Introduction1. 1. Aims To look into the harmonic gesture of a torsional pendulum upon angular warp. 1. 2. Background In a torsional pendulum. a phonograph record molded mass is suspended from a thin wire. Angular warp about the axis of the wire was introduced to the phonograph record. When released. the tortuosity minute of the wire will try to untwist itself. doing the phonograph record to hover back and Forth around the wire axis. making a harmonic gesture. This experiment will find the relation between the magnitude of angular warp and the end point harmonic gesture by mensurating the period T of oscillation on assorted angular warp.
The 2nd portion of the experiment will look into the relation between the mass of the pendulum phonograph record and the end point harmonic gesture by mensurating the period T of oscillation on different disc mass. The last portion of the experiment involves the debut of a muffling medium. in the signifier of an oil bath. and thenceforth finding the effects of muffling.
Experimental Procedure2. 1. Apparatus The torsional pendulum set-up consist of a pendulum phonograph record. suspended from a thin steel wire that is fixed to the terminal of a support construction. An unfastened top handbill container incorporating an oil bath. is mounted straight below the suspending phonograph record. The oil bath can be raised to envelop the pendulum phonograph record by the agencies of a lever. The measuring equipment used are viz. the angular graduated table on the phonograph record ( read off a arrow ) . a steel swayer mensurating to 1mm. a digital Vernier calliper mensurating to 0. 01mm and a digital micron screw gauge measurement to 0. 001mm. A digital stop watch was used for clip measuring. entering up to 0. 01seconds. 2. 2. Procedure Before the start of the experiment. measurings were done on the Torsional Pendulum to obtain the length of the pendulum wire ( L ) . diameter of pendulum phonograph record ( dw ) and radius of pendulum phonograph record ( R ) . All dimensions were measured in millimetres. The footing of the experiment involves writhing of the pendulum phonograph record to a predetermined angle ( ? ) . and thenceforth let go ofing it to let the phonograph record to untwist itself.
The oscillation induced will be timed till the completion of the 5th rhythm. and the mean clip ( T ) in seconds peroscillation will be recorded. The first set of measurings for ( T ) will be taken utilizing merely the pendulum phonograph record with initial inactiveness of I0 with amplitudes of ? to be 20o. 30o. and 40 O severally. Three measurings of each amplitude are taken. and the mean reading will be recorded. The 2nd set of measurings taken will be utilizing a default ? of 30 o. but with increasing mass added to the pendulum phonograph record ( Mn ) . A sum of six assorted Mn will be used. and their measurings taken twice and averaged. Once we obtain the mean T. the square of mean period of oscillation ( T2 ) can be calculated. The concluding portion of the experiment involves the debut of muffling by the agencies of an oil bath. The effects of muffling shall be recorded with the same process and parametric quantities as the first set of measurings.
3. Consequences3. 1. Processing of Data The first measurings taken were the dimensions of the Torsional Pendulum. All measurings are taken in millimetres ( millimeter ) . and are measured twice and mean out for better truth. The wire length was measured with a simple steel swayer. with increase of 1mm. The wire diameter was measured with a micron screw gage with increase of 0. 001mm. while the phonograph record diameter are taken with a vernier calliper with increase of 0. 01mm.
Table 2: Variation of T with ? In the 2nd portion of the experiment. extra mass ( Mn ) was introduced to the pendulum phonograph record. A control amplitude of 30o was used throughout all the assorted mass. and their corresponding clip taken for a period T were recorded in Table 3. The square of period of oscillation T2 were calculated from the mean reading squared.
Mass Added to Pendulum Disc
Square of Average Period of 1st Oscillation Mn ( kilogram ) 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 consequence of increasing Pendulum Mass on its Time periods for undamped oscillation with an initial warp of 30o
Time period of Oscillation T ( s ) . Initial Amplitude 30?
In the concluding portion of the experiment. the oil bath was raised to be in contact with the pendulum phonograph record. and it is meant to move as a signifier of muffling to the oscillation. Due to clip restraint. no quantitative measurings were taken. However. an observation was made. When damped. the amplitude of oscillation deteriorated quickly over clip. doing the phonograph record stopped after merely a few oscillation. 3. 2. Analysis of Results With the dimension dw. the polar 2nd minute of country of cross subdivision of wire. J. can be obtained. This value would be required to find other characteristics/ parametric quantities of the Torsional Pendulum. J = ? dw4 32 = ? x 0. 075844 32 = 2. 359 ten 10-13 m4 where. dw: Diameter of wire
Consequences gathered in Table 2 showed that even when there is an addition in angular warp ? of the phonograph record. it will non ensue in a similar alteration in period of oscillation T. This suggest that T is non dependent of the initial amplitude.
The factors that would impact T will be discuss in farther inside informations in subdivision 5. 1. The consequences of Table 3 were represented in graphical signifier. 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 tendency line. the relation of T2 to the mass of the phonograph record can be quantified by the gradient of the incline. In this instance. 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 phonograph record. the initial polar minute of inactiveness. I0. can besides be determined. I0 = M0 R2 = x 2. 22kg ten 0. 075842 thousand = 6. 37 ten 10-3 kgm2 3. 3. Mistake Analysis The estimated absolute mistake ( EAE ) that may happen during the dimensional measuring of the Torsional Pendulum are included into Table 1. These figures came from the premise that when the users are taking readings off the equipment. the highest possible mistake caused by inaccurate appraisal would non be more than half of the lower limit measureable graduated table. With the EAE of each measuring. we can specify the possible magnitude of mistake in per centum ( % ) for the measuring of L. R and dw by spliting the EAE over their several measured consequences. The method used to clip the oscillation was a simple stop watch.
Assuming that the mean human reaction clip to be about 0. 1seconds. this mistake can be reduced by clocking N figure of oscillation alternatively. and thenceforth obtaining the T by spliting the reading by N. Therefore in this instance. the timing of 5 oscillation was taken alternatively of one. this will practically cut down the uncertainness of T by a factor of 5. dT = human reaction clip ? 0. 1s = 0. 02s per reading 5 Though the estimated mistake of every T measuring was reduced to a mere 0. 20s. we have to presume it to be negligible for our following mistake analysis. If we take it that T has no mistake. therefore the gradient of T2 vs Mn can be assumed right. The estimated mistake in G can so be obtained with the undermentioned equation. where. decigram: mistake in G decigram = deciliter + 2dR + 4ddw deciliter: mistake in L G L R dw dR: mistake in R ddw: mistake in dw decigram = ( 7. 08 ten 1010 ) ten ( 1. 215?10-3 + 4. 014?10-4 + 6. 593?10-5 ) = 0. 0021 % = 2. 1 ten 10-3 % where. M0: Initial mass of disc R: Radius of phonograph record
Discussion of Consequences4. 1. Comparison with theory It is known in theory that the period of oscillation T is independent to the magnitude of angular warp in a Torsional Pendulum. Mentioning to the consequences in Table 2. with the ? of 20o. 30o. and 40 o. the T obtained was 2. 51s. 2. 51s and 2. 52s severally. From this Numberss. we can see that the period of oscillation remains reasonably changeless. despite the of all time increasing initial amplitude. This merely implies that T is so independent of the angular warp. Naturally the following inquiry in head is what will do T to alter? From the equation used to deduce T in a Torsional Pendulum. we know that: T = 2? The length of the wire L. Moment of Initial of phonograph record I. Shear Modulus of wire G. and the polar 2nd minute of country of cross subdivision of wire J. are the parametric quantities that will impact the period of oscillation. Because the steel wire used for the experiment was predetermined. the stuff of the wire. its length and its diameter are hence fixed. This meant that J. which was fundamentally a map of the wire diameter will non alter.
Similarly for G. which is a feature of the wire determined by its stuff ( steel ) . will besides be fixed. In the context of this experiment. the last variable in the equation. I. will be tested to turn out its consequence on T. As mentioned earlier in subdivision 4. 2. I is a map of mass and phonograph record radius ( fixed ) . Therefore by increasing the mass of the phonograph record ( Mn ) . the polar minute of inactiveness will besides increase. The consequences 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 consequence of I on the period of oscillation becomes clear. From Graph 1. we are able to find that the gradient ( ?T2 / ?Mn ) . was 2. 8s2/kg. which means to state that for every kg addition in mass of phonograph record. T2 would increase by about 2. 8 seconds.
This relationship is changeless. because it is a consecutive line graph. It is now proven that the period of oscillation T is dependent on polar minute of inactiveness of the mass suspended in a Torsional Pendulum. With respects to the debut of muffling to the experiment. the simple observation made. mention to subdivision 4. 1. was plenty to foreground the of import effects of muffling. All the old parts of the experiments involves the measuring in an undamped status. Which means to state the phonograph record are allowed to move of course. with air clash being the lone force moving against it as it rotates. When the oil bath was introduce. it basically meant that the clash moving on the surface of the phonograph record will increase significantly. due to the difference in viscousness of oil as compared to air. Damping is a manner of cut downing the energy of the oscillation by dispersing some energy through clash. The consequence of an oil bath is an exponential decay in amplitude across clip. 4. 2. Discussion The consequences and analysis obtained from this experiment was successful in turn outing the known theories. In the first portion of the experiment. the information strongly supports the theory that T is independent of the initial amplitude.
Although there are one amplitude ( 40o ) which had a consequence of 0. 01s slower than the remainder. we can accept the fact that this is likely due to human reaction clip and incompatibility. The 2nd portion of the experiment went on to obtain the T2 V Mn graph. from which. we were able to deduce adequate information to finally cipher values of G and M0. Having seen from the mistake analysis from subdivision 4. 3. the possible mistake during the calculation for G was simply 0. 0021 % . little plenty to be considered negligible. The eventual value of G was found to be about 71GPa. which confirms the cogency of the computation. because this value falls into the typical shear modulus scope for steel. The initial mass was calculated to be 2. 22kg. which is a reasonable value for the given size of the pendulum phonograph record.
All the computation did non take into history the possible mistakes in clip measuring. as mentioned earlier. To farther cut down the sum of human reaction related job. the Torsional Pendulum apparatus can be modified or redesign to integrate a better system for numbering and clocking the oscillation. for illustration. we can see the usage of magnetic pickup detector with an interrupter to exactly clip the oscillation. The magnetic pickup detector can be placed above the angular graduated table. while a little ferric metallic home base ( interrupter ) can be place on the zeroo grade on the pendulum phonograph record. This experiment merely allows the varying of the angular warp and mass of the pendulum phonograph record. and observations can merely be made with regard to these alterations. Possibly if the experiment were to let pupils to alter the wire stuff. diameter or length. so we will be able to analyze and verify the relation of these elements on the attendant harmonics gesture.
5. DecisionIn a torsional pendulum. the initial angular amplitude will non impact the period of oscillation. However. when we increase the mass of the phonograph record. the clip period of an oscillation will besides increases. Damping will impact the harmonic gesture of the oscillation. and depending on the muffling ratio. The oil bath used in this experiment resulted in a quickly disintegrating amplitude of oscillation.
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