B1 Moments of Inertia 2

Kabir Bhasin Prof. Andrews

SN: 918601

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B1: Moments of Inertia 2

Introduction:...................................................................................................................2!

Aim: ...............................................................................................................................2!

Method: ..........................................................................................................................2!

Equipment Required: .................................................................................................3!

Rolling Disc Down Slope ..........................................................................................3!

Pendulum Swing Test/ Oscillating Disc ....................................................................3!

Calculations of Moment of Inertia .............................................................................3!

Results:...........................................................................................................................4!

Calculations Rolling Disc Down a Slope:...................................................................6!

a.)! Volume of Material in Discs..............................................................................6!

Small Disc: .............................................................................................................6!

Large Disc: .............................................................................................................6!

b.)! Mass of Discs:....................................................................................................7!

Small Disc ..............................................................................................................7!

Large Disc ..............................................................................................................7!

c.)! Moment of Inertia of Discs ................................................................................7!

Small Disc ..............................................................................................................7!

Large Discs ............................................................................................................8!

Calculations Pendulum Swing Test/ Oscillating Disc ................................................8!

Small Disc ..................................................................................................................8!

Large Disc ..................................................................................................................8!

Calculations Theoretical Moments of Inertia..............................................................9!

Small Disc ..................................................................................................................9!

Large Disc ..................................................................................................................9!

Discussion: ...................................................................................................................10!

Small Disc ................................................................................................................10!

Large Disc ................................................................................................................10!

Overall......................................................................................................................10!

Bibliography.................................................................................................................11!


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!"#$%&'(#)%"*+When making machinery, sometimes it is important to know the value of the moment

of inertia (the resistance to rotation of a body) of part. This can be calculated

theoretically for simple shapes, but for more advanced shapes, tests have to be carried

out and an experimental value would have to be calculated.

,)-*+The aim of this experiment is to work out the moment of inertia of 2 discs using two

different methods (rolling discs down a slop and a pendulum swing test) and

comparing the value obtained with the theoretical values.

./#0%&*+

Set up the apparatus as shown:

FIGURE 1 Rolling Disc Experiment

FIGURE 2 Oscillating Disc Experiment

Inclined Plane

(1.2m)

Large Steel Disc

Small Steel Disc

Measuring

Instruments

Knife Edged

Apparatus

Attachable

Pendulum


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12')3-/"#+4/2')$/&*+ Inclined plane (as shown above)

Large and Small steel discs (made from Steel)

Attachable pendulum (made from Cast Iron)

Stop-watch

Ruler

Spirit level

Digital callipers

Firstly, the Measurements of the Large and Small Disc, and the pendulum, need to be

recorded, using digital callipers (as Shown in the Results section below).

4%55)"6+7)8(+7%9"+:5%3/+(Use Discs when pendulum is not attached)

1. Place the inclined plane on a firm bench, and level it (using a spirit level), so

that when the disc is placed on it, it will not roll.

2. Adjust the plane (using a spirit level) so that when a disc rolls down it, it will

stay centre.

3. Measure the distance from the edge of the disc (at the starting position) to the

point where it stops (in this case it was 1.2m).

4. Now raise the starting end of the plane by 30mm, so that it is angled. Then

place the large disc at this point, and let it go, measuring the time taken for it

to reach the bottom (without the disc hitting the sides).

5. After taking the readings, do the same for the small disc.

6. Then repeat the procedure for a height up to 105mm, in increments of 15mm.

;/"&'5'-+:9)"6+8()55?#)"6+7)8(+(For this attach pendulum to the discs)

1. Place the apparatus on a firm bench, and level it using a spirit level.

2. Attach Pendulum to the Large Disc, and make sure the axel is perpendicular to

the knife-edge, and make sure the pendulum can swing freely.

3. Displace the pendulum by less than 30, and measure the time taken for the

pendulum to swing 5 oscillations.

4. Repeat this 5 times.

5. Then attach the pendulum to the small disc, and repeat the procedure.

@?5('5?#)%"8+%A+.%-/"#+%A+!"/$#)?+1. Calculate volume of material in discs

2. Use this information and density of the discs to calculate mass.

3. Use the mass and information obtained to calculate moment of inertia for the 2

discs.

4. Use the exact measurements of the discs and axels to calculate theoretical

moment of inertia.


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4/8'5#8*+

First, the Measurements of the Discs were taken:

FIGURE 3 Small Disc Dimensions

FIGURE 4 Large Disc Dimensions


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FIGURE 5 Pendulum Dimensions

After Taking the Dimensions, the result could be taken down (as shown in the tables

below):


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@?5('5?#)%"8+B+4%55)"6+7)8(+7%9"+?+:5%3/*+After obtaining the raw data, some calculations could be done (as shown below). But

First a graph of the results were plotted, so that the Moment of Inertia for the Discs

could easily be calculated.

[In all calculations the Pendulum rod is considered light, i.e. mass is negligible]

?CD E%5'-/+%A+.?#/$)?5+)"+7)8(8+

:-?55+7)8(*+

!

VS

= ["rrod

2(lrod1 + lrod 2)] + ["rdisc

2# l

disc] + [

1

3"r

rod

2# h

cone]

= [" (6.285)2 # (65.0 + 65.36)]+ [" (50.005)2 # (20.12)] + [1

3" (6.285)2 # (8.02)]

=174534.2763 mm3

=1.7453#10$4 m3 (4dp)

+ F?$6/+7)8(*+

!

VL

= ["rrod

2(lrod1 + lrod 2)] + ["rdisc

2# l

disc] + [

1

3"r

rod

2# h

cone]

= [" (6.245)2 # (63.49 + 64.56)]+ [" (75.115)2 # (22.56)] + [1

3" (6.245)2 # (7.32)]

= 400190.6064 mm3

= 4.0019 #10$4 m3 (4dp)


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+ ;/"&'5'-+G%H+

!

VBOB

= ["r2h1] +1

3"r2h2

#

$ % &

' (

= [" (14.34)2(16.14) + 21

3" (14.34)2(8.05)

#

$ % &

' (

=13893.80756

=1.3894 )10*5 m3 (4dp)

HCD .?88+%A+7)8(8*+

:-?55+7)8(+

!

MassS,MS = "steel #VS

= 7850kg /m3 # (1.7453#10$4 )m3

=1.3701kg (4dp)

+ F?$6/+7)8(+

!

MassL ,ML = "steel #VL

= 7850kg /m3 # (2.1593#10$4 )m3

=1.6951kg (4dp)

;/"&'5'-+G%H+

!

MassBOB ,MBOB = "CastIron #VBOB

= 7300kg /m3 # (1.3894 #10$5)m3

= 0.1014kg (4dp)

(CD .%-/"#+%A+!"/$#)?+%A+7)8(8+

:-?55+7)8(+

!

Moment of Inertia, IS = MSrS2 gk

2L2"1

#

$ % &

' (

= (1.3701)(0.006285)2(9.81)(0.0415)

2(1.2)2"1

#

$ %

&

' (

= "4.6470 )10"5 kg.m2 (4dp)

Assume the bob attached to

rod is symmetrical hence 2

cones for volume

Density of Cast Iron ranges

from 6800 to 7800 kg/m3.

Therefore use average of 7300

kg/m3


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F?$6/+7)8(8+

!

Moment of Inertia, IL = MLrL2 gk

2L2"1

#

$ % &

' (

= (1.6951)(0.00645)2(9.81)(0.0981)

2(1.2)2"1

#

$ %

&

' (

= "4.6956 )10"5 kg.m2 (4dp)

@?5('5?#)%"8+B+;/"&'5'-+:9)"6+8()55?#)"6+7)8(+

Using the Swing Test, the Theoretical Moment of Inertia can be calculated using the

following equation:

+

!

Moment of Inertia of Disc, I =T2mgLO

4" 2# IBOB #mLO

2

IBOB = 0.35mrs2

IBOB = 0.35 $castiron % "rs2h + 2

1

3"rs

2h

&

' (

)

* +

&

' (

)

* +

&

' (

)

* + rs

2

= 0.35 7800 % " (0.01434)2(0.01614) + 21

3" (0.01434)2(0.00805)

&

' (

)

* +

&

' (

)

* +

&

' (

)

* + (0.01434)

2

= 0.35(0.1084)(0.01434)2

IBOB = 7.8018 %10#6 kg.m2(4dp)

+ :-?55+7)8(+

!

IS =(0.92)2(1.3701)(9.81)(0.07557)

4" 2# (7.8018 $10#6) # (1.3701)(0.07557)2

= 0.0139 kg.m2(4dp)

+ F?$6/+7)8(+

!

IS =(1.88)2(1.6951)(9.81)(0.07557)

4" 2# (7.8018 $10#6) # (1.6951)(0.07557)2

= 0.1028 kg.m2(4dp)


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@?5('5?#)%"8+B+


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7)8('88)%"*+

:-?55+7)8(+The experiment showed two different ways of measuring moments of inertia, rolling

discs down a slope and by oscillating the disc using a pendulum. By using the results

from both the experiments to calculate the moments of inertia for the large and small

discs, different values were obtained. Firstly, the slope experiment gave negative

moments of inertia, which would suggest that the inertia was aiding the disc to roll

down, which is not possible, as moment of inertia is described as the resistance to

change in rotation. The problem with the value could be due to errors in the

measuring apparatus. However, when working out the moment of inertia using the

values from the oscillating disc experiment, a reasonable value of 0.0139 kg.m2 was

achieved. This would have meant that there was resistance to the change in rotation,

just as the definition of inertia describes.

However, according to the theoretical moment of inertia, the calculated inertia was

too large. This could be due to the fact that the theoretical value simplifies the inertia

too much, and doesnt take account for all variables.

F?$6/+7)8(+The calculated values for this experiment had the same general pattern as that of the

small disc. The rolling slope experiment gave a negative value, whereas, the

oscillating disc experiment gave a decent value for the moment of inertia of 0.1028

kg.m2. However, once again, the experimental value was too large for the theoretical

value.

>I/$?55+All results do have one correlation: the larger, heavier disc has a greater moment of

inertia than the smaller, lighter disc. This means that the large disc has a greater

resistance to change in rotation, meaning that the larger disc took longer to make one

oscillation, to roll down the slope, or just generally to rotate. This is supported by the

general equation of Moment of Inertia,

!

I = r2dm"

= mr2

. This shows that as the mass, and

radius of the disc increases (like for the large disc), then the Moment of Inertia, I, will

increase.

However, there was a problem with the results obtained. By using such a simple

shape in the experiment, the theoretical values, and both the experimental values

should not have been very different from each other. This could be due to errors done

during the experiment or due to errors in the measuring equipment. Ways of

improving this experiment could be:

Rolling Slope Experiment:

Using laser and touch sensitive sensors connected to stopwatch to measure the

time take for the disc to be let go and to reach the end.


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Also an electronic gate could be attached to hold the disc, so when it is

required to be let go, it will do so without human error (in the form of reaction

time, and altering direction of the disc).

Furthermore, supports/ guiders could be attached to the apparatus to prevent

the disc from moving sideways.

Oscillating Discs Improvements:

Using a laser sensor attached to a stopwatch to measure the time when the

pendulum reached its maximum point.

Using a protractor to make sure the angular displacement of the pendulum was

the same every time.

Using mechanical arms to hold the pendulum at its displaced angle to prevent

human error effecting the time, when releasing the pendulum.

Another improvement that could have been made is that instead of measuring the

volume by using a calliper, a eureka can could be filled with water, and the discs and

pendulums could be put in it individually, and the volume of water displaced would

provide a more accurate value for the volume. The callipers would not have been very

reliable due to the human error, when measuring the dimensions, due to the points

where the measurements were taken from, and also the callipers could have had a

slight error in them or in the digital read out.

@%"(5'8)%"*+To conclude, the experiments carried out did give a general correlation where the

greater the mass and greater the radius of the object, gave a greater moment of inertia,

meaning that the resistance to change in rotation was greater. This also means that 1

oscillation, and the time to roll down a slope, is greater than that for one where an

object is lighter and has a smaller radius.

I believe that these two methods would be a very good way of calculating the moment

of inertia of objects, however, in my case, there may have been too many errors,

causing problems in the calculations. Although the oscillating disc experiment gave

better results in my experiment, I believe it would be better to use the method where

the object is rolled down a slope. I believe this because it would be easier to measure,

because trying to measure the maximum displacement of an oscillation is harder.

G)H5)%6$?30J+ Notes Provided see Appendix

Wikipedia moments of inertia,

, [Accessed 21/02/10]

Wikipedia List of Moments of inertia,

, [Accessed

21/02/10]

Hyperphysics moments of inertia, , [Accessed 21/02/10]

B1 Moments of Inertia 2

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