceleration and the acceleration and the square of the radius of the shaft but inversely proportional to the acceleration which was the reading from the smart timer. After three trials, the average inertia was computed and followed by the computation of the percentage difference using equation (7):

I TOTAL =

116703.3656 +103345.2906+11655.7219 3

=113902.4324

¿ 141721.9881+113902.43 ( 2 %diff =¿

¿ 141721.9881−113902.4324∨

x100% =21.77% Based from the computed percentage difference, the data obtained for the experimental value of inertia is accurate. Table 3: Moment of Inertia of Disk (rotated about the center) Trial Mass,m Acceleration Inertia 1 00 g 00 cm/s2 gcm2 2 2 00 g 00 cm/s gcm2 3 00 g 00 cm/s2 gcm2 The actual value of inertia of the disk rotated about the center was computed using equation (1):

1 I DISK = (1430)(11.35)2 =92108.0875 2 The mass used to overcome friction was 10 g. No trend can be observed from the data, but it is understood that the data is less compared to the data from table 2 since only the inertia of the disk was computed and the ring was removed. Though the computed percentage difference is quite larger than table 2, the data obtained for the value of inertia is still accurate. Determining the actual value of inertia of the ring is the same as the one done with the disk’ inertia, it is computed using equation (4):

5.37 ¿ 1 I RING = (1428.2)¿ +6.3752) 2 ¿

= 49613.9006

However, the experimental moment of inertia for the ring alone cannot be performed because of its hollow center, but is can be computed mathematically using the total experimental 3|Page

inertia as a minuend and having the experimental disk inertia as the subtrahend or simply using equation (5) followed by the computation of the percentage difference:

I RING(EXPTL)=113902.4324−76861.09 =37041.3424

¿ 76861.09+113902.4324 ( ) 2 %diff =¿

¿ 76861.09−113902.4324∨

x100% =18.05% Table 4: Moment of Inertia of Disk (rotated about the diameter) Trial Mass,m Acceleration Inertia 1 00 g 00 cm/s2 gcm2 2 2 00 g 00 cm/s gcm2 3 00 g 00 cm/s2 gcm2 The mass needed to overcome friction was 10 g. The actual value of the inertia of the disk was computed using equation (6):

1 I DISK = M DISK R 2 4 From the data above, it can be observed that the inertia decreases with increasing mass and acceleration. Compared with the data from table 2, the data is decreased by half. The disk is positioned vertically affecting the weight distribution much less like the horizontal position. When the disk is positioned vertically, the weight distribution is concentrated onto the shaft, but when the disk is positioned horizontally, the weight distribution is equal. The experimental value of the moment of the disk is computed using equation (2). The average inertia was computed then the percentage difference using equation (7):

I DISK (EXPTL) ¿ 76861.09+113902.4324 ( ) 2 %diff =¿

¿ 76861.09−113902.4324∨

x100% =18.05%

The data is still accurate though it was the part of the experiment where the computed percentage difference was the highest. DISCUSSION & CONCLUSION In this experiment, we need to determine the moment of inertia in any change of direction. In the first part, we need to determine the Moment of Inertia of Disk and Ring. In the second part, we need to determine the Moment of Inertia of Disk (rotated about the center) and in the last part, we need to determine the Moment of Inertia of Disk (rotated about the diameter). Inertia is the property of a body to resist any change in its uniform motion. This uniform motion was determined by the frictional mass, in which is the mass that can have a body on a uniform motion. The moment of inertia depends on the mass of the body and the chosen axis of rotation. Inertia varies with position acquired by the object, and proves its point. The moments of inertia of a solid disk rotated at two different axes proved its point. From the results, the disk that is rotated about the center gives inertia larger than the one rotated about the diameter. The inertia of the disk rotated about the diameter is reduced by half due to the weight distribution across the disk onto the shaft. ACKNOWLEDGMENT & REFERENCE I would like to express my sincere gratitude to my group mates who made the experiment a success, to the institute that provided the instruments we needed for this experiment in order for us to learn, to our professor that guided us and provided us the instructions and techniques in order to execute the experiment and obtained a realistic and decent data, and lastly to God for providing us the knowledge we needed and the chance to be able to perform the experiment. Halliday, D., Resnick, R., & Walker, J. (2014). Principles of Physics, 10th Ed, John Wiley & Sons Singapore Pte. Ltd., Singapore. (ABSTRACT) Young, H., Freedman, R., Sears, F., Zemansky, M. (1949). University Physics, 12th Ed, Pearson Education, United States. (MATERIALS AND METHODS) 4|Page

5|Page

View more...
DE VEGA, Kim Lawrence M.

OBJECTIVE The purpose of this experiment is to measure the object’s resistance to change in a rotation direction. For the first part, the solid disk was rotated at the center. The moment of the disk and ring was determined as well as the moment of the disk alone. From the data gathered, the moment of the disk alone is smaller compared to the moment of the disk and ring. For the latter part of the experiment, the solid disk is rotated along its diameter. The computed moment was reduced in half due to the distribution of weight was less compared when the disk was positioned horizontally. MATERIALS AND METHODS The materials used for this experiment were a set of rotating platform, disk, ring, a photo gate, a smart timer, a smart pulley, set of weights, a mass hanger and a vernier caliper. The experiment was divided into four parts. Before the experiment started, the mounting rod was attached to the smart pulley and photo gate head. The mass hanger was connected to a thread that passed over the smart pulley and was looped around the cylinder along the vertical shaft. The smart timer was connected to the photo gate head and was set to accel, linear

pulley. The first part was the determination of moment of inertia of disk and ring. the ring was placed on the disk and a vernier caliper was used to measure the diameter of the shaft, from the diameter measured the radius was computed. A small amount of mass was added to the hanger to overcome kinetic friction which was the friction mass. Different masses were added to the hanger and the reading from the smart timer was considered its acceleration. Three trials were made. The actual value of moment of inertia of the disk and ring was computed using the formula:

R

2 1

2 1 2 1 I TOTAL = M DISK R + M RING ¿ +R2 ) 2 2

(1)

The experimental value of moment of inertia was computed using the formula:

I=

m ( g−a ) r a

2

(2)

For the second part, the moment of inertia of disk rotated about the center was determined. For this part the ting was removed from the disk and same procedure was done. The experimental value was calculated using equation (2) while the actual value was computed using the formula: 1|Page

1 I DISK = M DISK R 2 2

(3)

For the determination of the actual moment of inertia ring, it was calculated using the formula: 2

R1 2 1 I RING = M RING ¿ +R2 ) 2

(4)

The experimental value for the moment of inertia ring was calculated using the formula:

I RING(EXPTL)=I TOTAL(EXPTL)−I DISK( EXPTL)

(5)

For the last part which was the determination of moment of inertia disk rotated about its diameter, the disk was positioned vertically and same procedure as the first part was done. The experimental value was computed using equation (2).The actual value was computed using the formula:

1 I DISK = M DISK R 2 4

Fig. 2 Moment of Inertia of the Disk rotated about the center set-up

(6)

The mean experimental inertia was compared with the actual value using the percentage difference formula:

¿ actual+ experimental ( ) 2 %diff =¿

¿ actual−experimental∨

x100%

Fig. 3 Moment of Inertia of the Disk & Ring Setup

(7)

Fig. 1 Materials used

Fig. 4 Moment of Inertia of the Disk rotated about the diameter set-up

2|Page

OBSERVATIONS AND RESULTS Table 1: Weights and Lengths Measured Constant Values MDISK 1480 g MRING 1428.2 g RDISK 11.35 in R1 5.37 in R2 6.375 in Radius, r Table 2: Moment of Inertia of Disk and Ring Trial Mass,m Acceleration Inertia 1 00 g 00 cm/s2 gcm2 2 00 g 00 cm/s2 gcm2 2 3 00 g 00 cm/s gcm2 average gcm2 % difference % The actual value of inertia of the disk and ring was computed using the values on table 1 and using equation (1):

5.37 ¿

1 1 I TOTAL = (1480) ( 11.35 )2+ (1428.2) ¿ +6.3752) 2 2 ¿ =141721.9881 The mass needed to overcome the friction was 20 grams. The reading from the smart timer was recorded and the experimental value of inertia was computed using equation (2):

I=

35 ( 980−0.2 ) 0.2252 0.2

=116703.3656

It can be observed from the table that as the mass added increases, the acceleration increases. But for the computed inertia, as the mass and acceleration increases, it decreases. From equation (2), inertia is directly proportional to the product of the mass, the difference between the gravitational acceleration and the acceleration and the square of the radius of the shaft but inversely proportional to the acceleration which was the reading from the smart timer. After three trials, the average inertia was computed and followed by the computation of the percentage difference using equation (7):

I TOTAL =

116703.3656 +103345.2906+11655.7219 3

=113902.4324

¿ 141721.9881+113902.43 ( 2 %diff =¿

¿ 141721.9881−113902.4324∨

x100% =21.77% Based from the computed percentage difference, the data obtained for the experimental value of inertia is accurate. Table 3: Moment of Inertia of Disk (rotated about the center) Trial Mass,m Acceleration Inertia 1 00 g 00 cm/s2 gcm2 2 2 00 g 00 cm/s gcm2 3 00 g 00 cm/s2 gcm2 The actual value of inertia of the disk rotated about the center was computed using equation (1):

1 I DISK = (1430)(11.35)2 =92108.0875 2 The mass used to overcome friction was 10 g. No trend can be observed from the data, but it is understood that the data is less compared to the data from table 2 since only the inertia of the disk was computed and the ring was removed. Though the computed percentage difference is quite larger than table 2, the data obtained for the value of inertia is still accurate. Determining the actual value of inertia of the ring is the same as the one done with the disk’ inertia, it is computed using equation (4):

5.37 ¿ 1 I RING = (1428.2)¿ +6.3752) 2 ¿

= 49613.9006

However, the experimental moment of inertia for the ring alone cannot be performed because of its hollow center, but is can be computed mathematically using the total experimental 3|Page

inertia as a minuend and having the experimental disk inertia as the subtrahend or simply using equation (5) followed by the computation of the percentage difference:

I RING(EXPTL)=113902.4324−76861.09 =37041.3424

¿ 76861.09+113902.4324 ( ) 2 %diff =¿

¿ 76861.09−113902.4324∨

x100% =18.05% Table 4: Moment of Inertia of Disk (rotated about the diameter) Trial Mass,m Acceleration Inertia 1 00 g 00 cm/s2 gcm2 2 2 00 g 00 cm/s gcm2 3 00 g 00 cm/s2 gcm2 The mass needed to overcome friction was 10 g. The actual value of the inertia of the disk was computed using equation (6):

1 I DISK = M DISK R 2 4 From the data above, it can be observed that the inertia decreases with increasing mass and acceleration. Compared with the data from table 2, the data is decreased by half. The disk is positioned vertically affecting the weight distribution much less like the horizontal position. When the disk is positioned vertically, the weight distribution is concentrated onto the shaft, but when the disk is positioned horizontally, the weight distribution is equal. The experimental value of the moment of the disk is computed using equation (2). The average inertia was computed then the percentage difference using equation (7):

I DISK (EXPTL) ¿ 76861.09+113902.4324 ( ) 2 %diff =¿

¿ 76861.09−113902.4324∨

x100% =18.05%

The data is still accurate though it was the part of the experiment where the computed percentage difference was the highest. DISCUSSION & CONCLUSION In this experiment, we need to determine the moment of inertia in any change of direction. In the first part, we need to determine the Moment of Inertia of Disk and Ring. In the second part, we need to determine the Moment of Inertia of Disk (rotated about the center) and in the last part, we need to determine the Moment of Inertia of Disk (rotated about the diameter). Inertia is the property of a body to resist any change in its uniform motion. This uniform motion was determined by the frictional mass, in which is the mass that can have a body on a uniform motion. The moment of inertia depends on the mass of the body and the chosen axis of rotation. Inertia varies with position acquired by the object, and proves its point. The moments of inertia of a solid disk rotated at two different axes proved its point. From the results, the disk that is rotated about the center gives inertia larger than the one rotated about the diameter. The inertia of the disk rotated about the diameter is reduced by half due to the weight distribution across the disk onto the shaft. ACKNOWLEDGMENT & REFERENCE I would like to express my sincere gratitude to my group mates who made the experiment a success, to the institute that provided the instruments we needed for this experiment in order for us to learn, to our professor that guided us and provided us the instructions and techniques in order to execute the experiment and obtained a realistic and decent data, and lastly to God for providing us the knowledge we needed and the chance to be able to perform the experiment. Halliday, D., Resnick, R., & Walker, J. (2014). Principles of Physics, 10th Ed, John Wiley & Sons Singapore Pte. Ltd., Singapore. (ABSTRACT) Young, H., Freedman, R., Sears, F., Zemansky, M. (1949). University Physics, 12th Ed, Pearson Education, United States. (MATERIALS AND METHODS) 4|Page

5|Page

Thank you for interesting in our services. We are a non-profit group that run this website to share documents. We need your help to maintenance this website.

To keep our site running, we need your help to cover our server cost (about $400/m), a small donation will help us a lot.