CONTENTS

TOPIC PAGE

1.0 Abstract

2.0 Title

3.0 Objective

4.0 Introduction

5.0 Theory

6.0 Apparatus

7.0 Procedure

8.0 Result

9.0 Discussion

10.0 Conclusion

11.0 References

ABSTRACT

This experiment was conducted in order to determine the mass moment of inertia at the centre of

gravity, IG and at the suspension points, IO1 and IO2 by oscillation. From the experiment

conducted, the finding is that there are some differences between the values of IO and IG 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 IO2.Based on the experiment, it is found

out that the value of IG and IO 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, T1 and T2 are obtained from the two

different suspension points. Hence, after getting T value, then the value of IG and IO 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 IG and IO are able to be calculated by using the theory.

TITLE

Physical Pendulum - Wooden Pendulum

OBJECTIVE

To determine the mass moment of inertia at the centre of gravity, IG and at the suspension load, IO

by oscillation.

INTRODUCTION

A simple pendulum consists of a point-mass hanging on a length of a string assumed to be

weightless. A small weight hanging by a string from a retort stand illustrates this condition. If the

mass is displaced slightly from its equilibrium position, the mass will perform simple harmonic

oscillation. An extended solid object that is 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.

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

1930’s. Pendulums are used to regulate pendulum clocks, and are used in scientific instruments

such 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.

THEORY

The simple gravity pendulum is an idealized mathematical model of a pendulum. This is a

weight or bob on the end of a weightless cord suspended from a pivot, without friction. When

given an initial push, it will swing back and forth on constant amplitude. Real pendulums are

subject to friction and air drag, so the amplitude of their swings declines. 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, θ.

The component Ft = m.g.sin θ of the force due to the weight applied with the lever arm, rG at the

centre of gravity, G, likewise attempts to return the pendulum to its initial position.

Given the Mass Moment of Inertia, IO is about the suspension point, o, and this result in :

Substitution, linearization and normalization then produce Equation of Motion for the physical

pendulum.

Calculation of the natural frequency and the period of oscillation is analogous to the

mathematical pendulum.

APPARATUS

Universal Vibration System Apparatus

Wooden Pendulum

Vee support Stopwatch

Protractor Ruler

Results:

1. Time taken for 10 oscillations.

a. At first suspension point, O1.

T1, s T2, s T3, s Tave

From left side 14.19 14.17 14.210 14.19

From right side 14.130 14.090 14.180 14.133

b. At second suspension point, O2.

T1, s T2, s Tavg, s Tave

From left side 14.060 14.560 14.400 14.340

From right side 14.050 14.460 14.300 14.27

2. Dimension of wooden pendulum.

a. At first suspension point, O1.

b. At second suspension point, O2.

c.

1 cm 8 cm

70 cm

80 cm

70 cm

1 cm

2.4 cm

45 cm

Sample Calculations:

1. Volume of each component.

a. Component 1.

V 1=0.08 × 0.01× 0.8

¿6.4 × 10−4 m3

b. Component 2.

V 2=0.01× 0.01× 0.45

¿4.5× 10−5 m3

c. Component 3.

V 3=π (0.012 )2× 0.01

¿4.524 ×10−6 m3

d. Total volume of the wooden pendulum.

V pendulum=V 1−V 2−V 3

¿6.4× 10−4−4.524 × 10−6−4.5 ×10−5

¿5.9 ×10−4 m3

2. Total density of wooden pendulum.

ρ pendulum=mtotal

v pendulum

¿ 0.6 kg5.9× 10−4 m3

¿1016.13 kgm3

3. Mass of each component.

a. m1=ρ ∙ v1

¿1016.13 ∙(6.4 ×10−4)

¿0.651 kg

b. m2=ρ ∙ v2

¿1016.13 ∙(4.5 ×10−5)

¿0.046 kg

c. m3=ρ ∙ v3

¿1016.13 ∙(4.524 × 10−6)

¿5 ×10−3kg

4. Moment of Inertia about point O1 and point O2 (Experimental Calculation)

a. Point O1.

Component Area, A (m2) y (m) y A (m3)

1 0.8 x 0.08 = 0.064 0.4 0.0256

2 0.45 x 0.01 = 4.5 x 10-3 0.275 1.238 x 10-3

3 π (0.012)3 = 4.909 x 10-4 0.762 3.620 x10-4

∑A = 0.069 ∑y A = 0.0272

yO1 = ∑ y A

∑ A=

0.02720.0689

=0.394 m

Component 1:

I1 = 1/12 m l² + m d²

= 1/12 (0.65)(0.8)2 + (0.6)(0.406 – 0.4)2

= 0.0347 kgm3

Component 2:

I2 = 1/12 m l² + m d²

= 1/12 (0.046)(0.45)2 + (0.046)(0.525-0.406)2

= 1.428 x 10-3 kgm3

Component 3:

I3 = 1/4 m r² + m d²

= 1/4 (5 x10-3)(0.0125)2 + (5 x10-3)(0.406-0.0625)2

= 5.902 x 10-4 kgm3

Total inertia of moment

IG1 = I1 – I2 – I3

= 0.0347 – 1.428 X 10-3 – 5.902 X10-4

= 0.0327 kgm3

Io1 = IG1 + md2

= 0.0327 + (0.6)(0.394)

= 0.126 kgm3

b. Point O2.

Component Area, A (m2) y (m) y A (m3)

1 0.8 x 0.08 = 0.064 0.4 0.0256

2 0.45 x 0.01 = 4.5 x 10-3 0.525 2.3265 x 10-3

3 π (0.012)3 = 4,524 x 10-4 0.038 1.719 x10-5

∑A = 0.0689 ∑y A = 0.028

yO2 = ∑ y A

∑ A=

0.0280.0689

=0.4061m

Component 1:

I1 = 1/12 m l² + m d²

= 1/12 (0.65)(0.8)2 + (0.65)(0.4061 – 0.4)2

= 0.0347 kgm3

Component 2:

I2 = 1/12 m l² + m d²

= 1/12 (45.73 x10-3)(0.45)2 + (45.73 x10-3)(0.525 – 0.4061)2

= 1.418 x 10-3 kgm3

Component 3:

I3 = 1/4 m r² + m d²

= 1/4 (4.6 x10-3)(0.012)2 + (4.6 x10-3)(0.4061-0.038)2

= 6.234 x 10-4 kgm3

Total:

IG2 = I1 – I2 – I3

= 0.0347 – 1.418 X 10-3 – 5.902 X10-4

= 0.0327 kgm3

Io2 = IG2 + md2

= 0.03266 + (0.6)(0.4061)

= 0.2763 kgm3

IO

mgrg

5. Moment of Inertia about point O1 and point O2 (experimental Calculation)

a. O1

Tavg = 14.162 s

Therefore:

T1 oscilations = 14.162 / 10 =1.4162 sec

T1 average = 2π

I01 = T1 oscilations

2 π

= 1.4162

2π

I01 = 0.105 kg/m3

IG1 =I01 – mrg2

= 0.105 – 0.6(0.35)2

= 0.0315kg m³

(mgrg)

(0.6x9.81x0.35)

IO

mgrg

b. O2

Tavg = 14.31 s

Therefore:

T2average = 14.31 / 10 =1.431 sec

T2 average = 2π

I02 = T1 oscilations

2 π

= 1.431

2π

I02 = 0.107kg/m3

IG2 =I02 – mrg2

= 0.107 – 0.6(0.35)2

= 0.0335kg m³

(mgrg)

(0.6x9.81x0.35)

6. Percentage Error.

For IO1, percentage of error % = (0.126-0.105) x 100% 0.105

= 20 %

For IG1, percentage of error % = (0.0315– 0.0347) x 100% 0.0347

= 9.22 %

For I02, percentage of error % = (0.0335 – 0.107) x 100% 0.107

= 68.69%

For IG2, percentage of error % = (0.0315 – 0.0335) x 100% 0.0335

= `5.97%

Point Moment of Inertia

Theoretical Value (kg m³)

Experimental Value(kg m³) Percentage Error (%)

O1

IO1 0.105 0.126 20

IG1 0.0347 0.0315 9.22

O2

IO2 0.107 0.0335 68.69

IG2 0.0335 0.0315 17.33

5.1 DISCUSSION

            Based on the experiment conducted, all the values of mass moment of inertia at the centre of gravity, IG and at the suspension point, IO on different end, O1 and O2 have been determined according to the experiment and theory. The values of I01,Ig1, I02 and Ig2 are theoretically calculated using formulae and finding the volume of each component exist in the non-homogeneous wooden pendulum. The values of I01, Ig1, I02 and Ig2 are experimentally determined by taking time for 10 complete oscillation of the wooden pendulum on different angle for each suspension point. Comparing all the values of I01, Ig1, I02 and Ig2 in theoretical and experimental calculation, it is found out that each value is slightly different from each other. The percentage error between the theoretical and experimental values can be observed in Table 4.1. The percentage errors are merely less than 20% and therefore can be considered as acceptable. The difference in values may be caused by several errors during the experiment and calculation. The dimension of the wooden pendulum may be taken under parallax and precision errors as only a ruler is used to take the dimensions of the wooden pendulum including the circular parts. Therefore, this might affect the reading taken. During the oscillation, a stopwatch is used and therefore, there might be zero error as the starting of the swing is not precisely parallel with the starting of the time taken. This may cause the time to be slower or faster than it is supposed to be recorded. The oscillation of the wooden pendulum especially on the smaller circle, O2 is wobbling as the supporting part is so small and this cause disturbance during the oscillation. In the calculation procedure, only several decimal points are considered and this also affected all the values calculated. All of this disturbance and errors has affected the values of I01, Ig1, I02 and Ig2 obtained.

6.1 CONCLUSION

In a nutshell, it is found out that a pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced from its equilibrium position, it is subjected 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. It is also found out that the values of IO1, IG1, IO2 and IG2 are affected by errors and disturbance during experiment and even decimal points consideration during calculation is a contributing factor. The error percentage between the theoretical and experimental values can be considered as a slight error as the values of the error are just less than 20%. 

REFERENCES

BIBLIOGRAPHY A.Russell, D. (2011). The Simple Pendulum. Retrieved from Acoustics and Vibration

Animations: http://www.acs.psu.edu/drussell/Demos/Pendulum/Pendula.html

Amrita. (2013). Kater's Pendulum. Retrieved from Vlab: http://amrita.vlab.co.in/?

sub=1&brch=280&sim=518&cnt=1

Andrea. (n.d.). The Pendulum. Retrieved from muse.tau:

http://muse.tau.ac.il/museum/galileo/pendulum.html

Vector Mechanics for Engineers: Dynamics, Eight Edition in SI units, Ferdinand P. Beer, E.

Russell Johnston, Jr, and William E. Clausen, Singapore, 2007.

Lee, P. (2011). Simple Pendulum. Retrieved from NVCC:

http://www.nvcc.edu/alexandria/science/physics/lab231hybrid/pendulum/

pendulum_instructions.htm

Appendix