Notes: the numbers at the end of each of the controlled variables refer to their illustration counter part on the next page. (Diagrams drew by pencil and ruler on a separate page)
Apparatus:
- Two clamp stands both the same making and 50 cm high
- 1 metal bar, 17 cm long and 14.1 grams of a thin cylinder shape
- Two strings of the same fabric, both 40cm long at their extension limit
- Digital timer
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30cm ruler to measure distance l
- Meter ruler to measure the height of the clamp stands
Having conceived my experiment and design , I was ready to collect results. Before anything else I measured the distance l at 2 cm interval after each try starting at 2 cm (7.5 cm on the right and left of the two strings), 4 cm (6.5 cm on the end of each string), 6 (5.5 on the end of each string till the end of the metal bar), 8 cm (4.5 on the end of each string), 10 cm (3.5 cm on the end of each string) and finally 12cm (2.5 cm on the end of each string). I did not carry out my trials until distance l was 17cm for precision and accuracy reasons (discussed in evaluation). The second thing to avoid human error, I was doing 5 trials for each distance l so that I could have a reasonable and fair average of the values. I also decided to measure not the time taken for 3 or 5 waves but for 10 waves (thus getting the time taken for the metal bar to swing back and forth from its original 90 degree position for 10 waves) thus I was able to eliminate some of the human error.
From the description above, I drew a table for my future results
The digital timer I used only had 2 decimal places and it therefore had an error of ±0.01 seconds. In addition to this I also had to account for my human error reaction speed of ±0.2 seconds making a grand total of ±0.21 seconds to each trial.
Having figured out these factors, I was ready to begin collecting results. The first thing I did was to set up my apparatus according to my Diagram 1. I then measured the distance l as 2 cm with 30cm ruler (used for precision and accuracy) to make the centre of the bar meaning 1 cm from each side of the centre of the bar with 7.5cm on each side of the strings.
Then I pulled back the left side of the metal bar with my left hand until it was 90 degrees from its original position completely perpendicular to my body facing my apparatus. Whilst with my right hand I got ready to press the ‘start’ button of the digital timer. I then released the metal bar without exerting any other force and this caused to oscillate back and forth from the 90 degrees position to its original or resting position and back again. Then in my head, I started to count the number of waves (swing of the metal bar back and forth from its original position) and stopped the timer when I had counted 10 waves. I repeated this very step 5 times to minimize human error and get a fair average of the recorded values. After inscribing the values recorded for the five tries, I added them all together and divided the resultant by five to get average.
I then proceeded to repeat the above steps for the different distances of l (specific measurements in the first paragraph of my method) for 4 cm, 6 cm, 8cm, 10 cm and 12 cm. Each time I proceeded with a try I was careful of the position of the clamps and the strings making sure the metal bar was always parallel to the clamps and perpendicular to the strings with the sae constant length of the strings (30cm).
From all the recorded values I emanated with this table
Time taken for 10 complete waves on a double string pendulum
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However I was interest in how the distance l affected the period of the metal bar therefore I divided all the averages for each distance l by 10 (number of waves counted) to get the period.
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The graph here shows a trend line with the formula y=-0.461n(x)+1.970 (logarithmic trend line) with a high correlation of value 0.998. The fact that my trend line does not touch my data points exactly is due to several errors that could not be controlled in my design. The instrument I was using for time, a digital timer, had an error of ±0.01 seconds and human reaction time of an error of ±0.2 seconds making a grand total of ±0.21 seconds. This error will be demonstrated in the graph where I will plot the equation y=logx (base 10) for every value I have recorded through the use of error bars. The next graph should have data points, which would give me a fairly straight line, negatively downward sloping.
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As you can see from the graph of y=logx (base 10) it gives a straight negative downward sloping trend line identifying the relationship between the distance l and the period of the metal bar as inversely proportional. As you can see from the arrow bars representing the potential error of ±0.21 s on the y-axis, there are three different trend lines that could have be drawn one with the error of + 0.21 seconds, a trend line with an error of -0.21 seconds and finally a polynomial trend line with the data points shown on the graph. This last trend line has a formula of y=0.114x2 -1.238x+2.016 with correlation of 0.999. Nevertheless, the other aim of my investigation was to look at the frequency and its relation with distance l enclosed between the two strings. Below is the table that I will be using in my graph.
In this graph, the trend line is polynomial and has a formula y=-0.001x2 + 0.082x +0.449 with a correlation of 0.998. However since the trend line based on the data points are polynomial I can produce another graph to demonstrate the direct relationship as is would be impossible to find the equation (in the form of ax2+bx +c). However since my values recorded are fairly accurate and precise with a fairly high correlation thus the trend line show a positive upward sloping trend line indicating that the distance l is directly proportional to the frequency through the formula line equation of the graph (thus one increased the other increases).
Note: In all graphs the line equation (or formula) stated below and on the graph represents the relationship Y and X (vertical and horizontal components of the graph or the relationship between the distance l and the period or the frequency of the metal bar)
From my recorded values throughout the experiment, the graph and the analysis of each graph or table I was able to find a reasonable conclusion to my investigation.
I conclude that in almost all certainty that my hypothesis was correct and sound. The second graph you can see the trend line has an equation of y=0.114x2 -1.238x+2.016 corresponding with the negative downward sloping of trend line establishing that the relationship between the distance l and the period of the oscillating metal bar are inversely related to one another. Thus as the distance l increases then the period (time taken for one complete wave) decreases. My third and final graph also proves my hypothesis is correct as the trend line is positively upward sloping corresponding with its equation y=-0.001x2 + 0.082x +0.449. Thus I can determine that as the distance l between the two strings increases so does the frequency (Hz). Generally all the relationships between the distance l between the two strings and the period or frequency can be written in the form of f= kl2+gl+c. However the formulas expressed above are only true for my metal bar of 17cm long, thin cylinder shape and 14.1 grams, as other metal bars will have different formula between their distance l and its period and frequency. Nevertheless, for any metal bar its will be inversely related to its period and directly proportional to is frequency. Thus only the formulas between the variables change but the general relationship (inversely or directly proportional) do not change for any different bars.
The general explanation of this phenomenon is possibly that as I increase the distance between the two strings whilst keeping the other factors constant and controlled, I increase the tension in the strings. So as the strings get further away from the center of mass of the metal bar, the more tension they have and as they are pulled back at a certain angle (90 degrees in this case) thus they have a higher displacement, higher frequency as more force acts upon the bar (tension ins strings not angle) and a lower period. This explanation is in correlation with what I have found through my investigation and seems the more plausible one.
In conclusion, this experiment supports my hypothesis that the frequency of the metal bar and the distance l are related and are directly proportional to one another. And the period of the metal bar is inversely proportional to the distance l between the two strings.
Finally, if I had used a wider range of distances between the two strings (l in cm) then I would have been able to continue my graphs and perhaps find a different type of trend line for both the frequency and period such as exponential and not polynomial. It is because the small total length of the metal bar that this limited the range of values that I could have recorded. I could have until 17 cm (total length of the bar) but I would have been very hard to measure this with a 30 cm ruler possibly giving inaccurate and imprecise values. This contributes to be a major limitation to the analysis of my investigation.
If I had a choice I would used a bigger metal bar perhaps of 50 cm or an 85 cm bar long (ideal length as I could have related it to 1m bar) to have a larger range of values. In addition, if I was to redo this experiment with the 17 cm bar I could have used a 15cm ruler to measure the distance between the two strings, for more accurate and precise readings of the period of the metal bar. Thus it would essential for me repeat this investigation with different types of metal bars, different length and mass to then compare the different formulas between each relationship of the distance between the two strings and the period or frequency of the oscillating metal bar.
Another major limitation to my experiment was my human reaction time and speed on counting the number of waves and pressing the ‘stop’ button on the digital timer after counting 10 waves. In addition, the total number of values recorded and trials done and waves counted would have affected my human reaction time and speed, as it would be long and tedious making me more prone towards error at the end of my value recording session. Also my human reaction time and speed is also due to my perception of where a wave started and finished as I have bad eyesight I might have over counted or undercounted some waves. The factor that mainly affected my human reaction speed was also the independent factor of my experiment as it increased I had to move my hand quicker to stop the timer. Nevertheless I accounted all these errors in the period graph of y=logx and the frequency of a metal bar graph as ±0.2 seconds for my human error and reaction time and speed. However I would need to bring the total number of waves to be counted to 5 and the number of tries to 3 to make it less tedious and long for me and to increase my human reaction time and speed if I was to redo this investigation. However for human error due to my perception and a decreasing period, I would need to employ a second person that will concentrate solely on the timing and digital timer of the period. However co-ordination and good interaction would be needed between the two persons would be needed for this work.
However this factor of error was not the only to be a limitation to my investigation as the digital timer had only two decimal places there was an error of ±0.01 seconds therefore the grand total error of ±0.21 seconds to be taken into account in all my tables and graphs. In addition to this the sue of the 30cm ruler was fairly inaccurate and I should have taken a 15 cm ruler to do my measurements, thus the measurement of the distance between the two strings might have incorrect to a few mm. This was not due poor equipment or instruments but my lack of judgement on using better material. Another important limitation was my inaccuracy at tying knots to hold the metal bar in the horizontal plane parallel to the clamps, thus perhaps my strings were a few cm off from one another. Tying knots and better judgment regarding equipment are two things I should do before starting a knew investigation further on in this domain.
Taking these errors into account, I may have been able to improve on them in a number of ways in order to further support my conclusion, or even disprove it.
