Oscillation of Torsional Pendulum
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1.1. Objectives To investigate the harmonic motion of a torsional pendulum upon angular deflection. 1.2. Background In a torsional pendulum, a disc shaped mass is suspended from a thin wire. Angular deflection about the axis of the wire was introduced to the disc. When released, the torsion moment of the wire will attempt to untwist itself, causing the disc to oscillate back and forth around the wire axis, creating a harmonic motion. This experiment will determine the relation between the magnitude of angular deflection and the resultant harmonic motion by measuring the period T of oscillation on various angular deflection. The second part of the experiment will investigate the relation between the mass of the pendulum disc and the resultant harmonic motion by measuring the period T of oscillation on different disc mass. The last part of the experiment involves the introduction of a damping medium, in the form of an oil bath, and thereafter determining the effects of damping.
2.1. Apparatus The torsional pendulum set-up consist of a pendulum disc, suspended from a thin steel wire that is fixed to the end of a support structure. An open top circular container containing an oil bath, is mounted directly below the suspending disc. The oil bath can be raised to enclose the pendulum disc by the means of a lever. The measurement equipment used are namely the angular scale on the disc (read off a pointer), a steel ruler measuring to 1mm,a digital vernier caliper measuring to 0.01mm and a digital micrometer screw gauge measuring to 0.001mm. A digital stopwatch was used for time measurement, recording up to 0.01seconds. 2.2. Procedure Before the start of the experiment, measurements were done on the Torsional Pendulum to obtain the length of the pendulum wire (L), diameter of pendulum disc (dw) and radius of pendulum disc (R). All dimensions were measured in millimeters. The basis of the experiment involves twisting of the pendulum disc to a predetermined angle (θ), and thereafter releasing it to allow the disc to untwist itself.
The oscillation induced will be timed till the completion of the fifth cycle, and the average time (T) in seconds per oscillation will be recorded. The first set of measurements for (T) will be taken using only the pendulum disc with initial inertia of I0 with amplitudes of θ to be 20o, 30o, and 40 o respectively. Three measurements of each amplitude are taken, and the average reading will be recorded. The second set of measurements taken will be using a default θ of 30 o, but with increasing mass added to the pendulum disc (Mn). A total of six various Mn will be used, and their measurements taken twice and averaged. Once we obtain the average T, the square of average period of oscillation (T2) can be calculated. The final part of the experiment involves the introduction of damping by the means of an oil bath. The effects of damping shall be recorded with the same procedure and parameters as the first set of measurements.
3.1. Processing of Data The first measurements taken were the dimensions of the Torsional Pendulum. All measurements are taken in millimeters (mm), and are measured twice and average out for better accuracy. The wire length was measured with a simple steel ruler, with increment of 1mm. The wire diameter was measured with a micrometer screw gauge with increment of 0.001mm, while the disc diameter are taken with a vernier caliper with increment of 0.01mm.
Table 2: Variation of T with θ In the second part of the experiment, additional mass (Mn) was introduced to the pendulum disc. A control amplitude of 30o was used throughout all the various mass, and their corresponding time taken for a period T were recorded in Table 3. The square of period of oscillation T2 were calculated from the average reading squared.
Mass Added to Pendulum Disc
Square of Average Period of 1st Oscillation Mn (kg) Meas. 2nd Meas. Average T2 (s2) 0 2.51 6.30 0.490 2.76 2.78 2.77 7.67 0.980 3.02 2.98 3.00 9.00 1.450 3.20 3.10 3.15 9.92 1.935 3.38 3.38 3.38 11.42 2.420 3.58 3.58 3.58 12.82 2.904 3.86 3.80 3.83 14.67 Table 3: The effect of increasing Pendulum Mass on its Periods for undamped oscillation with an initial deflection of 30o
Period of Oscillation T(s), Initial Amplitude 30ο
In the final part of the experiment, the oil bath was raised to be in contact with the pendulum disc, and it is meant to act as a form of damping to the oscillation. Due to time constraint, no quantitative measurements were taken. However, an observation was made. When damped, the amplitude of oscillation deteriorated rapidly over time, causing the disc stopped after just a few oscillation. 3.2. Analysis of Results With the dimension dw, the polar second moment of area of cross section of wire, J, can be obtained. This value would be required to determine other characteristics/ parameters of the Torsional Pendulum. J = π dw4 32 = π x 0.075844 32 = 2.359 x 10-13 m4 where, dw: Diameter of wire
Results gathered in Table 2 showed that even when there is an increase in angular deflection θ of the disc, it will not result in a similar change in period of oscillation T. This suggest that T is not dependent of the initial amplitude.
The factors that would affect T will be discuss in further details in section 5.1. The results of Table 3 were represented in graphical form, with mass added (Mn) being the x-axis and square of period of oscillation T2 as the y-axis (Graph 1). When plotted with a best fit trend line, the relation of T2 to the mass of the disc can be quantified by the gradient of the slope. In this case, the gradient obtained from the graph was 2.796s2/kg.
M0 = 6.192 s2 2.796 s2/kg = 2.22 kg Having known the original mass of the pendulum disc, the initial polar moment of inertia, I0, can also be determined. I0 = M0 R2 = x 2.22kg x 0.075842 m = 6.37 x 10-3 kgm2 3.3. Error Analysis The estimated absolute error (EAE) that may occur during the dimensional measurement of the Torsional Pendulum are included into Table 1. These figures came from the assumption that when the users are taking readings off the equipment, the highest possible error caused by inaccurate estimation would not be more than half of the minimum measureable scale. With the EAE of each measurement, we can define the possible magnitude of error in percentage (%) for the measurement of L, R and dw by dividing the EAE over their respective measured results. The method used to time the oscillation was a simple stopwatch.
Assuming that the average human reaction time to be approximately 0.1seconds, this error can be reduced by timing N number of oscillation instead, and thereafter obtaining the T by dividing the reading by N. Therefore in this case, the timing of 5 oscillation was taken instead of one, this will practically reduce the uncertainty of T by a factor of 5. dT = human reaction time ≈ 0.1s = 0.02s per reading 5 Though the estimated error of every T measurement was reduced to a mere 0.20s,we have to assume it to be negligible for our next error analysis. If we take it that T has no error, therefore the gradient of T2 vs Mn can be assumed correct. The estimated error in G can then be obtained with the following equation. where, dG: error in G dG = dL + 2dR + 4ddw dL: error in L G L R dw dR: error in R ddw: error in dw dG = (7.08 x 1010) x (1.215×10-3 + 4.014×10-4 + 6.593×10-5) = 0.0021% = 2.1 x 10-3 % where, M0: Initial mass of disc R: Radius of disc
Discussion of Results
4.1. Comparison with theory It is known in theory that the period of oscillation T is independent to the magnitude of angular deflection in a Torsional Pendulum. Referring to the results in Table 2, with the θ of 20o, 30o, and 40 o, the T obtained was 2.51s, 2.51s and 2.52s respectively. From this numbers, we can see that the period of oscillation remains fairly constant, despite the ever increasing initial amplitude. This simply implies that T is indeed independent of the angular deflection. Naturally the next question in mind is what will cause T to change? From the equation used to derive T in a Torsional Pendulum, we know that: T = 2π The length of the wire L, Moment of Initial of disc I, Shear Modulus of wire G, and the polar 2nd moment of area of cross section of wire J, are the parameters that will affect the period of oscillation. Because the steel wire used for the experiment was predetermined, the material of the wire, its length and its diameter are therefore fixed. This meant that J, which was basically a function of the wire diameter will not change.
Similarly for G, which is a characteristic of the wire determined by its material (steel), will also be fixed. In the context of this experiment, the last variable in the equation, I, will be tested to prove its effect on T. As mentioned earlier in section 4.2, I is a function of mass and disc radius (fixed). Therefore by increasing the mass of the disc (Mn), the polar moment of inertia will also increase. The results tabulated in Table 3 shows that T will increase when we add more mass (higher I). When we plot T2 against the mass on a graph, the effect of I on the period of oscillation becomes clear. From Graph 1, we are able to determine that the gradient(∆T2 / ∆Mn), was 2.8s2/kg, which means to say that for every kilogram increase in mass of disc, T2 would increase by approximately 2.8 seconds.
This relationship is constant, because it is a straight line graph. It is now proven that the period of oscillation T is dependent on polar moment of inertia of the mass suspended in a Torsional Pendulum. With regards to the introduction of damping to the experiment, the simple observation made, refer to section 4.1, was enough to highlight the important effects of damping. All the previous parts of the experiments involves the measurement in an undamped condition. Which means to say the disc are allowed to act naturally, with air friction being the only force acting against it as it rotates. When the oil bath was introduce, it essentially meant that the friction acting on the surface of the disc will increase significantly, due to the difference in viscosity of oil as compared to air. Damping is a way of reducing the energy of the oscillation by dissipating some energy through friction. The result of an oil bath is an exponential decay in amplitude across time. 4.2. Discussion The results and analysis obtained from this experiment was successful in proving the known theories. In the first part of the experiment, the data strongly supports the theory that T is independent of the initial amplitude.
Although there are one amplitude (40o) which had a result of 0.01s slower than the rest, we can accept the fact that this is probably due to human reaction time and inconsistency. The second part of the experiment went on to obtain the T2 vs Mn graph, from which, we were able to derive enough information to eventually calculate values of G and M0. Having seen from the error analysis from section 4.3, the possible error during the computation for G was merely 0.0021%, small enough to be considered negligible. The eventual value of G was found to be around 71GPa, which confirms the validity of the calculation, because this value falls into the typical shear modulus range for steel. The initial mass was calculated to be 2.22kg, which is a sensible value for the given size of the pendulum disc.
All the calculation did not take into account the possible errors in time measurement, as mentioned earlier. To further reduce the amount of human reaction related problem, the Torsional Pendulum setup can be modified or redesign to incorporate a better system for counting and timing the oscillation. for example, we can consider the use of magnetic pickup sensor with an interrupter to precisely time the oscillation. The magnetic pickup sensor can be placed above the angular scale, while a small ferrous metallic plate (interrupter) can be place on the zeroo mark on the pendulum disc. This experiment only allows the varying of the angular deflection and mass of the pendulum disc, and observations can only be made with respect to these changes. Perhaps if the experiment were to allow students to change the wire material, diameter or length, then we will be able to study and verify the relation of these elements on the resultant harmonics motion.
In a torsional pendulum, the initial angular amplitude will not affect the period of oscillation. However, when we increase the mass of the disc, the time period of an oscillation will also increases. Damping will affect the harmonic motion of the oscillation, and depending on the damping ratio. The oil bath used in this experiment resulted in a rapidly decaying amplitude of oscillation.
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