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Experiment No DYN4 : Moment of Inertia of a rotating diskMUHAMMAD FAZLI BIN MOHAMAD ASRIL(BK17110141)GROUP 11HK08 MECHANICAL ENGINEERINGUNIVERSITY MALAYSIA SABAHFACULTY OF ENGINEERINGTABLE OF CONTENTS3 EXPERIMENTAL PROCEDURE 4-3 EXPERIMENTAL PROCEDURE3 APPARATUSThe following are the required apparat...

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Experiment No DYN4 : Moment of Inertia of a rotating disk

MUHAMMAD FAZLI BIN MOHAMAD ASRIL (BK17110141) GROUP 11 HK08 MECHANICAL ENGINEERING UNIVERSITY MALAYSIA SABAH FACULTY OF ENGINEERING

TABLE OF CONTENTS NO.

TOPIC

PAGES

1.0

INTRODUCTION

3

2.0

OBJECTIVES

3

3.0

EXPERIMENTAL PROCEDURE

4-5

4.0

RESULTS

6-7

5.0

CALCULATION

8-9

6.0

DISCUSSION

9

7.0

CONCLUSION

10

8.0

REFERENCES

10

1.0 INTRODUCTION Rotating an object requires that one overcomes that object’s rotational inertia, better known as its moment of inertia. For highly symmetrical cases it is possible to develop formulas for calculating an object’s moment of inertia. In this experiment, we are going to determine the moment of inertia of a rotating disk. The different of mass of disk will change the time taken and the trace length and thus we can calculate the velocity and acceleration of the rotating disk. The velocity and acceleration for both linear and angular can also be determined from the result that we obtained. Newton’s second law for rotating objects states that the net torque on an object equals moment of inertia multiplied by angular acceleration 𝛴𝜏 = 𝐼𝛼 .In this expression, 𝜏 is the torque, 𝐼 is the moment of inertia, and 𝛼 is the angular acceleration. The moment of inertia, 𝐼, of a rigid body depends on the mass of the body as well as how the mass is distributed around the rotation axis. Mass distribution is calculated based on the shape of the object. For example, the moment of inertia for a solid cylinder or disk about an axis through its centre of mass is 𝐼 =1/2𝑀𝑅 where 𝑀is the mass of the disk, and 𝑅 is the radius. When a force is applied to rotate an object about some axis, we produce a torque defined as 𝜏 = 𝑟×𝐹 where 𝑟 is the position vector from the rotation axis to the point where the force is applied. The magnitude of the torque vector is given by 𝜏 = 𝑟𝐹𝑠𝑖𝑛𝜃 where 𝜃 is the angle between the force and position vector 𝑟. The torque is usually defined as positive for counterclockwise rotation and negative for clockwise.

2.0 OBJECTIVE 

To determine the moment of inertia of a rotating disk

3.0 EXPERIMENTAL PROCEDURE 3.1 APPARATUS The following are the required apparatus used for the experiment : 

Angular acceleration apparatus



Cord



Weights



Adhesive tapes



Ink



Ruler and Scissor



Paper strips

The apparatus and materials were then setup as shown in figure 1:

Figure 1 shows the angular accelertaion apparatus of rotating disk

3.2 PROCEDURE 

The mass and diameter of the rotating disk was recorded before the experiment started.



The disk was check whether it free to rotated and the top pivot of the angular acceleration apparatus was adjusted to let the disk rotated smoothly.



A paper strip was fastened to the periphery of the disk by adhesive tape. And the paper was fastened correctly on the positioned with the start of the paper just before the brush.



A cord was connected to the pulley on the axle and passes the cord over the pulley on the vertical pillar and located the weight hanger with the require number of weight on the end of the cord.



The mass of the weight,𝑀 and the radius of the turning pulley, 𝐷 was recorded in Table 1.



The brush was inked and the position the brush was adjusted such that the tip of the brush just touches the surface of the paper.



Before the brush was vibrated, a line was drawing on the paper strip by rotating the disk with contact the brush.



The brush and weight was released simultaneously, and the disk was stopped when it almost completed one revolution to avoid the line overlapping happened on the paper strip.



Lastly, the paper strip was removed carefully and analysis was done on it. The readings was recorded and tabulated in Table 1 and the angular velocity and angular acceleration against time was plotted on a graph paper.

4.0 RESULTS Natural frequency of the vibrator, f = 5Hz Period of an oscillation, T = 0.2s Diameter of the disk, d = 0.3m Disk of mass, m = 3.25kg Weight mass, M = 0.31kg & 0.41kg Diameter of pulley, D = 0.055m

Table 1 : Results of using weight of mass of 0.31kg and 0.41kg Mass,

Time,

Trace

Angle, θ

m(kg)

t(s)

length(m)

(radian)

0.31

0.41

Velocity

Acceleration

Linear,

Angular,

Linear,

Angular,

v(m/s)

w(rad/s)

(m/s2)

(rad/s2)

0.0

0.000

0.000

0.000

0.000

0.000

0.000

0.2

0.008

0.053

0.040

0.265

0.200

1.325

0.4

0.022

0.147

0.055

0.368

0.137

0.913

0.6

0.041

0.273

0.068

0.455

0.114

0.758

0.8

0.066

0.440

0.083

0.550

0.104

0.688

1.0

0.096

0.640

0.096

0.640

0.096

0.640

1.2

0.131

0.873

0.109

0.728

0.091

0.607

1.4

0.172

1.147

0.123

0.819

0.088

0.585

1.6

0.219

1.460

0.137

0.913

0.086

0.571

1.8

0.271

1.807

0.151

1.004

0.084

0.558

0.0

0.000

0.000

0.000

0.000

0.000

0.000

0.2

0.012

0.080

0.060

0.400

0.300

2.000

0.4

0.033

0.220

0.083

0.550

0.208

1.375

0.6

0.065

0.433

0.108

0.722

0.180

1.203

0.8

0.107

0.713

0.134

0.891

0.168

1.114

1.0

0.160

1.067

0.160

1.067

0.160

1.067

1.2

0.225

1.500

0.188

1.250

0.157

1.042

1.4

0.300

2.000

0.214

1.429

0.153

1.021

1.6

0.387

2.580

0.242

1.613

0.151

1.008

1.8

0.489

3.260

0.272

1.811

0.151

1.006

Table 1 For Mass = 0.31kg

Angular velocity, w(rad/s) and angular acceleration, (rad/s2) against time, t(s)

1.4

1.325 1.2 1

1.004 0.913

0.913

0.819

0.8

0.758

0.728

0.688 0.6

0.64

0.607

0.55

0.585

0.571

0.558

0.455

0.4

0.368 0.265

0.2 0

0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Table 2 For Mass = 0.41kg

Angular velocity, w(rad/s) and angular acceleration, (rad/s2) against time, t(s)

2.5

2

2 1.811 1.613

1.5

1.429

1.375 1.25

1.203

1.114

1.067 1.067

1

1.042

1.021

1.008

1.006

0.891 0.722 0.55

0.5

0.4

0

0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

5.0 CALCULATIONS Experimental value : For weight of mass 0.31kg 1 =

1.325+0.913+0.758+0.688+0.640+0.607+0.585+0.571+0.558 9

= 0.738 rad/s2

For weight of mass 0.41kg 2 =

2.000+1.375+1.203+1.114+1.067+1.042+1.021+1.008+1.008 9

= 1.204 rad/s2

Theoretical value : Moment of inertia of rotating disk was calculated by using equation : 1

I = 2mr2 =

1 2

(3.25) (0.15)2

= 0.03656 kgm3 For weight of mass 0.31kg, the value was calculated by using equation : mgr = I (0.31) (9.81) (0.028) = (0.03656)  T1 = 2.329 rad/s2 For weight of mass 0.41kg, the value was calculated by using equation : mgr = I (0.41) (9.81) (0.028) = (0.03656) T2 = 3.080 rad/s2

Percentage of error For the weight of mass 0.31kg, the percentage error is : Percentage error =

𝑎𝑇1−𝑎1 x 𝑎𝑇1

=

100%

2.329−0.738 x 2.329

100%

= 68.31%

For the weight of mass 0.41kg. the percentage error is : Percentage error =

𝑎𝑇2−𝑎2 𝑎𝑇2

=

x 100%

3.080−1.204 3.080

x 100%

= 60.91%

6.0 DISCUSSION In the experiment that was conducted, for the weight of mass 0.31kg, the theoretical value of the angular acceleration is 2.329 rad/s2 while the experimental value of angular acceleration is 0.738 rad/s2. The percentage error between these two values is 68.31%. For the weight of mass 0.41kg, the theoretical value of the angular acceleration is 3.080 rad/s2 while the experimental value of angular acceleration is 1.204 rad/s2 . The percentage error between these two values is 60.91%. In this experiment, there are some errors that deviate the results of the experiment such as the pulley is normally assuming smooth pulley but in reality, the pulley was not smooth enough that it changes the result. Apart from that, when the weight start to release, it will exist kinetic friction between the rope and the pulley. This will affect the angular acceleration of the rotating disk. Besides that, the vibrating arm will loosen each time the paper strips of the trolley was drawn by the brush, this mostly is caused by the screw line that worn off. The experiment can be improved by changing the pulley to a pulley that having the smoother surface. This will reduce the friction between the rope and the pulley. In addition to that, the screw must be checked each time an experiment is conducted.

7.0 CONCLUSION The objective of this experiment was achieved. However, there were some possible errors that had shifted the results of the experiment, steps of improvement must be taken wisely in order to have a better results in the future experiment as mentioned in discussion above.

8.0 REFERENCES 1. KM20801 Experiment No.: DYN4 Moment of Inertia of a rotating disk. Taken from lab sheet. 2. Theory of Machines by J.K. Gupta and R.Ss Khurmi...