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ABSTRACT

This experiment was conducted in order to determine the mass moment of inertia at the centre
of gravity, I
G
and at the suspension points, I
O1
and I
O2
by oscillation. From the experiment
conducted, the finding is that there are some differences between the values of I
O
and I
G
from
the experiment data and also from theoretical value. The potential factors that cause to the
differences in values are further discussed. The finding is that the wooden pendulum oscillates
in non-uniform motion especially when it is suspended at I
O2.
Based on the experiment, it is
found out that the value of I
G
and I
O
from both suspension points is totally different although
they share the same value of mass of the wooden pendulum. The period is also different for
both points setting. After the data was taken, the period of oscillation, T
1
and T
2
are

obtained
from the two different suspension points. Hence, after getting T value, then the value of I
G
and
I
O
can be measured. The errors that occur might be due to disturbing from surrounding and
human error. The time for 10 oscillations was taken manually by using stopwatch. By the end
of this experiment, the values of I
G
and I
O
are

able to be calculated by using the theory.



OBJECTIVES

- Determine the mass moment of inertia (at the center of gravity, I and at suspension
point, I) by oscillation.









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INTRODUCTION

The idea of a simple pendulum consists of a point mass hanging on a length of mass less sting
supported rigidly. A small weight hanging by a light string from a retort stand approximates
these conditions. If displaced slightly from equilibrium the mass will perform simple harmonic
oscillation. An extended solid object free to swing on an axis is called a physical pendulum,
whose period is now dependant on the mass moment of inertia about the rotational axis and it
distance from the centre of mass
2
.

A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is
displaced from its resting equilibrium position, it is subject to a restoring force due to gravity
that will accelerate it back toward the equilibrium position. When released, the restoring force
combined with the pendulum's mass causes it to oscillate about the equilibrium position,
swinging back and forth. The time for one complete cycle, a left swing and a right swing, is
called the period. A pendulum swings with a specific period which depends (mainly) on its
length. From its discovery around 1602 by Galileo Galilei the regular motion of pendulums was
used for timekeeping, and was the world's most accurate timekeeping technology until the
1930s.
[2]
Pendulums are used to regulate pendulum clocks, and are used in scientific
instruments such as accelerometers and seismometers. Historically they were used as
gravimeters to measure the acceleration of gravity in geophysical surveys, and even as a
standard of length. The word 'pendulum' is new Latin, from the Latin pendulus, meaning
'hanging'.
The simple gravity pendulum is an idealized mathematical model of a pendulum. This is a
weight (or bob) on the end of a massless cord suspended from a pivot, without friction. When
given an initial push, it will swing back and forth at a constant amplitude. Real pendulums are
subject to friction and air drag, so the amplitude of their swings declines.







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THEORY
A physical pendulum is a pendulum where the pendulum mass is not concentrated at
one point. In reality all pendulums are physical, since it is not possible to achieve the ideal
concentration of mass at a single point.
An equilibrium moment is formed about the suspension point to establish the equation
of motion.
The pendulum is deflected about the angle, . The component
r


of the
force due to the weight applied with the lever arm, r
G
, at the center of gravity, G, likewise
attempts to return the pendulum to its initial position.


Physical pendulum with extensive mass distribution

Given the Mass Moment of Inertia (MMI), I
O
, about the suspension point, o, and this result in:

G
sin
Substitution, linearization and normalization then produce Equation of Motion for the physical
pendulum.

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Calculation of the natural frequency and period of oscillation is analogous to the mathematical
pendulum.

- Where :

= the rotational inertia of the pendulum about its rotation axis.

m= is the total mass of the pendulum.
g= is the acceleration of gravity

= distance from the rotation axis to the center of mass

- The equation of a harmonic oscillator is :

- And is related to the period T by :

T depends only on the distribution of mass within the object. T can be used to compute
g, if the moment of inertia is known. The moment of inertia about the pivot point is related to
the moment of inertia about the center of mass I
o
by the parallel axis theorem,


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I = I
o
+ mr
2
- Moment at O :

u sin .L mg M
O
=
clockwise direction.

u L mg M
O
. =
if angle
u
is small.

- By using Newtons Law for rotation:

O O
I L mg o u = .

= ( .
O
I L mg u
u

)

0 . = + u u L mg I
O

- The natural frequency of this system can be obtained by:
O
I
mgL
=
2
e

s r
I
mgL
O
/ = e


- Periodic time (T) is the time taken to complete one cycle.(or radian). :
e
t 2
= T

mgL
I
T
O
t 2 =


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- For rigid body,moment of inertia at point O is:

2
mL I I
G O
+ =
where
G
I
is the moment of inertia at the center of
gravity.
2 2
mL mk I
O
+ =
where
k
is the radius of gyration at the center of gravity
mgL
mL mk
T
) (
2
2 2
+
= t

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EQUIPMENT AND APPARATUS
: Universal Vibration System Apparatus.
1. Wooden pendulum.
2. Vee support.
3. Ruler.
4. Stopwatch.
5. Rod support.
6. Internal calliper


Universal Vibration System Apparatus.



Wooden pendulum.

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Vee support.


Rod support.


Stopwatch.





Internal calliper
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PROCEDURE

1. The apparatus was setup.
2. The dimensions of the wooden pendulum were taken using a ruler and a caliper.
3. Mass of wooden pendulum was record.
4. By using the Vee support, the wooden pendulum was placed at the pivot point.
5. The pendulum was placed at a 10 angle to the left and then release to let it oscillates
freely for 10 oscillations.
6. Time for 10 oscillations was taken by using stopwatch.
7. Step 3 and 4 is repeated three times and the average result was recorded in table.
8. The pendulum was placed at a 10 angle to the right and then release to let it oscillates
freely for 10 oscillations.
9. Steps 6 and 7 were repeated and the results were record.
10. The wooden pendulum was turned upside down for the other point and the wooden
pendulum was placed at the pivot point by using rod support.
11. Step 5 to 9 is repeated for the other point at wooden pendulum. Then the result was
recorded.










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DATA AND RESULT

Experimental Data:



Moment of Inertia:

Suspension
Point
Experimental Value of
Mass Moment of Inertia
Theoretical Value of
Mass Moment of Inertia
Percentage error (%)
I
o
(kgm)

I
g
(kgm)

I
o
(kgm)

I
g
(kgm)

I
o
I
g

Point 1 0.03658 0.0287 0.0342 0.02914 6.96 1.51
Point 2 0.03589 0.0280 0.0275 0.02935 30.51 4.6





Suspension
Point

Angle
(n
0
)
Experimental Time of 10 Oscillations (s) Time of 1
Oscillation
(s)
T1 T2 T3 Average Average
Point 1

Left 10 14.63 14.50 14.57 14.57 14.64 1.464
Right 14.69 14.72 14.70 14.70
Point 2

Left 10 14.50 14.53 14.52 14.52 14.50 1.450
Right 14.53 14.47 14.44 14.48
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Wooden Pendulum dimensions

Mass of wooden pendulum = 0.6kg
Thick of wooden pendulum = 0.01m




















0.8m
0.078cm
2.5cm
mm
PART A
PART B
PART C
0.013cm 0.012cm
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At Point 1 :











At point 2 :











0.078m
0.078m
0.012m
0.012m
D=0.025m
D=0.025m
0.3m
0.3m
0.45m
0.45m
0.013m
0.013m
0.8m
0.8m
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DISCUSSION

In this experiment, we are calculate the mass moment of inertia(I
G
and I
O
) by oscillation. The
moment of inertia for I
G
is

0.0287 kgm
2
at suspension point 1, and 0.0280 kgm
2
at suspension
point 2. As we can see from result the theoretical value are not same but different a little bit.
For the mass moment of inertia I
0
at suspension 1 is 0.03658 kgm
2
and 0.03589 kgm
2
at
suspension point 2. These value are slightly same with the value of theoretical of I
0
that we
gain.
Error that can being detected in this experiment are the parallax error which is the angle of
oscillation are not constant due to position of eye but this error being reduced by take the
result twice and we take the average, another error which is the oscillation are not stable due
to the oscillation move forward and backward a little bit.
As we can see, the mass moment of inertia for I
g
and I
0
are different due to different in
suspension point. It is also because the center of gravity of wooden pendulum in state for
suspension point 1 are diferrent with suspension point 2.












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CONCLUSION

In a nutshell, the objective in this experiment to find the moment of inertia (I
g
and I
0
) by using
oscilation method was achieved. The wooden pendulum serves as an example of
inhomogeneous pendulum with a complicated geometry. It used to show how the mass
moment of inertia of an unknown body could be determined by way of oscillation.
An example of an application are Foucault pendulum, which is being used to demonstrate
rotation of earth, another example are the clock that use pendulum to move the gear, because
of the characteristic of mass moment inertia, this mechanism works.






Foucault pendulum Pendulum clock
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REFERENCES

Engineering Mechanics Dynamics Twelfth Edition In SI Units, R.C.Hibbeler, Prentice Hall, Twelfth
Edition, 2010. Pearson Education, Inc. In Jurong, Singapore.
Physical Pendulum. (n.d.). Retrieved April 3, 2013, from
http://faculty.wwu.edu/vawter/PhysicsNet/Topics/SHM/PhysicalPendulum.html
Physical Pendulum. (n.d.). Retrieved April 4, 2013, from
http://www.colorado.edu/UCB/AcademicAffairs/ArtsSciences/physics/phys1140/phys1140_f
a04/Expts/M3Fall04.html
Docstoc (2011), Simple and Physical Pendulum, Retrieved April 4, 2013, from
http://www.docstoc.com/docs/27251888/Simple-and-Physical-Pendulum
Singh, S. K. (2007, December 02). Simple and physical pendulum. Retrieved April 3, 2013,
from http://cnx.org/content/m15585/latest/