Abstract:
In this experiment a bifilar suspension methods are applied to determine
the mass momentum of inertia of a beam, platform, cylindrical mass and a body with an
irregular mass.
First with the bifilar method a body was suspended by two parallel cords
with the same length and by displacing the system through a small angle and using few
equations which will be given further in the report we get the time period and then
calculating the mass moment of inertia.
Moreover, and for the trifilar method a platform was suspended by three
cords having the same length and also by getting the time period and the moment of
inertia of the platform we calculated the moment of inertia of a cylindrical mass and a
body with an irregular mass.
1
Table of contents:
Abstract …………………………………………………………………………...….. (3)
Table of contents ………………………………………………………………....…..... (4)
List of figures and tables……………………………………….………………………. (5)
Nomenclature…………………….……………………….…………………....……..... (6)
Chapter 1
1.1 Introduction ………………………………………………………… …………(7)
1.2 objectives ………………………………………………………………………..(8)
Chapter 2
Theoretical background ……………………………………………………..… (9)
Chapter 3
Experimental setup and procedure …………………………………………….... (15)
3.1 Equipments …..……………………………………………………………… (15)
3.2 Procedure……………………………………………………………………. (16)
Chapter 4
Results and discussion………………………………………………………..… (18)
4.1 Sample calculations…………………………………………………..……. (19)
4.2 Discussion ………………………………………………………………….. (39)
4.3 Uncertainty Analysis ……………………………………………………....... (40)
Chapter 5
Conclusions……………………………………………………..………………. (42)
References………………………………………………………………………..….. (43)
2
List of figures and tables:
Figures:
Figure (1): Bifilar suspension………………………….………………………… (7)
Figure (2): (a) Bifilar suspension, (b) Trifilar suspension……..………………. (10)
Figure (3): The relation between log L and log τ………………………………. (38)
3
List of tables:
Table 1: Bifilar suspension with cord length equal to 0.4 (m) ..........................................21
Table 2: Bifilar suspension with cord length equal to 0.45 (m) ........................................22
Table 3: Bifilar suspension with cord length equal to 0.48 (m) ........................................23
Table 4: Trifilar suspension with cord length equal to 0.4 (m) ..... Ошибка! Закладка не
определена.
Table 5: Trifilar suspension with cord length equal to 0.44 (m) ... Ошибка! Закладка не
определена.
Table 6: Trifilar suspension with cord length equal to 0.5 (m) ..... Ошибка! Закладка не
определена.
4
Nomenclature
Symbols
Scientific Name
d
Distance between the cords (m)
F
Restoring force (N)
g
Acceleration of gravity (m/𝑠 2 )
I
Mass-Moment of inertia of the body (𝑚4 )
𝑘
Radius of gyration
𝐿
Length of the cords (m)
m
Mass (kg)
𝑀𝑅
Restoring moment (N.m)
T
𝜃
Tension in the cords (N.m)
Angle of twist (degrees)
5
̈
𝜃(𝜔)
Angular velocity (rad/s)
𝜏
Time period (s)
Chapter 1
Introduction
In this experiment will introduce method for determining the mass moment of inertia
which is the bifilar suspension.
1.1 Bifilar suspension method
The bifilar suspension method is a way to determine the mass momentum of
inertia. (Fig. a) shows a uniform rod of mass M and length L suspended horizontally by
two vertical strings. The length of each string is l and they are attached symmetrically
about the center O, a distance R apart. If the bat is now twisted horizontally, it will
undergo SHM. We wish to analyze this motion, and in particular to find the period.
l
B’
φ
A
O
B
A
R
L
A’
B
θ
Figure 1: Bifilar suspension.
And by calculating the time period we can calculate the mass moment of inertia
by using some formulas we will provide later in the report.
6
1.2 Objectives
The objectives of this experiment are to introduce the bifilar
suspension methods as options for determining the mass moment of inertia of a
rigid body or group of rigid bodies. this should lead to the capability of designing
experiments for determining moments of inertia of rigid bodies using either bifilar
or trifilar pendulum systems.
The other objective is to compare theoretical results obtained from
pendulum modeling with experimental results and determining accuracy.
Chapter 2
7
Theoretical Background
Inertia Determination
Mass- moment of inertia I, about an axis of rotation is theoretically can be determined by:
I= ∫𝑴 𝒓𝟐 𝒅𝒎
(1)
Where:
r : Distance between the location of mass element, dm, and the axis of rotation
The integration is performed over the entire mass of the body if the axis of rotation
coincides with the body centroidal axis, the moment of inertia is given by Ī.
The parallel axis theorem is given by:
I= Ī + m · 𝒅𝟐
(2)
Where:
I: The inertia of the body about the axis of rotation.
Ī: the inertia of the body about its own centroidal axis.
m: Mass of the body.
d: the perpendicular distance from the centroidal axis of the body to the axis of rotation.
Bifilar suspension:
8
Figure 2: (a) Bifilar suspension, (b) Trifilar suspension.
In figure1 (a) shows the Bifilar suspension and the body is suspended by two
parallel cords of length L at a distance R=d apart, the tensions in the cords are
respectively:
𝑻𝟏 = 𝒎𝒈
𝒅𝟐
𝒅
𝒅𝟏 + 𝒅𝟐 = 𝒅
and
and
𝑻𝟐 = 𝒎𝒈
𝒅𝟏
𝒅
(4)
𝑻𝟏 + 𝑻𝟐 = 𝒎𝒈 (𝟓)
If the system is displaced through a small angle θ about its central axis, then
angular displacement 𝛷1 and 𝛷2 will result at the supporting chords. If both angles are
small, then:
L𝛟𝟏= 𝒅𝟏 Ɵ
𝛟𝟏 =
𝒅𝟏
𝑳
Ɵ
L𝛟𝟐= 𝒅𝟐 Ɵ
and
𝛟𝟐 =
and
The horizontal restoring forces at the points of attachment with the plate are:
9
𝒅𝟐
𝑳
Ɵ
(6)
(7)
𝐅𝟏 = 𝐓𝟏 𝐬𝐢𝐧𝚽𝟏
(8)
𝐅𝟐 = 𝐓𝟐 𝐬𝐢𝐧𝚽𝟐
(9)
And using equation (7) to substitute into (8) and (9) and then get:
𝑴𝑹 = ( 𝑭𝟏 𝒅𝟏+ 𝑭𝟐 𝒅𝟐 ) = −
𝑻𝟏 𝒅𝟏 𝟐 + 𝑻𝟐 𝒅𝟐 𝟐
𝑳
Ɵ
(10)
Where the minus sign is related to the restoring effect.
Using equation (4), the restoring moment is rewritten as:
𝑴𝑹 = -mg
𝒅𝟏 𝒅𝟐
𝑳
Ɵ
(11)
Using Newton’s second law and summing moment about the vertical centroidal axis give
equation of motion of the body can be written as:
Ī ϴ΄΄ = - mg
𝑑1 𝑑2
𝐿
Ɵ → ϴ΄΄ + mg
𝑑1 𝑑2
Ī𝐿
Ɵ=0
(12)
Where:
Ī = Mass-moment of inertia of the body about the centroidal axis
The natural frequency of the system can be directly obtained from as:
𝝎 = √𝐦𝐠
𝒅𝟏 𝒅𝟐
(13)
Ī𝑳
The period that measured experimentally is related to the natural frequency by:
𝝉=
𝟐𝝅
𝝎
=2𝝅√
Ī𝑳
𝐦𝐠 𝒅𝟏 𝒅𝟐
Using equation (14) the experimental mass-moment of inertia is given by:
10
(14)
Ī=
𝒎𝒈𝒅𝟏 𝒅𝟐 𝝉𝟐
(15)
𝟒 𝝅𝟐 𝑳
Equation (15) can be rewritten in terms of the radius of gyration:
𝝉 = 𝟐𝝅𝒌√
𝑳
𝒈𝒅𝟏 𝒅𝟐
(16)
Where:
k = Radius of gyration defined by
Ī = m 𝒌𝟐
(17)
To reduce the error, different readings at different cords length, L, should be taken and a
relation between the cord length and the corresponding mass moment of inertia is to be
obtained. In this regard, equation (15) can be written as:
L=
𝒎𝒈𝒅𝟏 𝒅𝟐 𝝉𝟐
𝟒 𝝅𝟐 Ī
(18)
This can be rewritten as,
L = k 𝝉𝟐
(19)
With k defined as:
k=
𝒎𝒈𝒅𝟏 𝒅𝟐
𝟒 𝝅𝟐 Ī
11
(20)
Chapter 3
Experimental setup and procedure
First of all, the needed equipments for the Bifilar and experiment will be illustrated with
some figures for more details.
the procedure involved in the Bifilar experiment will be explained briefly. Beginning
with adjusting the length of the cords until recording the needed data such as the
oscillation time.
Finally, after the set of procedure is done for the both cases, The study of the
relationship between the mass-moment of inertia of the body and the time period is
simply investigated.
3.1 Equipments:
In order to perform the bifilar and trifilar experiment, a set of equipments is
needed and is illustrated as follows:
1. Stop watch.
2. Ruler.
3. Electronic scale for weighting parts.
For the previous set of equipments, some helpful notes should be considered
such as being careful while recording the cord length and weighting parts
12
3.2 Procedure:
the procedure related to the bifilar apparatus is illustrated
3.2.1 Bifilar apparatus procedure:
In order to perform this case of this experiment sufficiently and achieve the needed
results with a great possibility to eliminate the human errors as much as possible, a
specific set of procedures should be followed and is illustrated as follows:
1. Measure the dimension and the mass of the beam.
2. Suspend the beam by the cords and adjust their common length L.
3. Displace the beam by small angle, release it and determine the periodic time of free
oscillations by timing 20 oscillations.
4. Repeat step 3 for three different values of L.
5. Measure the dimensions of the cylindrical masses.
6. Place the cylindrical masses of known weight either side of the center of gravity of the
beam, maintaining a specified distance between the masses.
7. Repeat the procedure with an irregularly shaped body that can be fixed on the beam.
8. In each test, record L, time period and the mass of the beam. Then use equation (13) and
(14) to calculate the mass-moment of inertia and radius of gyration of the beam alone and
of the beam with masses placed on it.
13
Results and discussion:
some sample calculations are obtained in order to achieve the desired results. In the other
hand, these results which are obtained are discussed briefly.
for bifilar setups, the time period, theoretical and experimental mass-moment of inertia
of different cases and the percentage error between them, and the radius of gyration of the
beam alone and of the beam with masses placed on it.
4.1 Sample calculations.
14
Bifilar Apparatus:
1. Total mass of frame hanging with wires = 1.444 kg.
2. Width of frame = 25.25 mm.
3. Height of frame = 12.75 mm.
4. Length of frame = 503 mm.
5. Distance between the wires = 15 mm.
6. Number of holes in hanging rectangular x-section bar = 15.
7. Distance between each hole from center to center = 25 mm.
8. Diameter of each hole = 10.42 mm.
9. Diameter of each cylinder fixed at the end of rectangular section = 20 mm.
10. Length of each cylinder fixed at the end of rectangular section = 50 mm.
11. Length of wires = 400mm. 450mm. 480mm.
12. Diameter of cylindrical mass 76.25.
13. Height of cylindrical mass = 50.9 mm.
14. Mass of each cylindrical mass = 1.793.
15. Distance between two masses 150mm. 200mm. 250mm.
16. Mass of the irregular shape = 2.513 kg.
For the Bifilar apparatus case, the following three cases are to be investigated
where the cord length was 400 (mm) as follows:
(a) The beam only:
15
Theoretical:
𝐼𝑧𝑧 = 𝐼 (𝑏𝑒𝑎𝑚) − 𝐼 (ℎ𝑜𝑙𝑒𝑠) + 𝐼(𝑓𝑖𝑥𝑒𝑑 𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟𝑠)
𝐼𝑧𝑧 = (((1/12) ∗ 1.444 ∗ (0.503^2 + 0.02525^2)) − (2 ∗ (1.444 ∗ (0.00521^2) −
(0.00008528 ∗ (0.025^2)))) − (2 ∗ (1.444 ∗ (0.00521^2) − (0.00008528 ∗ (0.05^2)))) −
(2 ∗ 1.444 ∗ (0.00521^2)) − (0.00008528 ∗ (0.075^2))) − (2 ∗ (1.444 ∗ (0.00521^2) −
(0.00008528 ∗ (0.1^2)))) − (2 ∗ (1.444 ∗ (0.00521^2) − (0.00008528 ∗ (0.125^2)))) − (2 ∗
(1.444 ∗ (0.00521^2) − (0.00008528 ∗ (0.15^2)))) − (2 ∗ (1.444 ∗ (0.00521^2) −
(0.00008528 ∗ (0.175^2)))) + (2 ∗ (0.5 ∗ 0.1233 ∗ (0.01^2) + 0.1233 ∗ 0.2525^2))
= 0.45721497 (𝑚4 )
Experimental:
16
For first trail when the length of the cord = 400 mm with having 20
oscillations, the calculated mass-moment of inertia of the platform and the time
period were found to be:
τ=
𝒕𝒊𝒎𝒆
𝒄𝒚𝒄𝒍𝒆𝒔
=
𝟏𝟕.𝟗𝟏
𝟐𝟎
= 0.8955 (s)
And the mass-moment of inertia of the platform was calculated
using equation (13) as follows:
𝐼𝑝 =
(𝑚)𝑑1 𝑑2 𝑔 𝜏2
4 𝜋2 𝐿
=
1.444∗0.2525∗0.2525∗ 0.76752 ∗ ( 9.81)
4 𝜋2 ( 0.4)
Also, the percentage error was found as shown:
% Error =
𝑡ℎ𝑒𝑜𝑟𝑖𝑡𝑖𝑐𝑎𝑙−𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑚𝑡𝑎𝑙
𝑡ℎ𝑒𝑜𝑟𝑖𝑡𝑖𝑐𝑎𝑙
∗ 100 = 0.482%
(b) The beam with one cylindrical mass placed on it.
Theoretical:
17
= 0.45501 (𝑚4 )
𝐼 = 𝐼 (𝑝𝑙𝑎𝑡𝑒) − 𝐼(ℎ𝑜𝑙𝑒𝑠) + 𝐼(𝑓𝑖𝑥𝑒𝑑 𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟𝑠 ) + 𝐼(𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟 𝑝𝑙𝑎𝑐𝑒𝑑)
= (((1/12) ∗ 1.444 ∗ (0.503^2 + 0.02525^2)) − (2 ∗ (1.444 ∗ (0.00521^2) −
(0.00008528 ∗ (0.025^2)))) − (2 ∗ (1.444 ∗ (0.00521^2) − (0.00008528 ∗
(0.05^2)))) − (2 ∗ 1.444 ∗ (0.00521^2)) − (0.00008528 ∗ (0.075^2))) − (2 ∗
(1.444 ∗ (0.00521^2) − (0.00008528 ∗ (0.1^2)))) − (2 ∗ (1.444 ∗ (0.00521^2) −
(0.00008528 ∗ (0.125^2)))) − (2 ∗ (1.444 ∗ (0.00521^2) − (0.00008528 ∗
(0.15^2)))) − (2 ∗ (1.444 ∗ (0.00521^2) − (0.00008528 ∗ (0.175^2)))) + (2 ∗
(0.5 ∗ 0.1233 ∗ (0.01^2) + 0.1233 ∗ 0.2525^2)) + 2 ∗ (0.5 ∗ 1.793 ∗ 0.038125^2)
= 0.048327651 (𝑚4 )
Experimental:
Using equation (13), the following experimental data of the massmoment of inertia was found as follow:
τ=
𝒕𝒊𝒎𝒆
𝒄𝒚𝒄𝒍𝒆𝒔
𝟏𝟐.𝟏𝟑
=
𝟐𝟎
𝟐
(𝒎)𝒅𝟏 𝒅𝟐 𝝉
𝑰̅ =
𝟐
= 𝟎. 𝟔𝟎𝟔𝟓 (𝐬)
𝒈
𝟒𝝅 𝑳
=
(𝟏.𝟒𝟒𝟒+𝟏.𝟕𝟗𝟑) ( 𝟎.𝟓𝟐)𝟐 (𝟎.𝟐𝟓𝟐𝟓)(𝟎.𝟐𝟓𝟐𝟓) ( 𝟗.𝟖𝟏)
𝟒 𝝅𝟐 ( 𝟎.𝟒)
= 0.045155684 (𝑚4 )
%Error =
𝒕𝒉𝒆𝒐𝒓𝒊𝒕𝒊𝒄𝒂𝒍−𝒆𝒙𝒑𝒆𝒓𝒊𝒎𝒆𝒎𝒕𝒂𝒍
𝒕𝒉𝒆𝒐𝒓𝒊𝒕𝒊𝒄𝒂𝒍
= 𝟔. 𝟓𝟔𝟑𝟒 %
( c ) The beam with two cylindrical masses spaced 250 (mm) .
18
Theoretical:
I = I (plate) – I (holes) +I (fixed cylinders) + I (cylindrical masses placed)
== (((1/12) ∗ 1.444 ∗ (0.503^2 + 0.02525^2)) − (2 ∗ (1.444 ∗
(0.00521^2) − (0.00008528 ∗ (0.025^2)))) − (2 ∗ (1.444 ∗ (0.00521^2) −
(0.00008528 ∗ (0.05^2)))) − (2 ∗ 1.444 ∗ (0.00521^2)) − (0.00008528 ∗
(0.075^2))) − (2 ∗ (1.444 ∗ (0.00521^2) − (0.00008528 ∗ (0.1^2)))) − (2 ∗
(1.444 ∗ (0.00521^2) − (0.00008528 ∗ (0.125^2)))) − (2 ∗ (1.444 ∗
(0.00521^2) − (0.00008528 ∗ (0.15^2))
= 0.104358901 (𝑚4 )
Experimental:
In this case, with the two cylindrical masses placed 250 (mm), the massmoment of inertia was calculated using equation (13) as follows:
τ=
̅𝑰 =
=
𝒕𝒊𝒎𝒆
𝒄𝒚𝒄𝒍𝒆𝒔
=
𝟏𝟓.𝟎𝟗
𝟐𝟎
= 0.7545 (s)
(𝒎)𝒅𝟏 𝒅𝟐 𝒈 𝝉𝟐
𝟒 𝝅𝟐 𝑳
(𝟓.𝟎𝟑) ( 𝟎.𝟓𝟕𝟑𝟓)𝟐 (𝟎.𝟐𝟓𝟐𝟓)(𝟎.𝟐𝟓𝟐𝟓)( 𝟗.𝟖𝟏)
𝟒 𝝅𝟐 ( 𝟎.𝟒)
= 0.104606813 (𝑚4 )
% Error =
𝒕𝒉𝒆𝒐𝒓𝒊𝒕𝒊𝒄𝒂𝒍−𝒆𝒙𝒑𝒆𝒓𝒊𝒎𝒆𝒎𝒕𝒂𝒍
𝒕𝒉𝒆𝒐𝒓𝒊𝒕𝒊𝒄𝒂𝒍
∗ 𝟏𝟎𝟎 = 0.294 %
As the previous cases were discussed briefly on the
theoretical and experimental basis, the same previous steps were
19
repeated for different cord length as illustrated in the next pages
tables.
20
Table 1: Bifilar suspension with cord length equal to 0.4 (m)
System
Only beam
Beam with one
mass in the center
Beam with two
masses of distance
150 mm
Beam with two
masses of distance
200 mm
Beam with two
masses of distance
250 mm
Irregular shape
Cord
Length L
(m)
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
No. of
Oscillations
(cycles’)
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
Time of
Oscillations
(sec)
17.85
17.72
17.91
12.06
12.13
12.09
11.85
11.78
11.94
13.28
13.32
13.14
14.85
15.09
15.09
16.35
16.25
16.44
Period,
τ
0.8925
0.886
0.8955
0.603
0.6065
0.6045
0.5925
0.589
0.597
0.664
0.666
0.657
0.7425
0.7545
0.7545
0.8175
0.8125
0.822
21
Average time
(s)
Average
Period
Plate Moment of inertia
Theoretical
17.82666667
0.891333333
0.045721497
12.09333333
0.604666667
0.048327651
11.85666667
0.592833333
0.068498901
13.24666667
0.662333333
0.084187651
15.01
0.7505
0.104358901
16.34666667
0.817333333
0.048773142
Experimental
0.045196926
0.044540993
0.045501282
0.044636018
0.045155684
0.044858364
0.064508686
0.063748809
0.065492285
0.081017277
0.081506068
0.079318085
0.101305821
0.104606813
0.104606813
0.061356248
0.060608008
0.062033588
% Error
1.147318216
2.581945984
0.481645472
7.638759785
6.563461263
7.178679599
5.825225218
6.934552391
4.389290021
3.765841915
3.185244993
5.784180162
3.626517664
0.294475985
0.294475985
14.94649851
14.94649851
14.94649851
Table 2: Bifilar suspension with cord length equal to 0.45 (m)
System
Only beam
Beam with one mass
in the center
Beam with two
masses of distance
150 mm
Beam with two
masses of distance
200 mm
Beam with two
masses of distance
250 mm
Irregular shape
Cord
Length L
(m)
0.45
0.45
0.45
0.45
0.45
0.45
0.45
0.45
0.45
0.45
0.45
0.45
0.45
0.45
0.45
0.45
0.45
0.45
No. of
Oscillations
(cycles)
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
Time of
Oscillations
(sec)
18.63
19
19
13
12.82
12.78
12.68
12.72
12.53
15.03
15.06
15.03
15.75
15.85
15.82
17.35
17.16
17.16
Period,
τ
0.9315
0.95
0.95
0.65
0.641
0.639
0.634
0.636
0.6265
0.7515
0.753
0.7515
0.7875
0.7925
0.791
0.8675
0.858
0.858
22
Average time
(s)
Average
Period
Plate Moment of inertia
Theoretical
18.87666667
0.943833333
0.045721497
12.86666667
0.643333333
0.048327651
12.64333333
0.632166667
0.068498901
15.04
0.752
0.084187651
15.80666667
0.790333333
0.104358901
17.22333333
0.861166667
0.048773142
Experimental
0.043762855
0.045518416
0.045518416
0.046102556
0.044834709
0.044555366
0.06565495
0.066069831
0.064110788
0.09224587
0.092614485
0.09224587
0.101295485
0.10258586
0.10219789
0.061414329
0.060076597
0.060076597
% Error
4.283854264
0.4441707
0.4441707
4.604185043
7.227626401
7.805646014
4.151819235
3.546145034
6.406107612
9.571735346
10.009584
9.571735346
3.638795325
2.106058254
2.566897967
15.01548924
15.01548924
15.01548924
Table 3: Bifilar suspension with cord length equal to 0.48 (m)
System
Only beam
Beam with one
mass in the center
Beam with two
masses of distance
150 mm
Beam with two
masses of distance
200 mm
Beam with two
masses of distance
250 mm
Irregular shape
Cord
Length L
(m)
0.48
0.48
0.48
0.48
0.48
0.48
0.48
0.48
0.48
0.48
0.48
0.48
0.48
0.48
0.48
0.48
0.48
0.48
No. of
Oscillations
(cycles)
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
Time of
Oscillations
(sec)
19.53
19.56
19.72
13.44
13.28
13.41
13.22
13.25
12.63
14.63
14.66
14.47
16.28
16.13
16.25
17.66
17.47
17.41
Period,
τ
0.9765
0.978
0.986
0.672
0.664
0.6705
0.661
0.6625
0.6315
0.7315
0.733
0.7235
0.814
0.8065
0.8125
0.883
0.8735
0.8705
23
Average time
(s)
Average
Period
Plate Moment of inertia
Theoretical
19.60333333
0.980166667
0.045721497
13.37666667
0.668833333
0.048327651
13.03333333
0.651666667
0.068498901
14.58666667
0.729333333
0.084187651
16.22
0.811
0.104358901
17.51333333
0.875666667
0.048773142
Experimental
0.045087453
0.045226077
0.045968998
0.046196398
0.045103031
0.045990395
0.066905706
0.067209707
0.061067051
0.081938668
0.082275056
0.080156235
0.101463315
0.099602212
0.101089716
0.059651783
0.058375127
0.057974842
% Error
1.386753778
1.083561808
0.541321699
4.410006579
6.672411469
4.836271348
2.325869495
1.882064854
10.84959004
2.671393071
2.271823432
4.788607411
3.439442497
5.650103475
3.883211714
12.92189689
12.92189689
12.92189689
Linear relation for case (a) :
By drawing the relation between log τ and log l, we obtained with the following
linear graph, and it is intersection with the ordinate is equal to the constant C. and it slope
is equal 2.
log l
0,45
0,4
0,35
0,3
0,25
0,2
0,15
0,1
0,05
0
0,035
0,03
0,025
0,02
0,015
0,01
0,005
0
-0,005
-0,01
-0,015
log τ
Figure (3): The relation between log L and log τ
The pervious graph indicates that the relation between log L and log τ is
linear. And it is intercept with y-axis at value equal to = 0.388 which is the value
of constant C that we had calculated in each case. And the linear graph has a slope
equal to 2. And that’s what we expected in the theoretical equation.
24
-0,02
Chapter 5
Conclusion
The conclusion for this experiment is that the bifilar suspension methods for determining
the mass-moment of inertia can be considered as successful methods applied for that
purpose.
Moreover, the errors involved in this experiment whether in the case of bifilar
suspensions are not large which can be considered as a good indicator of how successful
this experiment was.
Finally, when this experiment was performed, some recommendations and
notices should be taken in mind for the purpose of improving the obtained results such as:
25