MEC424-Wooden Pendulum

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, T₁ and T₂ 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.

1.0       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 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.   

2.0       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.  

PROCEDURE:

1. Wooden pendulum is hanged by a steel pin at first suspension point, O₁.
2.  The pendulum is allowed to oscillate from left side to right side.  
3. The time taken for the pendulum to complete 10 oscillations is recorded by stopwatch.
4.  Steps 2 and 3 are repeated 3 times to obtain 3 readings in order to get the average time taken for 10 oscillations. 
5. Next, the pendulum is allowed to oscillate from right side to left side.
6.  The time taken for the pendulum to complete 10 oscillations is recorded by stopwatch.   
7. Steps 5 and 6 are repeated 3 times to obtain 3 readings in order to get the average time taken for 10 oscillations. 
8. The wooden pendulum is hanged by a steel pin at second suspension point, O_2.
9. Step 2 to 7 is repeated for the second suspension point, O_2.   
10. Finally, the Mass Moment of Inertia (MMI), I_G and I_O are determined by using the equations of Mathematical and Physical Pendulum..   
11. The value obtained is compared with the value obtained from manual calculation.

Results:

1.       Time taken for 10 oscillations.

a.       At first suspension point, O_1.

	T1, s	T2, s	Tavg, s
From left side	14.33	14.21	14.27
From right side	14.16	14.13	14.15

b.      At second suspension point, O_2.

	T1, s	T2, s	Tavg, s
From left side	14.10	14.50	14.30
From right side	14.25	14.35	14.30

Sample Calculations:

1.       Volume of each component.

a.       Component 1.

b.      Component 2.

c.       Component 3. 

d.      Total volume of the wooden pendulum.

2.       Total density of wooden pendulum.

3.       Mass of each component.

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

a.       Point O_1.

Component	Area, A (m2)	(m)	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	(0.012)3 = 4,524 x 10-4	0.762	3.45 x10-4
	∑A = 0.0689		∑ A = 0.0272

_O1 =

Component 1:

                            I₁ = 1/12 m l² + m d²

                                = 1/12 (0.65)(0.8)² + (0.6)(0.4 – 0.395)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)(0.395-0.275)²

                                = 1.43 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)(0.762-0.395)²

                                = 6.2 x 10^-4 kgm³

Total:

                            I_G1 = I₁ – I₂ – I₃

                                                = 0.0347 – 1.43 X 10^-3 – 6.2 X10^-4

                                                   = 0.03265 kgm³

                                           I_o1 = I_G1 + md²

                                  = 0.03265 + (0.6)(0.395)

                                  = 0.1092 kgm³

b.      Point O_2.

Component	Area, A (m2)	(m)	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	(0.012)3 = 4,524 x 10-4	0.038	1.719 x10-5
	∑A = 0.0689		∑ A = 0.028

_O2 =

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:

                            I_G2 = I₁ – I₂ – I₃

                                                = 0.0347 – 1.418 X 10^-3 – 6.234 X10^-4

                                                   = 0.03266 kgm³

                                           I_o2 = I_G2 + md²

                                  = 0.03266 + (0.6)(0.4061)

                                  = 0.2763 kgm³

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

a.       O₁

                                               T_avg = 14.21 s

                               Therefore:                                              

                               T_{1 oscilations} = 14.21 / 10 =1.421 sec

                               T_{1 oscilations}  = 2π                          

                              

                               L₀₁ = g            T_{1 oscilations}  

                                                            2 π

                                     

                                      = 9.81            1.421

                                                                  2π

                                 L₀₁ = 0.5018 m

                               I_G1 = m r_G   ( L₀₁ - r_G )                        

                                       = 0.6(0.355) (0.5018 - 0.355)

                                       = 0.0313kg m³

                               I_O1 = I_G1 +   m r_G²

                                     = 0.0313 + 0.6 (0.355) ²

                                     = 0.1069 kg m²

b.      O₂

                       T_avg = 14.30s

                               Therefore:                                              

                               T_{1 oscilations} = 13.98/ 10 =1.398 sec

                               T_{2 oscilations}  = 2π                          

                              

                                              

                               L₀₂ = g             T_{2 oscilations}  

                                                                2 π

                                     

                                      = 9.81     1.398

                                                            2π

                                  L₀₂   = 0.4857 m

                               I_G2 = m r_G   ( L₀₂ - r_G )                                        

                                                          = 0.6(0.355) (0.4857 - 0.355)

                                      = 0.0278 kg m³

                                I₀₂  =  I_G02 +  m r_G²

                                      = 0.0278 + 0.6 (0.355) ²

                                      = 0.1034 kg m²

6.       Percentage Error.

For I_{O1, percentage} of error % = (0.1092– 0.1042)     x 100%

                        0.1092

      = 4.58%

For I_G1, percentage of error % = (0.0326 – 0.0285)     x 100%

                        0.0326

                                                                                = 12.71 %

For I_02, percentage of error % = (0.2763 – 0.1034)     x 100%

                       0.2763

                                                                                = 62.58%

For I_G2, percentage of error % = (0.0326 – 0.0278)     x 100%

                        0.0326

                                                                                = `14.88%

Point	Moment of Inertia	Theoretical Value (kg m³)	Experimental Value(kg m³)	Percentage Error (%)
O1	IO1	0.1092	0.1042	5.39
	IG1	0.0326	0.0285	15.16
O2	IO2	0.2763	0.1034	18.76
	IG2	0.0326	0.0278	17.33

5.1 DISCUSSION

            Based on the experiment conducted, all the values of mass moment of inertia at the centre of gravity, I_G and at the suspension point, I_O on different end, O₁ and O₂ have been determined according to the experiment and theory. The values of IO1, IG1, IO2 and IG2 are theoretically calculated using formulae and finding the volume of each component exist in the non-homogeneous wooden pendulum. The values of IO1, IG1, IO2 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 IO1, IG1, IO2 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 IO1, IG1, IO2 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%. 

links for lab report:
full lab- http://www.mediafire.com/view/?0o9ey6j8rfsaunb
result by blogger- http://www.mediafire.com/view/?b82tb62ilctnq1a
abstract to theory - http://www.mediafire.com/view/?2yr3qkb6bbj57h7