AIM:
To determine the moment of inertia of a horizontal rectangular drop bar about
its center of mass using the bifilar suspension technique.
INTRODUCTION
The bifilar suspension is a technique
used to determine the moment of inertia of any type of object about any point
on the object. This is done by suspending two parallel cords of equal length
through the object examined. However, the approach taken for this experiment is
to determine the moment of inertia of a drop by suspending the cords through
the mass centre of bodies, obtaining an angular displacement about the vertical
axis through the centre of mass by a sensibly small angle.
SYSTEM
LAYOUT AND APPARATUS USED
The apparatus used for this
experiment consists of a uniform rectangular drop bar suspended by fine wires
(assumed to have negligible weight contribution to the system). This
rectangular bar contains holes equidistant from each other and two extra with
equal masses of 1.85kg, are made to peg through these holes. Chucks are also in
place to alter length of suspended wires.
In addition an enclosed-type
measuring tape and weight balance were also used for the experiment.
A diagram on the arrangement of the
apparatus used is illustrated below:
PROCEDURE
1.
With
the bar suspended by the wires, the length L was adjusted to a convenient
extent and then distance, b, between the wires was measured.
2.
The
bar was then tilted through a very small angle about the vertical axis and time
taken for 20 oscillations of the bar, was recorded. From this, the periodic
time was also calculated.
3. The length L
was further adjusted and the time taken for another 20 oscillations was
recorded.
4. The inertia of the
rectangular bar was then increased by including the “two 1.85kg masses”
symmetrically on either side of the centerline distance x apart.
5. Then step 3 was repeated but
with different values of L
6. The length of the wire L was
then fixed at a value and the time taken was recorded for 20 oscillations at
varying distances x, between the two
1.85kg masses.
7. The rectangular bar was then
detached from the apparatus arrangement and taken to weight balance in order to
determine the mass of the bar.
8. The internal diameters of the
holes, the thickness of the rectangular were also measured.
9. All measurements and data
recorded were collated for experimental analysis.
THEORY
THE
GOVERNING EQUATION – (THE EXPERIMENTAL ANALYSIS)
The equation of the angular
motion is:
However, Introduction of the two equal
masses into the bifilar suspension system can gives rise to a modification of
its mechanism and the equation of the angular motion. This also provides
another approach to the determination of the moment of inertia of the body. In
this situation the length of the wires are kept constant and the distances
between the two equal masses are varied. This is called the auxiliary mass
method but for this experiment, our
analysis would be on just the generic bifilar suspension approach.
Nonetheless, a description of its angular motion is enumerated below;
{r
is the radius of the mass (it has a circular cross-section) and x is the distance between the centre of
the bar and this mass)
From
equation (5)
THE
ANALYTICAL APPROACH
The
general approach for calculating the moment inertia of any type of body about
any axis on the body is given as:
Whereby individual moments of inertia
from individual differential mass value 
and distance X between the axis point and
the object are summed up and its
summations gives rise to the final moment of inertia of that body.
and distance X between the axis point and
the object are summed up and its
summations gives rise to the final moment of inertia of that body.
For a solid object with a rectangular
cross-section, the same general approach gave rise to determination of the
moment of inertia about its centre of mass G
and this is expressed as;
Since
the rectangular bar has bored holes within it, therefore its moment of inertia
is;
I = Is (solid rectangular bar) - Io (bored holes).
Where
Io is the moment of inertia of the
bored holes.
Where
Mo is the mass of the hole bored, Ro
is the radius of each hole bored and X
is the distance from each hole (on either side) to the centre of the
rectangular bar. Note the there are 15 holes bored, hence its introduction in
equation (8)
Therefore, Moment of inertia of the bar, I is
TABLE OF RESULTS
Basic
Parameter Information;
Parameter
|
Value
|
Parameter
|
Value
|
Length of bar l
|
50.8cm
|
Thickness of the bar
|
1.2cm
|
Mass of the bar m
|
1.4kg
|
Distance between the wires, b
|
0.48m
|
Mass of each added mass
|
1.85kg
|
Density of the bar (steel bar)
|
7850kg/m3
|
Radius of each added mass
|
3.8cm
|
Radius of each bored hole
|
0.5cm
|
Data
at various L (no mass included) TABLE 1
Test
|
L (m)
|
X (m)
|
t (s)
|
τ=
 (s) |
k (m) (from equation
above)
|
k2
(m)
|
m (kg)
|
I = mk2
(kgm2)
|
1
|
0.3
|
-
|
13.54
|
0.677
|
0.148
|
0.0219
|
1.4
|
0.0306
|
2
|
0.4
|
-
|
15.38
|
0.769
|
0.145
|
0.0212
|
1.4
|
0.0296
|
Data at various L (mass inclusive) TABLE
2
Test
|
L (m)
|
X (m)
|
t (s)
|
τ= t/20 (s)
|
k (m) (from equation
above)
|
k2
(m)
|
m (kg) {bar}
|
M (kg) {the added
mass}
|
I = (m+2M)*k2
(kgm2)
|
1
|
0.3
|
0.355
|
13.77
|
0.689
|
0.150
|
0.0225
|
1.4
|
1.85
|
0.1153
|
2
|
0.4
|
0.355
|
15.29
|
0.765
|
0.145
|
0.0209
|
1.4
|
1.85
|
0.1067
|
3
|
0.5
|
0.355
|
18.58
|
0.929
|
0.157
|
0.0247
|
1.4
|
1.85
|
0.1260
|
4
|
0.6
|
0.355
|
22.05
|
1.103
|
0.170
|
0.0289
|
1.4
|
1.85
|
0.1478
|
Data at various x and at a fixed L (i.e. L = 0.4m) TABLE 3
Test
|
L (m)
|
X (m)
|
t (s)
|
τ= t/20 (s)
|
1
|
0.4
|
0.355
|
16.59
|
0.689
|
2
|
0.4
|
0.305
|
14.72
|
0.765
|
3
|
0.4
|
0.255
|
13.26
|
0.929
|
4
|
0.4
|
0.205
|
11.77
|
1.103
|
*Note: the extra parameters are not included
in this table 3 because the radius of gyration and moment of inertia are to be
determined with the auxiliary mass method but we are streamlined to work with
the generic bifilar suspension equation of angular motion.
In
addition, the moment of inertia can be further determined by graphical
representation from data collated in the experiment.
Equation
4 aforementioned above can be modified to:
Additional
table 4 (extrapolated from table 2)
T
|
T2
|
L
|
0.689
|
0.4747
|
0.3
|
0.765
|
0.5852
|
0.4
|
0.929
|
0.8630
|
0.5
|
1.103
|
1.2166
|
0.6
|
From the graph,
Slope (sec2/m)
|
0.303
|
ANALYTICAL
DETERMINATION OF THE MOMENT OF INERTIA
From
equation 10 above,
The
moment of inertia calculated analytically is 0.01068 kg m2
PRECAUTION(S)
1.
Measurement
taken from the rule and the weight balance was done such that the line of sight
and the markings of the measuring equipments were in alignment in to reduce
errors due to parallax.
2.
When
taking down the time for the oscillations at various distances ‘x’ and a fixed
length ‘L’ for table 3 data, the length L was periodically checked after each
test in order to maintain the fixed length value of 0.4m
3.
Precautionary
methods were in place to keep the masses at a very comfortable position so as
to avoid slip or fall which could in-turn cause harm to our feet.
4.
The
experiment is done such that the oscillation was not dampened by carefully
tilting the bar before release for oscillations
CONCLUSIONS
OBSERVATION(S):
1. The periodic time significantly increased
when the length of the wires also go increased.
2. The periodic time also increased when the
distances between the masses added to system reduced
3. The moment of inertia determined using
the analytical approach was approximately equal to the value determined from test 2 in table 2 above.
4. The moment of inertia determined from the
graph representation was greater than the value gotten from the analytical
approach indicating that the two masses added during the experiment had a part
to play in the increment of the moment of inertia and also unavoidable human
errors caused a variation in their values.
5. The radius of gyration and moment of
inertia reduced after the length of wire was increased from test 1 to test 2
but increased right after till test 4.
FINAL
DEDUCTIONS:
The
bifilar suspension technique offers the opportunity to determine the radius of
gyration of a body by relating the readings gotten from the procedure in the
techniques and relating that into the equation of angular and this invariably
provides the determination of the moment of inertia for the same body. These readings encompasses the distance
between the wires used for the suspension, the length of the wires, the time
for the required number of oscillations, the distance between the masses
introduced into the experiment, and so on. All these and lots more provide the avenue
for determining the radius of gyration and the moment of inertia
Below
is a tabular representation of the final value of the moment of inertia
determined from the analytical approach, the graphical approach and a selected
value of the moment of inertia from test 2 in table 2.
Methodology
|
Moment
of inertia values
|
Analytical
approach
|
0.01068 kg m2
|
Value from test 2 in table 2
|
0.01067 kg m2
|
Graphical
approach
|
0.02210 kg m2
|
