I = Moment of Inertia of torsion bar
κ = Torsion constant
𝑇=2𝜋,-,𝐼-𝜅..
𝜅=,𝐼-,,,𝑇-2𝜋..-2..
After implementing my measurements into this formula I was able to plot graphs of the values of κ against the variables of thickness and length- which will be discussed at greater length later in this document.
Initial Predictions
From the outset I predicted that the torsion constant would decrease with increasing length, and increase with increasing thickness. A popular and effective demonstration of this effect is the act of twisting a wooden meter ruler; it is far easier to twist it when holding both ends of the ruler, and far more difficult when trying to twist, say, 10cm of the ruler. Similarly, if one attempts to twist a relatively thick ruler, the difficulty will again be increased. Whilst this analysis is fairly rudimentary, it gave me an idea of the sort of results to expect, and provided me a context which effectively chose my variables for me- length of wire and thickness of wire. In addition, these predictions are purely qualitative, so it was important to justify my analysis with numerical data.
Risk assessment
As with any experimentation, it was important for me to assess the risk involved in my experimentation and decide how to prevent accidents from happening. The risks and possible solutions are found below.
In fact, this investigation is relatively safe as long as these guidelines are followed, there is no electrical or chemical danger involved and most of the risks are those generally found in any laboratory.
Practical Investigation
Initial Results
Initially my results were poor for a number of reasons: Firstly, I had been taking far too many readings for each length & thickness of wire- 10 repeats of experimentation lasting in excess of 20 seconds each time. Furthermore, I encountered the problem that the wire was oscillating vertically, affecting the horizontal torsion and thus distorting the torsion constant. In addition, I had not secured the wire into the torsion bar tightly enough, leading to slipping.
However, I overcame these issues by reducing the amount of repeats to a more manageable, but still accurate, three repeats- thus improving the efficiency of my experimentation and providing me with more time to improve the method. In terms of the vertical oscillation, I managed to come up with a solution whereby a small excess of wire passing through the torsion bar was held in place by a pen lid; whilst crude, this arrangement successfully prevented the vertical oscillation, thereby improving the reliability of the experiment. The refined apparatus is displayed below.
Refined Technique and Results
After I was satisfied with the new arrangement for the investigation I began to test the effect of varying the thickness of a wire with the same length. This process was achieved by taking different thicknesses of wire and testing the same length by attaching them to the same point in the torsion bar and the top of the retort stand. Of course there was some uncertainty in the measurement due to the specimens of wire having varying lengths and uniformity; I adjudged this to be an uncertainty of approximately 5mm, not entirely negligible but still a factor in the investigation in terms of quality of measurement. The measurement of the thickness of the wire can be adjudged as being accurate to 3 significant figures as I used a digital micrometer which gave the thickness to 4 significant figures but may have been slightly inaccurate at that degree of detail- therefore deciding to use 3 significant figures seemed to be the most appropriate level of accuracy. These results are given below.
These results clearly show a negative correlation between Period and Thickness, a relationship which supports my earlier prediction that the torsion in the wire would increase with thickness. A decrease in Period conveys an increase in torsion as it means the wire is oscillating faster, and thus has greater torsional stiffness. This assertion is corroborated by the measurement of the torsion constant for this data, given below.
Here, one can clearly see there is a positive correlation between thickness and torsion constant. However, this data doesn’t fully convey the relationship until it is presented graphically, accompanied by an analysis of what the graph demonstrates. Below, I have implemented the data into a scatter graph.
The graph clearly demonstrates that there is an exponential relationship between thickness of wire and torsion constant, a conclusion which makes sense in terms of physics, as an infinitely thick material would be infinitely difficult to twist, just as an infinitely thin material would be infinitely easy to twist. I have also included the equation of the line, and the r² value of the correlation,
y = 0.0003e4309.5x
R² = 0.9863.
The equation of the line demonstrates that it is an exponential relationship, whilst the R² value conveys that there is a very strong positive correlation between thickness and torsion constant.
I was very satisfied with the investigation of the thickness of wire as it provided me with strong evidence for the exponential relationship between the two factors. In terms of quality of measurement, I feel the values of the torsion constant are very accurate as the impact of human reaction time can be considered as negligible as I repeated the experiment numerous times over a long period of time, (i.e. timing the period of 10 oscillations and dividing through by 10, thus lessening the percentage uncertainty of reaction time), also the measurement of I was very accurate as I used high quality scales and accurately measured the length of the torsion bar many times, each time getting the same length. The thickness of the wire can also be considered highly accurate as I used a digital micrometer which gave the thickness to 4 significant figures, and only used 3, thus lessening the uncertainty inherent in using too many significant figures.
The investigation of length was less enlightening and detailed than the investigation of thickness, mainly due to time constraints caused by the amount of time spent refining the apparatus and resolving the issues which surfaced during the initial experimentation. However, I was still able to take some accurate measurements and begin to draw some conclusions about the effect on torsion constant when varying length and keeping thickness consistent. In terms of quality of measurement there was some uncertainty in the measurement of length due to the arrangement of fastening the wire to the retort stand, and to a lesser extent, the torsion bar. I have estimated this uncertainty at 5mm, and included this in the form of error bars. The results of the length experimentation are given below.
These results clearly show a positive correlation between length and period, thus demonstrating that my initial prediction is true, torsion in the wire does decrease with length. The greater period indicates a lower torsional constant in the wire as it means the wire is oscillating slower. This is supported by the values of the torsional constant given by each period, listed below.
Whilst a negative correlation is clear from this data, it was necessary to represent the data graphically in order to get a better understanding of the relationship. Included in the graph are the horizontal error bars mentioned earlier, visually one can see that the quality of measurement was strong for this investigation as the error bars are relatively small and barely alter the curved line.
The graph of torsion constant against length seems to convey an inverse relationship between length and torsion constant, but the fact that there are only 4 points weakens the strength of the conclusion. Despite this fact, there does seem to be a strong negative correlation, as evidenced by the R² value for the data, in addition to the equation of the line.
y = 0.0131e-3.673x
R² = 0.9711
In order to test whether there was a strong inverse relationship I then plotted a graph of the torsion constant against the inverse of the length of the wire. This graph further corroborates the possibility of a strong inverse relationship as there is a strong positive correlation between the values of torsion constant and inverse length- as evidenced by the high R² value for the line.
y = 0.0013x - 0.0001
R² = 0.9796
Despite the relative scarcity of data collected, I am still satisfied with the conclusions one can draw from this analysis of the relationship between torsion and length of wire as there is very little uncertainty, and strong graphical evidence for the conclusions I have drawn. These conclusions are also suggested by the physics involving torsion and length as an infinitely long piece of wire would undoubtedly be infinitely easier to twist than a piece of wire with equal thickness that was only one molecule long.
Conclusion
Overall, I was reasonably satisfied with the conclusions I was able to draw from this investigation, but I feel that it would have been much more successful had the initial experimentation been better planned, meaning I would not have had to spend time solving these problems when in fact I should have been taking more readings.
Despite the setbacks the investigation was successful as one is able to draw solid conclusions from the data collected and the analysis drawn from both the raw data tables and the graphical representations which visually convey the relationship between the torsion of wire and the variables I chose to investigate- length and thickness.
In terms of experimental technique I was very satisfied with my final apparatus and method as it reduced uncertainty to a minimum, with almost wholly negligible uncertainty rendering the impact on the final conclusions almost non-existent and certainly of no real importance. The experimentation was also very safe as I followed my risk assessment with care, ensuring I didn’t endanger myself or those around me. By the end of the investigation I had improved my experimental technique massively, working with efficiency and speed- completely at ease with both the equipment and the method used to take my results.
The general conclusions one can glean from this investigation are supported by both the data collected throughout the investigation and the underlying physics of this particular topic; the exponential relationship between torsion constant and thickness of wire was postulated in my original prediction, albeit merely a suggestion of a positive correlation rather than an exponential one, the same is also true for the inverse relationship between torsion constant and length. As discussed earlier, the physics behind these conclusions is fairly rudimentary on the surface (the ruler twisting experimentation etc.), but is in fact much more complicated when one investigates further. Were I to repeat this experiment I would take measurements at the extremes of both thickness & length in order to investigate how large the range of the torsion constant can be. I was unfortunately unable to undertake this experimentation during this investigation as it would require specialist equipment to measure the oscillations of both massive and minute lengths & thicknesses of wire. In addition, it would have been useful to have specialist equipment such as a purpose built Torsion Pendulum rather than having to rely on crude solutions such as the pen lid stopping the vertical oscillations.
In conclusion, I believe this investigation has begun to prove that there is an exponential relationship between torsion and thickness of wire, and an inverse relationship between torsion and length of wire. Whilst it is true these claims cannot be given as wholly true as there is a relative scarcity of data- however, given the time and equipment at my disposal I believe it was a successful investigation undertaken with experimental finesse resulting in conclusions free from any impacting uncertainty.
